{"generated":"2026-08-04","papers":[{"id":"2608.02970","title":"Finite groups that are the product of every pair of non-conjugate maximal subgroups are soluble","authors":["Richie Sater"],"abstract":"We prove that every finite group which coincides with the product of any two of its non-conjugate maximal subgroups is soluble, answering Problem 10.34 of the Kourovka Notebook (V. S. Monakhov, 1986) in the negative. The almost simple case was settled by Tikhonenko and Tyutyanov (2010); the obstacle to the general case was the socle S^k with k >= 2, which no counting bound can control. We remove it with a divisibility criterion: a single pair of automorphism-stable conjugacy classes of subgroups of the simple group S, subject to a valuation inequality, excludes the socle S^k for all k >= 2 and every admissible embedding at once. Such pairs are constructed uniformly for every infinite family of finite simple groups, with Zsigmondy primes as the arithmetic obstruction; the sporadic groups and all remaining small cases are settled by independently re-checkable certificates produced in GAP.","abs_url":"https://arxiv.org/abs/2608.02970","pdf_url":"https://arxiv.org/pdf/2608.02970v1","primary_category":"math.GR","published":"2026-08-04","tags":["formalization","verification","algebra"],"lean_note":"The minimal-counterexample structure theorem (unique minimal normal subgroup, faithful conjugation map) is kernel-checked in Lean.","repo":"https://github.com/RichieSater/kourovka-10-34","thumb":"thumbs/2608.02970.jpg"},{"id":"2608.02295","title":"MechGeo: Autoformalizing and Proving Euclidean Geometry in Lean 4","authors":["Hao Shen","Junyu Guo","Tian Cui","Yuxuan Xiao","Lihong Zhi"],"abstract":"We present MechGeo, a Mathlib native agentic framework that jointly addresses faithful autoformalization and certified proof construction for Euclidean geometry. In this framework, GeoFormalizer represents informal problems in GeoIR, deterministically translates them into Lean 4, and iteratively repairs candidate statements using structural diagnostics and semantic evaluation. GeoProver constructs geometric proof plans, derives intermediate lemmas, and selectively algebraizes suitable subgoals through a library verified in Lean. Singular or SymPy may generate algebraic certificates, but all resulting proofs and counterexamples are checked by Lean's kernel. Experiments across seven LLM backbones show substantial improvements in autoformalization, particularly for models with weaker direct translation performance. On 43 historical IMO geometry problems, GeoFormalizer generates formal statements that GeoProver proves in 29 cases; for the remaining 14, it constructs counterexamples verified in Lean and proves all repaired statements after expert correction. Together with IMO 2026 Problem 2, this yields, to the best of our knowledge, the largest reported collection of automated, kernel-checked Lean proofs for IMO geometry problems. On the 14 geometry statements in LEAP's Lean-IMO-Bench, MechGeo proves 12 for the first time, formally refutes the remaining two, and proves both repaired statements. These results establish counterexample guided diagnosis, geometric reasoning, and certified symbolic computation as a practical foundation for trustworthy formal geometry.","abs_url":"https://arxiv.org/abs/2608.02295","pdf_url":"https://arxiv.org/pdf/2608.02295v1","primary_category":"cs.AI","published":"2026-08-03","tags":["autoformalization","theorem-proving-ml","ai-agents","geometry"],"lean_note":"MechGeo autoformalizes and proves Euclidean geometry problems in Lean 4, with all proofs and counterexamples verified by Lean's kernel.","thumb":"thumbs/2608.02295.jpg"},{"id":"2608.02066","title":"Every quasiperfect number has at least eight distinct prime factors","authors":["Akira Toyohara","Ye Tao","Siqiong Yao"],"abstract":"No quasiperfect number ($σ(n) = 2n + 1$) is known, and its number of distinct prime factors is bounded below; the bound $ω\\ge 7$ of Hagis--Cohen has stood since 1982, obstructed by a family of ``deep leaves'' on which pure enumeration cannot terminate (the scan bound for the intermediate prime reaches $8 \\times 10^8$, and the exponent dimension is unbounded). This paper clears that obstruction with three lemmas at the level of secondary-school algebra --- a discriminant criterion, a quadratic-residue sieve, and a multilinear resolver --- which eliminate the last prime $q$, the intermediate prime $p$, and the exponent dimension respectively, turning a non-terminating search into a finite decision. On this basis all 381 stems of ``$3 \\mid n$ and $ω= 7$'' and their $79{,}751{,}212$ deep leaves are eliminated, with the ledger closing exactly and zero solutions throughout; the complementary case ``$3 \\nmid n$ and $ω= 7$'' collapses to a single stem, which is eliminated directly, so that the proof does not rest on any theorem whose published record we could not independently re-verify. Together with the machine elimination of $ω\\le 6$ (Theorem B4), this yields the main theorem: \\emph{any quasiperfect number, if one exists, satisfies $ω(n) \\ge 8$} --- the first advance of this bound since Hagis--Cohen 1982. The full computation has been reproduced by seven separately closed ledgers across three algorithmic architectures (CPU and GPU), all with zero solutions and exact ledger closure, and the lemma layer is formalized in Lean (259 theorems, zero \\texttt{sorry}). A 2023 preprint of Zemann reported the same bound by a different computation; our audit of its public code found a coverage gap of 35 feasible exponents, so the elimination given here is, to our knowledge, the first complete proof. Code, ledgers, and Lean sources are available from the authors.","abs_url":"https://arxiv.org/abs/2608.02066","pdf_url":"https://arxiv.org/pdf/2608.02066v1","primary_category":"math.NT","published":"2026-08-03","tags":["formalization","number-theory"],"lean_note":"The three-lemma system underlying the elimination is formalized in Lean with 259 theorems and zero sorry.","thumb":"thumbs/2608.02066.jpg"},{"id":"2608.01671","title":"The Godsil--McKay Asymptotic for Latin Rectangles in the Sublinear Range of Erdős Problem 725","authors":["Eric Li"],"abstract":"Erdős Problem 725 asks for an asymptotic formula for the number $L_{k,n}$ of ordered, labelled $k\\times n$ Latin rectangles. Godsil and McKay proved that $L_{k,n}\\sim (n!)^k((n)_k/n^k)^n(1-k/n)^{-n/2}e^{-k/2}$ for $k=o(n^{6/7})$. We provide a partial solution to Erdős Problem 725 by proving this asymptotic for every $k=o(n)$. More precisely, set $\\widetilde A_{k,n}=(n!)^k((n)_k/n^k)^n\\exp\\{[n(H_n-H_{n-k})-k]/2\\}$. For every $K(n)=o(n)$, uniformly for $0\\leq k\\leq K(n)$, we prove $\\log(L_{k,n}/\\widetilde A_{k,n})=O(k^2/n^2)$, with an absolute implied constant. The results of this paper have been formally verified in Lean.","abs_url":"https://arxiv.org/abs/2608.01671","pdf_url":"https://arxiv.org/pdf/2608.01671v1","primary_category":"math.CO","published":"2026-08-03","tags":["formalization","combinatorics"],"lean_note":"The asymptotic enumeration results for k×n Latin rectangles across the full sublinear range are formally verified in Lean.","repo":"https://github.com/ericlisg/erdos725partial-lean","thumb":"thumbs/2608.01671.jpg"},{"id":"2608.01515","title":"The Set of Correlated Equilibrium Payoffs for a Fixed Information Structure Need Not Be Closed","authors":["Michael Greinecker","Patrick Lahr","Christoph Schwerdtfeger"],"abstract":"Aumann (1974) showed that an atomless public randomization device makes the feasible- and equilibrium-payoff sets of a game with a fixed information structure convex, and asked whether they are closed. We show that, in every case the question leaves open, they need not be. One information structure drives all the examples: two sequences of fair signs whose coordinate correlations increase to a ceiling $ρ<1$ that no pair of separately measurable square-integrable rules attains. For every $0<ρ<1$ it yields a three-player game with a public randomization device whose equilibrium-payoff set is exactly the open interval $\\{(0,0,t):-ρ<t<ρ\\}$; a two-player game with a public randomization device whose equilibrium-payoff set is convex, full dimensional, and not closed; and, without any public device, nonclosed feasible- and equilibrium-payoff sets, the latter along equilibria with unique best replies modulo null events whose payoffs approach a vector that is not even feasible. With distinct but mutually absolutely continuous subjective priors, even the feasible-payoff set can fail to be closed in the presence of an objective public randomization device, together with every $\\varepsilon$-equilibrium payoff set and the set of induced law tuples. Our construction also allows us to resolve a conjecture of Stinchcombe (2011). The main results and the lemmas supporting them are formalized in the Lean proof assistant; an appendix records the exact coverage of each statement, including the clauses for which only a paper proof is given.","abs_url":"https://arxiv.org/abs/2608.01515","pdf_url":"https://arxiv.org/pdf/2608.01515v1","primary_category":"econ.TH","published":"2026-08-02","tags":["formalization","economics","probability"],"lean_note":"Main results and supporting lemmas on nonclosed correlated-equilibrium payoff sets are formalized in the Lean proof assistant, with an appendix recording coverage.","thumb":"thumbs/2608.01515.jpg"},{"id":"2608.01396","title":"Three Graffiti.pc Conjectures on Largest Induced Trees: Proofs of Conjectures 141, 142, and 143","authors":["Alper Ferudun"],"abstract":"For a finite simple graph $G$, let $t(G)$ be the largest order of an induced tree and let $g(G)$ be the girth. We prove three consecutive conjectures of DeLaViña's Graffiti.pc program. First, writing $\\ell(v)$ for the independence number of the subgraph induced by the neighbourhood of $v$, we prove $t(G) \\ge \\lfloor g(G)/2 \\rfloor - 1 + \\max_{v \\in V(G)} \\ell(v)$. Second, if $\\mathrm{Per}(G)$ is the periphery and $f(G) = \\max_x d(x, \\mathrm{Per}(G))$, we prove $t(G) \\ge \\frac{2}{3} g(G) + f(G)$, and establish the stronger integral bound $t(G) \\ge f(G) + \\lceil 2g(G)/3 \\rceil$ when $G$ contains a cycle. Third, if $δ'(G)$ is the second-smallest degree, counted with multiplicity, then every connected non-tree graph satisfies $t(G) δ'(G) \\ge g(G) + 1$. These are Conjectures 141, 142, and 143 of Written on the Wall II. Complete, machine-checked Lean 4 proofs of all three formal statements accompany the manuscript.","abs_url":"https://arxiv.org/abs/2608.01396","pdf_url":"https://arxiv.org/pdf/2608.01396v1","primary_category":"math.CO","published":"2026-08-02","tags":["formalization","combinatorics"],"lean_note":"Machine-checked Lean 4 proofs accompany three graph-theory conjectures on largest induced trees, contributed to the Formal Conjectures repository.","repo":"https://github.com/AlperTheKing/formal-conjectures","thumb":"thumbs/2608.01396.jpg"},{"id":"2608.01393","title":"Universal Triangle Covering Curve and Polygonal Chain: Escaping Forest and Fitting Worm","authors":["Zhipeng Deng"],"abstract":"In this paper, we present a general formulation to address the problems of covering curves and polygonal chains with triangle, and fitting these curves into triangle. These problems can be formulated as special cases of Bellman's lost-in-a-forest problem (escaping triangular forest) and Moser's worm problem (covered by triangle). We model and reformulate the problem by keeping the curve stationary while allowing the triangle to translate and rotate. Subsequently, we derive the functional minimization formulation with support function constraints to solve. We also prove the equivalence and convergence of the formulas. Finally, we employ numerical methods and present results for covering curves with arbitrary triangles of various angles. We also present some corollaries and variant results, including closed curves and closed polygonal chains.","abs_url":"https://arxiv.org/abs/2608.01393","pdf_url":"https://arxiv.org/pdf/2608.01393v1","primary_category":"math.OC","published":"2026-08-02","tags":["formalization","geometry"],"lean_note":"Theorems 4-8 characterizing support-function constraints for the triangular lost-in-a-forest problem are formalized in Lean 4 with Mathlib.","thumb":"thumbs/2608.01393.jpg"},{"id":"2608.01317","title":"Analytic Qubit Separation between POVMs and Projective Measurements","authors":["Lin Zhu","Ranyiliu Chen","Xin Wang"],"abstract":"Generalized measurements can be implemented projectively after enlarging the Hilbert space, but this dilation changes the available local dimension. We construct a Bell functional with rational coefficients that separates the two measurement models at local dimension two. An explicit three-outcome qubit positive-operator-valued measure with rational matrix entries attains $2\\sqrt2+1/100$. On the other hand, all qubit-projective strategies are bounded by $2\\sqrt2+\\sqrt5/250+\\sqrt2/32400$, giving a fully analytic certified gap greater than $1/1000$. To our knowledge, this is the first fully analytic Bell-functional separation between qubit POVMs and qubit projective measurements over arbitrary shared two-qubit states. Lean certificate for the separation theorem is provided for completeness. Separately, an exact level-3 noncommutative sum-of-squares certificate proves that the explicit qubit strategy attains the unrestricted finite-dimensional tensor-product quantum optimum.","abs_url":"https://arxiv.org/abs/2608.01317","pdf_url":"https://arxiv.org/pdf/2608.01317v1","primary_category":"quant-ph","published":"2026-08-02","tags":["verification","quantum"],"lean_note":"A Lean certificate is provided verifying the separation theorem bounding qubit projective strategies below the POVM Bell value.","repo":"https://github.com/real-lin-zhu/Exact-Fixed-Dimension-Bell-Separation-between-Qubit-POVMs-and-PVMs","thumb":"thumbs/2608.01317.jpg"},{"id":"2608.01277","title":"Discriminant Varieties for Stick Knots and Links","authors":["Alexander Kolpakov","Igor Rivin"],"abstract":"How many knot types can be built from a fixed budget of straight sticks? We prove that the answer has factorial-scale growth, settling its order for the first time. No previously published general upper bound improves on the exponential-in-the-square estimate obtained from crossing-number enumeration; we replace it with a factorial-scale upper bound, which is optimal at the level of growth order. The proof turns polygonal self-intersection into a sparse real-algebraic chamber problem in only linearly many dimensions, while a complementary braid construction supplies factorially many distinct knots. The result creates a direct bridge between knot topology, real algebraic geometry, fewnomial structure, and permutation combinatorics.","abs_url":"https://arxiv.org/abs/2608.01277","pdf_url":"https://arxiv.org/pdf/2608.01277v1","primary_category":"math.GT","published":"2026-08-02","tags":["formalization","geometry","combinatorics"],"lean_note":"A Lean 4 development verifies the algebraic and finite-combinatorial core of the stick-knot chamber bounds, taking topological inputs as named hypotheses.","thumb":"thumbs/2608.01277.jpg"},{"id":"2608.00983","title":"Plücker coordinates of finite-dimensional subspaces of $\\ell^p$ and its direct sums: summability, reconstruction, stratification","authors":["David Victor Feldman"],"abstract":"An $n$-dimensional subspace of $\\ell^p$ has Plücker coordinates indexed by the $n$-element subsets of $\\N$. We show these coordinates lie in $\\ell^p\\In{n}$ --- the exponent is preserved --- with multilinear norm exactly $1$ for $0<p\\le 2$; for $p>2$ the sharp constant exceeds $1$ and its determination contains the Hadamard maximal determinant problem. A reconstruction lemma shows every nonzero solution of the quadratic Plücker relations in $\\ell^p\\In{n}$ is decomposable with frame in $\\ell^p$; consequently $\\Gr_n(\\ell^p)$ is a closed Banach-analytic submanifold of $\\mathbb{P}\\big(\\ell^p\\In{n}\\big)$ cut out by the Plücker relations alone, with no auxiliary summability condition and no polarization. For mixed sums $\\bigoplus \\ell^{p_i}$ the exterior power is graded by compositions of $n$; the support of the grading is the lattice-point set of a generalized permutohedron determined by the intersection pattern of the subspace with partial sums, this stratification is canonical for the isometry group though not for $\\GL$, and each stratum admits a tubular neighborhood whose normal coordinates are precisely the Plücker blocks vanishing on it. We record what is proved and what is conjectured; the finitary and single-space core of the theory, including full proofs of Cauchy--Binet and Hadamard's inequality, has been formally verified in Lean~4.","abs_url":"https://arxiv.org/abs/2608.00983","pdf_url":"https://arxiv.org/pdf/2608.00983v1","primary_category":"math.FA","published":"2026-08-02","tags":["formalization","analysis"],"lean_note":"The finitary single-space core, including Cauchy–Binet and Hadamard's inequality, is formally verified in Lean 4.","thumb":"thumbs/2608.00983.jpg"},{"id":"2608.00851","title":"Block Sensitivity can exceed Spectral Sensitivity Squared","authors":["Alexander Meiburg"],"abstract":"The spectral sensitivity $λ(f)$ of a Boolean function is the largest eigenvalue of the adjacency matrix of its sensitivity graph. It lower-bounds every standard measure of query complexity, and Aaronson, Ben-David, Kothari, Rao and Tal, who introduced it, asked whether block sensitivity is at most quadratic in it: is $bs(f)=O(λ(f)^{2})$? We show that it is not. We construct a total Boolean function on $2017584$ variables with $bs(f)\\ge 14011$ and $λ(f)\\le 89.0162$, so that $bs(f)\\geλ(f)^{2.127}$, and hence by composition a family with $λ(f_n)\\to\\infty$ and $bs(f_n)=Ω(λ(f_n)^{2.127})$. The function is the indicator of a union of $k$ subcubes indexed by the vertices of a doubly regular tournament, and the freedom left in the construction is fixed by the Lovász local lemma. The main result has been formally verified in Lean. We also give numerical evidence that a member of the same family on $1255$ variables reaches an exponent near $2.20$, and exhibit a member on $30$ variables whose exponent already exceeds $2$ and whose spectral sensitivity can be computed exactly.","abs_url":"https://arxiv.org/abs/2608.00851","pdf_url":"https://arxiv.org/pdf/2608.00851v1","primary_category":"cs.CC","published":"2026-08-01","tags":["formalization","combinatorics"],"lean_note":"The main theorem constructing a Boolean function with block sensitivity exceeding spectral sensitivity squared is formally verified in Lean on Mathlib.","repo":"https://github.com/Timeroot/BS_Lam","thumb":"thumbs/2608.00851.jpg"},{"id":"2608.00323","title":"On a conjecture of Han and Xiong for fractional Gaussian binomial coefficients","authors":["Ken Ono"],"abstract":"Han and Xiong recently extended the Gaussian binomial coefficient $\\genfrac{[}{]}{0pt}{}{r+k}{k}_{q}$ to positive rational $r$ and conjectured that its integer trace, the integer-exponent part of the resulting power series, is coefficientwise largest at $r=1/2$. We prove a support-dominance theorem comparing rational parameters under an explicit divisibility condition. It settles the conjecture for every $r\\geq 1/2$ and reduces the full conjecture to the unit fractions $r=\\frac{1}{2m}$, only finitely many of which are nontrivial for each fixed $k$. A computer computation then verifies the conjecture for every positive rational $r$ and every $k\\leq 200$. The theoretical results were autonomously produced and verified in Lean by AxiomProver.","abs_url":"https://arxiv.org/abs/2608.00323","pdf_url":"https://arxiv.org/pdf/2608.00323v1","primary_category":"math.CO","published":"2026-07-31","tags":["formalization","ai-agents","combinatorics"],"lean_note":"Support-dominance theorem for fractional Gaussian binomial coefficients was autonomously produced and verified in Lean by the AxiomProver AI system.","repo":"https://github.com/AxiomMath/QBinomialTrace","thumb":"thumbs/2608.00323.jpg"},{"id":"2607.29681","title":"Lean-verified lower bounds for the Shannon capacity of odd cycles","authors":["Pjotr Buys","Sven Polak","Jeroen Zuiddam"],"abstract":"We give new lower bounds for the Shannon capacities of small odd cycles: $Θ(C_7)\\geq3.258805369885\\ldots$, $Θ(C_{11})\\geq5.294502522149\\ldots$, $Θ(C_{13})\\geq6.302455083464\\ldots$, $Θ(C_{15})\\geq7.301600534487\\ldots$, $Θ(C_{19})\\geq9.357192705918\\ldots$, $Θ(C_{21})\\geq10.342455853338\\ldots$, and $Θ(C_{23})\\geq11.328224257774\\ldots$. The bounds are obtained by an iterative procedure due to Gao (2026) which is based on a method by Itty, Rosin, Carstensen and Reichman (2026). The bounds are fully formalised in Lean.","abs_url":"https://arxiv.org/abs/2607.29681","pdf_url":"https://arxiv.org/pdf/2607.29681v1","primary_category":"math.CO","published":"2026-07-31","tags":["formalization","combinatorics"],"lean_note":"Base valid tuples and their validity proofs for odd-cycle Shannon capacity lower bounds are formalised and verified in Lean.","repo":"https://github.com/spectra-research/shannon-capacity-lean","thumb":"thumbs/2607.29681.jpg"},{"id":"2607.29349","title":"The $δ$-calculus: from distinction to arithmetic","authors":["Jonathan Washburn","Milan Zlatanović"],"abstract":"Let $δ$ denote the primitive act of distinction, formally realized as the one-step extension $r \\mapsto Sr$ of a finite record. We study the inductively generated $δ$-orbit and its first-order arithmetic presentation $\\mathbb{N}_δ$. The corresponding $δ$-calculus is an intuitionistic first-order proof system over the signature $\\{0,S,+,\\cdot\\}$. Every derivation carries a ledger recording the use of the law of excluded middle, the limited principle of omniscience, Markov's principle, and induction on quantified formulas. The last entry does not affect whether a derivation is forced. If a closed formula is derivable in the forced fragment, then it is true in the standard model. Starting from $δ$, we construct a choice-free number tower $δ\\leadsto \\mathbb{N}_δ\\hookrightarrow \\mathbb{Z}_δ\\hookrightarrow \\mathbb{Q}_δ$. The metatheoretic number systems $\\mathbb{N}$, $\\mathbb{Z}$, and $\\mathbb{Q}$ each admit an explicit injection into $\\mathbb{N}_δ$. We also classify the recognition quotients of the additive monoid $(\\mathbb{N}_δ,+,0)$. Assuming the law of excluded middle, every recognizer is either injective or has a kernel congruence $\\equiv_{i,p}$, with index $i$ and period $p$, for a unique pair $i\\ge 0$, $p\\ge 1$. In the noninjective case the quotient is isomorphic to the finite monogenic monoid $M(i,p)$. We price this classification with the ledger, determining which nonconstructive principles each form requires. If the congruence is decidable and an explicit pair of distinct related elements is given, the classification is forced. If the congruence is decidable and different from equality, Markov's principle is needed. For an arbitrary congruence, the dichotomy requires the law of excluded middle. The reverse implications show that the last two prices cannot be lowered. The main results are formalized in Lean 4.","abs_url":"https://arxiv.org/abs/2607.29349","pdf_url":"https://arxiv.org/pdf/2607.29349v1","primary_category":"math.LO","published":"2026-07-31","tags":["formalization","logic","number-theory"],"lean_note":"Main results on the δ-calculus, number tower, and recognition-quotient classification are formalized in Lean 4 over Mathlib with axiom audits.","repo":"https://github.com/jonwashburn/actual-mathematics","thumb":"thumbs/2607.29349.jpg"},{"id":"2607.29191","title":"A Proof of the Dittert Conjecture in Dimension 4 via an Agent-Guided Exact Sum-of-Squares Certificate","authors":["Jinhui Li","Beibei Xiong","Zhengfeng Yang"],"abstract":"The Dittert conjecture states that the Dittert functional on nonnegative $n\\times n$ matrices whose entries sum to $n$ is uniquely maximized by the uniform matrix. We prove the conjecture in dimension $4$. More precisely, let $K_4$ be the simplex of nonnegative $4\\times4$ real matrices whose entries sum to $4$, let $U_4$ be the uniform matrix, and let $φ$ denote the Dittert functional. We establish $\\frac{61}{32}-φ(A)\\geq \\frac{1}{52}\\lVert A-U_4\\rVert_F^2$ for every $A\\in K_4$. Consequently, $U_4$ is the unique maximizer of $φ$ on $K_4$. The proof reduces to certifying the nonnegativity of a structured quartic polynomial in sixteen variables on a simplex. We construct an exact rational constrained sum-of-squares certificate using an agent-guided symbolic-numeric procedure that combines template selection with sequential rational recovery. The main SOS consists of $152$ positively weighted rational squares, while each of the $136$ smaller SOS blocks consists of $16$ such squares. Exact $LDL^{\\mathsf{T}}$ decompositions certify positivity, and exact coefficient comparison over $\\mathbb{Q}$ verifies the complete polynomial identity. The resulting exact certificate is formally verified using the Lean proof assistant.","abs_url":"https://arxiv.org/abs/2607.29191","pdf_url":"https://arxiv.org/pdf/2607.29191v1","primary_category":"cs.SC","published":"2026-07-31","tags":["formalization","verification","combinatorics"],"lean_note":"An exact rational sum-of-squares certificate for the Dittert conjecture in dimension 4 is formally verified in Lean 4 with Mathlib.","repo":"https://github.com/123ljh0bot/Dittert_Conjecture_in_Dimension_4","thumb":"thumbs/2607.29191.jpg"},{"id":"2607.29111","title":"Fitting's Theorem and Semirings of Normal Subgroups","authors":["Damiano Testa"],"abstract":"We define a non-unital, generally non-associative, commutative semiring structure on the collection of normal subgroups of a group $G$. This viewpoint allows us to recast in ring-theoretic terms Fitting's classical theorem that the join of two nilpotent normal subgroups is nilpotent. From this perspective, the two key inputs are a binomial expansion in a non-associative setting and the fact that the commutator subgroup of two normal subgroups lies in each factor. The development is formalized in Lean, making essential use of Mathlib for the core definitions and results.","abs_url":"https://arxiv.org/abs/2607.29111","pdf_url":"https://arxiv.org/pdf/2607.29111v1","primary_category":"math.GR","published":"2026-07-31","tags":["formalization","mathlib","algebra"],"lean_note":"Fitting's theorem is reproved via a semiring structure on normal subgroups and fully formalized in Lean 4 using Mathlib.","repo":"https://github.com/adomani/FittingsTheorem","thumb":"thumbs/2607.29111.jpg"},{"id":"2607.28699","title":"WitCert: Sound Runtime Risk Observability and Gating for KV-Cache Quantization","authors":["Fanzhe Wei","Li Liu"],"abstract":"KV-cache quantization is validated today by offline benchmark averages; a deployed system cannot tell whether compression is damaging the request it is serving right now. We give it a provably sound runtime meter, a \"DTrace for KV quantization\": a per-(layer, head, step) upper bound on the total variation between exact and compressed attention. The meter has two tiers: a deterministic band-norm-witness bound, sound for any cache-preserving black-box quantizer and for any query (adaptive-safe, worst-case Cauchy-Schwarz plus RoPE band-unitarity), and a tighter probabilistic certificate for a controlled subtractively-dithered INT8 quantizer under an explicit request-level failure budget (stated for non-adaptive queries; core theorems machine-checked in Lean 4). Three results. Observability: the meter enters SGLang through an environment-guarded patch, and any scheme registered as one tensor function is measured in live serving. Repair: meter-driven gating, risk-ranked where the witness is saturated and certified where it is informative, empirically restores the quality floor at benchmark scale. For example, raw-cast FP8 improves from 22.8 back to 79.7 on hard RULER tasks, with the difference from uncompressed bounded at [+0.0, +0.8] by a paired test. Analysis: aggressive schemes survive on cross-layer error cancellation, not per-step fidelity. In a 28-layer sweep, no single layer's pollution alone loses anything (0/28), and the certified INT8 cache serves 1.88 times more KV tokens at the same memory in SGLang.","abs_url":"https://arxiv.org/abs/2607.28699","pdf_url":"https://arxiv.org/pdf/2607.28699v1","primary_category":"cs.AR","published":"2026-07-30","tags":["verification"],"lean_note":"Core soundness theorems for the KV-cache quantization total-variation bound are machine-checked in Lean 4 using Mathlib.","repo":"https://github.com/wayfind/witcert","thumb":"thumbs/2607.28699.jpg"},{"id":"2607.28459","title":"LeanCSP: A Framework for Certifying Constraint Reformulation and Solving in Lean","authors":["Pablo Manrique","Stefan Szeider"],"abstract":"Constraint programming is a core technology for solving complex combinatorial problems in scheduling, planning, configuration, and verification. Trusting its results therefore demands guarantees at two levels: that reformulations applied beforehand are semantics-preserving, and that solvers produce correct answers. In this work, we introduce a framework that addresses both verification levels in the Lean theorem prover: it can be used to prove formulation-level properties, such as equivalence, equisatisfiability, and the correctness of symmetry-breaking constraints, parametrically for entire problem families; and to check solver-produced certificates for individual instances via translation backends to external formats such as MiniZinc, SMT-LIB, and OPB. Combining both levels yields an end-to-end workflow that establishes the satisfiability or unsatisfiability of a constraint problem without trusting the external solver. Experimental results show that our framework's verified symmetry breaking also pays off in practice: a single parametric proof per problem family, reused across all instance sizes, reduces solver search effort by a factor of up to 2x10^7, while the entire in-Lean certification stays affordable, taking at most a few minutes for our largest instances.","abs_url":"https://arxiv.org/abs/2607.28459","pdf_url":"https://arxiv.org/pdf/2607.28459v1","primary_category":"cs.AI","published":"2026-07-30","tags":["verification","tooling","logic"],"lean_note":"LeanCSP formalizes constraint satisfaction problems in Lean, proving reformulation soundness and checking external solver certificates back inside Lean.","repo":"https://github.com/leansolving/leancsp","thumb":"thumbs/2607.28459.jpg"},{"id":"2607.28110","title":"BlueprintRepair: Typed Local Edits for Failed Lean Proof Blueprints","authors":["Ruslan Khrulev"],"abstract":"LLM-based Lean proving systems increasingly organize a proof as a blueprint: a dependency graph of formal statements. We introduce BlueprintRepair, a repair interface that lets a model change this graph through ten schema-checked local operations. An operation names the node it edits, so the target theorem cannot be changed. Lean checks every applied change, and an accepted repair must declare every blueprint lemma its proof uses. We also construct BlueprintTrace, a benchmark of 142 controlled failures with complete accepted and rejected repair trajectories. We compare typed edits, exact source patches, and complete module rewrites under matched source, feedback, model, and budget, one episode per state and interface. With DeepSeek-V4-Flash, the three interfaces solve almost the same number of the benchmark's localized failures. Typed repair is the cheapest per solved state (patching is 1.30x as expensive, rewriting 2.06x), and within 10,000 completion tokens per task it reaches almost all of its final coverage, while both free-form interfaces are well behind. A second model, Qwen3.6-Flash, solves fewer states but keeps typed repair cheapest, puts it ahead on the proof-authoring states, and repeats the localized pattern.","abs_url":"https://arxiv.org/abs/2607.28110","pdf_url":"https://arxiv.org/pdf/2607.28110v1","primary_category":"cs.AI","published":"2026-07-30","tags":["ai-agents","theorem-proving-ml","benchmark","tooling"],"lean_note":"Introduces a schema-checked typed local-edit interface for repairing failed Lean proof blueprints, plus BlueprintTrace, a benchmark of 142 controlled failures.","thumb":"thumbs/2607.28110.jpg"},{"id":"2608.02634","title":"A Dimension-Two Counterexample to the Separable Jacobian Conjecture in Characteristic Two","authors":["Romy Mondello"],"abstract":"Let k be the algebraic closure of F_2. We study the polynomial endomorphism F=(P,Q) of the affine plane, where P=x+x^2 y+x^4+x^6 y^2 and Q=y+x^5+x^6 y+x^7 y^2+x^8 y^3. Its Jacobian determinant is 1, while the three distinct points (0,1), (1,0), and (1,1) have the same image. We prove that [k(x,y):k(P,Q)]=3 and that the extension is separable. Thus the generic degree is prime to the characteristic although F is not an automorphism, giving a dimension-two counterexample to the separable Jacobian conjecture in characteristic two. The proof uses explicit recovery from a hidden cubic, irreducibility over the actual target field, and a bridge between function-field embeddings and the geometric generic fiber. We also give an explicit graph presentation proving that F is etale and derive the map from a coordinate-permuted form of a three-variable map of Irit Huq-Kuruvilla. An appendix records the precise scope and evidence boundaries of a Lean formalization and an independent Harmonic Aristotle replay.","abs_url":"https://arxiv.org/abs/2608.02634","pdf_url":"https://arxiv.org/pdf/2608.02634v1","primary_category":"math.AG","published":"2026-07-29","tags":["formalization","algebra"],"lean_note":"An appendix records a Lean formalization verifying parts of the characteristic-two counterexample to the separable Jacobian conjecture.","thumb":"thumbs/2608.02634.jpg"},{"id":"2608.00081","title":"Value distributions for read-once polynomials on finite fields","authors":["Alexey Yashunsky","Dmitrii Tabalin"],"abstract":"Consider read-once polynomial functions over a finite field of order $k$, i.e., functions defined by expressions built from field addition, multiplication, and constants, in which every variable occurs at most once. Let $p=(p_1,\\dots,p_k)$ be the distribution of the values of a read-once function on independent uniform inputs. We prove that for every $k\\ge4$ all such distributions belong to a body $\\mathcal{B}_k$, defined by the following relation on the sorted atoms $p_1^\\downarrow\\geq\\cdots\\geq p_k^\\downarrow$ of the distribution: \\[ \\mathcal{B}_k=\\left\\{p:p_k^\\downarrow\\ge \\frac{1-p_2^\\downarrow-(1-p_2^\\downarrow)^k}{k-1}\\right\\}, \\] or equivalently, $\\mathcal{B}_k = \\{ p \\colon 1-p_2^\\downarrow-(k-1)p_k^\\downarrow\\le(1-p_2^\\downarrow)^k\\}$. Our main theorem is that this body is stable under the convolutions corresponding to both field operations. More generally, convolution for any quasigroup operation on $k$ points preserves $\\mathcal{B}_k$; the multiplicative conclusion needs only an absorbing zero and a quasigroup operation on the nonzero elements. The body is full-dimensional and contains the uniform law and every point mass; its normalized volume is given by an exact one-dimensional integral, and the $k$th root of that volume tends to $0.2183305369\\ldots$. The complete development, including the two stability theorems, the exact volume formula, and its sharp exponential rate, has been formalized in Lean 4.","abs_url":"https://arxiv.org/abs/2608.00081","pdf_url":"https://arxiv.org/pdf/2608.00081v1","primary_category":"math.CO","published":"2026-07-29","tags":["formalization","combinatorics","probability"],"lean_note":"Lean 4 formalizes the two convolution-stability theorems, the exact volume formula, and its sharp exponential rate for the body of distributions.","repo":"https://github.com/Iluvmagick/rronce","thumb":"thumbs/2608.00081.jpg"},{"id":"2607.27452","title":"Counterexamples, Spectral Obstructions, and Deletion Stability for WOW-284","authors":["Samuil Petkov"],"abstract":"WOW-284 asserts that the minimum dual degree of every connected graph of order at least three and girth at least five does not exceed the negative of its least distance eigenvalue. We refute it with exact counterexamples of orders $38,39,40,42$, and $50$, and develop a structural theory of the failure. For a connected $k$-regular graph of girth at least five and diameter three, we prove $δ^*(G)+λ_{\\min}(D(G))=2k-2-\\max_{θ\\ne k}(θ+1)^2$. Here $θ$ ranges over the nonprincipal adjacency eigenvalues. We further prove that every regular strict counterexample has degree at least six and diameter at most four, while diameter four forces degree at least ten. We solve the associated one-variable nonbacktracking linear program exactly, including optimizer rigidity. For regular strict counterexamples of diameter three, the optimizer yields a positive-semidefinite slack matrix whose integral excess gives the stronger bound $|V(G)|\\le\\left\\lfloor 3(k+2)^2(k^2+3)/(18k+41)\\right\\rfloor$; this follows from a three-to-one quantization theorem for the integral excess. The slack matrix's principal minors also recover local cycle constraints. In particular, regular degree-six counterexamples have order at most $50$, and at the degree-six, order-$50$ boundary the associated signed complement is necessarily disconnected. We determine the distance spectra of one- and two-vertex punctures of Moore graphs and establish a uniform deletion-stability bound: every deletion of at most five vertices from the Hoffman--Singleton graph remains a strict counterexample, whereas an explicit six-vertex deletion does not. All theorem-level computations use exact arithmetic. Lean 4.31 kernel-checks the explicit $50$-vertex Hoffman--Singleton counterexample at graph level, finite spectral certificates at orders $38,39,40,42$, and the analytic LP optimum and rigidity for every integer $k\\ge4$.","abs_url":"https://arxiv.org/abs/2607.27452","pdf_url":"https://arxiv.org/pdf/2607.27452v1","primary_category":"math.CO","published":"2026-07-29","tags":["formalization","combinatorics"],"lean_note":"Lean 4.31 kernel-checks the 50-vertex Hoffman–Singleton counterexample, finite spectral certificates at orders 38–42, and the analytic LP optimum for all k≥4.","repo":"https://github.com/SamPetkov/wow284","thumb":"thumbs/2607.27452.jpg"},{"id":"2607.27298","title":"From Lecture Notes to Lean: Formalizing a Textbook on Probability Theory","authors":["Shuo Deng","Kenneth W. Shum"],"abstract":"As large language models become increasingly capable of generating mathematical arguments, mathematics is likely to face not a scarcity of proofs but an abundance of plausible ones. In such an environment, verification, exposition, and incorporation into reusable mathematical infrastructure become central tasks. We report on an ongoing Lean formalization of \"Measure-Theoretic Probability: With Applications to Statistics, Finance, and Engineering\", a fourteen-chapter upper-level undergraduate textbook covering topics from Riemann--Stieltjes integration to martingales and limit theorems. The project produces a machine-checked companion to the textbook and contributes reusable infrastructure for future formalizations involving probability theory. A Lean formalization provides computer-checked statements and proofs, makes hypotheses explicit, and allows readers to inspect the precise logical content of textbook results. A central challenge is to bridge textbook-facing statements with Mathlib's more general measure-theoretic interfaces. We reuse Mathlib results when possible and introduce reviewable interface lemmas when the textbook formulation and library abstraction differ. The project illustrates how formalized textbooks can support teaching, clarify mathematical assumptions, and help build the formal foundations needed for reliable AI-assisted mathematics.","abs_url":"https://arxiv.org/abs/2607.27298","pdf_url":"https://arxiv.org/pdf/2607.27298v1","primary_category":"cs.LO","published":"2026-07-29","tags":["formalization","mathlib","ai-agents","probability"],"lean_note":"Formalizes a fourteen-chapter measure-theoretic probability textbook in Lean, building bridge lemmas connecting textbook statements to Mathlib's general measure-theory interfaces via an agentic workflow.","repo":"https://github.com/wkshum/ProbabilityTheory","thumb":"thumbs/2607.27298.jpg"},{"id":"2607.27259","title":"CircuitProver: Agentic Lean 4 Theorem Proving with Reusable Circuit Proof Library for Hardware Verification","authors":["Ziyi Yang","Wenji Fang","Chen Chen","Zhiyao Xie","Hongce Zhang"],"abstract":"Modern integrated circuits (ICs) are becoming increasingly complex, making functional verification a major bottleneck. The dominant hardware formal verification methodology, model checking, verifies each design instance separately and exposes only pass/fail results, so the reasoning behind a proof stays locked inside solver heuristics and is repeatedly reconstructed across related designs. Interactive theorem proving instead yields explicit, reusable proof artifacts, but applying it to hardware remains largely manual, demanding expert effort for formalization, invariant discovery, and proof development. In this paper, we present CircuitProver, an agentic Lean 4-based verification framework supporting proof-accumulation and parameterized verification. CircuitProver automatically translates parameterized hardware designs and their natural language specifications into executable Lean 4 models. It then iteratively constructs machine-checked proofs through Lean feedback to establish that the hardware code complies with the specification. The proving traces and verified theorems are distilled into reusable libraries, where proving strategies guide future agent reasoning and verified lemmas support formal proof reuse across related hardware verification tasks. We further introduce the first benchmark suite for evaluating agentic hardware theorem proving, covering diverse parameterized hardware designs, specifications, proof tasks, and evaluation metrics. Across 63 tasks, CircuitProver successfully proves all benchmarks, while a vanilla agent solves 92.1% of them and requires twice as many proof rounds on average. Ablation studies show that accumulated proof knowledge reduces redundant proof construction across related verification tasks, reducing proof length by 16.3% and verification time by 23.2%.","abs_url":"https://arxiv.org/abs/2607.27259","pdf_url":"https://arxiv.org/pdf/2607.27259v1","primary_category":"cs.LO","published":"2026-07-29","tags":["verification","ai-agents","theorem-proving-ml","autoformalization"],"lean_note":"CircuitProver is an agentic Lean 4 framework that translates parameterized Chisel hardware designs into Lean models and proves correctness theorems.","thumb":"thumbs/2607.27259.jpg"},{"id":"2607.26997","title":"Estimating Size of the Union of Sets in Streaming Model","authors":["Kuldeep S. Meel","N. V. Vinodchandran","Sourav Chakraborty"],"abstract":"We study estimating the size of the union of sets $S_1,\\dots,S_M$, where each $S_i\\subseteqΩ$ is presented implicitly and arrives in a stream. We introduce Delphic sets, a class of streaming problems in which membership, sampling, and counting queries to each set are efficient, and show that this notion captures three well-known problems: Klee's measure problem (discrete version), test coverage estimation in combinatorial testing, and model counting of DNF formulas. Our primary contribution is a simple and efficient sampling-based algorithm that outputs an $(\\varepsilon,δ)$-approximation of the cardinality of the union of Delphic sets in the streaming setting. It has space complexity $O(R\\log|Ω|)$ and update time $O(R\\log R\\cdot\\log(M/δ)\\cdot\\log|Ω|)$, where $R=O(\\log(M/δ)\\cdot\\varepsilon^{-2})$. For the streaming Klee's measure problem, this gives the first algorithm whose update time depends linearly on the dimension $d$ for $d>1$, settling an open problem of Tirthapura and Woodruff (PODS 2012), and it directly yields efficient streaming algorithms for coverage estimation and DNF model counting. We further show that the space for coverage estimation can be made near-optimal at the cost of an update procedure in $\\mathrm{P}^{\\mathrm{NP}}$, revealing a time-space trade-off. A key strength of our approach is the simplicity of both the algorithm and its analysis, which makes it amenable to practical implementation. In this revised version, the algorithm and its correctness analysis have additionally been formalized and machine-checked in Lean 4. (Shortened for Arxiv)","abs_url":"https://arxiv.org/abs/2607.26997","pdf_url":"https://arxiv.org/pdf/2607.26997v1","primary_category":"cs.DS","published":"2026-07-29","tags":["verification","formalization","combinatorics"],"lean_note":"The union-of-sets streaming estimator and its correctness analysis are formalized and machine-checked in Lean 4.","thumb":"thumbs/2607.26997.jpg"},{"id":"2607.26413","title":"Machine-Checked Certificates for the Geometric Half of the Minimum Kochen-Specker Bound","authors":["Shayaan Siddique","Ibrahim Mian"],"abstract":"The best known lower bound for the minimum Kochen-Specker vector system in $\\mathbb{R}^3$ -- 24 vectors -- rests on a computational proof whose combinatorial half emits DRAT proofs but whose geometric half does not: the non-embeddability of thousands of candidate graphs is established by Z3's nonlinear real arithmetic, which produces no checkable proof objects. We close this gap for the proof's blocking database. We introduce exact rational case-tree certificates of real non-embeddability, whose splits are polynomial factorizations and rational sum-of-squares decompositions and whose leaves are discharged by injectivity, ideal-membership, or Positivstellensatz-shaped positivity arguments, and we certify all 291 source lines (180 distinct graphs) of the published pipeline's order-10 to order-13 blocking lists. Certificates are replayed by two independent checkers that share no code with the generator: a pure-Python replay over exact fractions, and a total checker implemented and proved sound in Lean 4. The soundness theorem -- acceptance implies that no injective-on-rays, orthogonality-respecting assignment of nonzero real vectors realizes the graph -- is kernel-checked with axiom closure {propext, Classical.choice, Quot.sound}, and a gcd-free rational arithmetic layer makes the entire verdict computation kernel-reducible, so each per-graph non-embeddability result is a closed kernel theorem proved by decide. The formalization surfaced findings about the published pipeline, including a load-bearing injectivity side condition in its embeddability notion, hidden WLOG case obligations invisible to Z3-based workflows, and an unreproducible candidate count that we resolve against the published artifacts. All certificates, checkers, and proofs are available and replayable from a single build.","abs_url":"https://arxiv.org/abs/2607.26413","pdf_url":"https://arxiv.org/pdf/2607.26413v1","primary_category":"cs.LO","published":"2026-07-29","tags":["verification","formalization","quantum","geometry"],"lean_note":"A checker for real non-embeddability certificates of Kochen-Specker candidate graphs is implemented and proved sound in Lean 4, with per-graph results as kernel theorems.","repo":"https://github.com/shayaansiddique06/kscert","thumb":"thumbs/2607.26413.jpg"},{"id":"2607.27246","title":"The 2-Domination Number and the Upper Median Degree: A Proof of Graffiti.pc Conjecture 387","authors":["Jun Qing"],"abstract":"Let G be a nonempty finite simple graph of order n, and let m(G) be the upper median of its degree sequence. We prove that the 2-domination number satisfies gamma_2(G) <= n - m(G) + 1. This proves Graffiti.pc Conjecture 387. In fact, the argument establishes the inequality for every nonempty finite simple graph, so the connectedness hypothesis in the original formulation is unnecessary. The proof uses the complement graph and a minimally linearly dependent family of polynomials encoding selected nonneighborhoods.","abs_url":"https://arxiv.org/abs/2607.27246","pdf_url":"https://arxiv.org/pdf/2607.27246v1","primary_category":"math.CO","published":"2026-07-28","tags":["formalization","combinatorics"],"lean_note":"Graffiti.pc Conjecture 387 on the 2-domination number bound was formalized and machine-checked in Lean 4 using Mathlib.","repo":"https://github.com/qscqesze/graffiti-pc-conjecture-387","thumb":"thumbs/2607.27246.jpg"},{"id":"2607.26344","title":"Automorphism-Induced Non-Canonicity in Top-k Explanations of Graph Neural Networks","authors":["Xin Xu","Siru Tao","Kaizhen Tan"],"abstract":"A gradient-based GNN explainer given a molecule with two chemically equivalent nitro groups assigns them attribution scores that are equal to the last bit. It cannot do otherwise: message passing is exactly permutation equivariant, so any automorphism of the input leaves every attribution invariant. Yet the standard report, the top-k edges, names one of the two, and which one is settled by the order of an array. We show this is a structural obstruction rather than an implementation slip. When no minimal valid explanation is fixed by the input's automorphism group, no rule can be single-valued, minimal and symmetry-respecting at once. For the exact-k reports used in practice we give a parameter-free criterion, mechanised in Lean 4 with no axiom dependencies, that decides from the graph alone whether every score-optimal report of that size must split an orbit. Across 21298 instance-budget decisions the criterion agrees with a mechanical model-equivalence check without exception, and no severing case we found admitted a neutral alternative. The obstruction is common. Nontrivial automorphisms occur in 93.4% of Mutagenicity, the dataset the seminal explainability papers use, so the measure-zero dismissal of symmetric inputs, sound on the continuous domains it was made for, collapses here. At the sparsity budget those papers report, 24.0% of molecules with two interchangeable nitro groups (6 of 25) surface exactly one of them, every one arbitrary under mechanical verification. A model's blindness also manufactures symmetry: every MUTAG molecule contains atoms chemistry separates and the network provably cannot, and a matched control shows the resolution is set by what the model reads rather than how it is parameterised. Reporting orbits removes the arbitrariness at 0.11 ms and 0.43 extra edges per graph.","abs_url":"https://arxiv.org/abs/2607.26344","pdf_url":"https://arxiv.org/pdf/2607.26344v1","primary_category":"cs.LG","published":"2026-07-28","tags":["verification","ml-theory"],"lean_note":"A parameter-free criterion deciding whether score-optimal top-k GNN explanations must split an automorphism orbit is mechanised in Lean 4 with no axiom dependencies.","thumb":"thumbs/2607.26344.jpg"},{"id":"2607.26331","title":"Quantum Turing Patterns","authors":["Kazuki Ikeda"],"abstract":"We construct quantum Turing patterns in Lindblad lattice dynamics and establish a rigorous theory of their nonlinear order and quantum fluctuations. For an explicit completely positive family with finite-range couplings, the first-moment equations undergo a supercritical instability at a nonzero wave number and admit analytic site- and bond-centered commensurate stripe branches. These branches are locally asymptotically stable in their reflection-fixed period-cell spaces, and projected coherent states exhibit extensive Bragg order on every bounded time interval in the semiclassical limit. We prove $O(N^{-1/2})$ convergence of microscopic covariances to a nonautonomous Gaussian Lyapunov flow, transferring strict partial-transpose uncertainty violations to sufficiently large $N$. In the homogeneous Gaussian sector, a single dimensionless ratio controls both the Turing stability determinant and the logarithmic negativity of opposite momenta, relating wavelength selection directly to quantum entanglement. Differential transport shifts the strongest opposite-momentum correlations from the infrared to the selected Turing scale. Numerical continuation and two-dimensional simulations display stripe, spot, and labyrinth morphologies whose Fourier modes and fluctuation spectra concentrate at the same selected wave numbers.","abs_url":"https://arxiv.org/abs/2607.26331","pdf_url":"https://arxiv.org/pdf/2607.26331v1","primary_category":"math-ph","published":"2026-07-28","tags":["formalization","physics"],"lean_note":"A Lean 4 formalization accompanies the numerical source and reference data for the quantum Turing pattern construction in the author's repository.","repo":"https://github.com/IKEDAKAZUKI/Quantum-Turing-Pattern","thumb":"thumbs/2607.26331.jpg"},{"id":"2607.26306","title":"Foundational Refinement Proofs for Deployed Bytecode, at the Price of Tokens","authors":["Lefteris Lazaropoulos","Zoe Paraskevopoulou"],"abstract":"Relating low-level executable code to a high-level account of its behavior has been a central concern of programming-language research for decades. From formally verified compilers to translation validators, certifying compilers, and proof-carrying code, each approach chooses between laborious but foundational mechanized proofs and automation that costs completeness, generality, and an increased trusted base. Recently, large language models (LLMs) have begun to change the economics of formal verification. Agentic proof development is now capable of producing machine-checked proofs at a scale and speed that were previously out of reach. In this paper, we evaluate the capabilities of LLMs to produce foundational, machine-checked proofs of refinement between executable code and its high-level specification, as post hoc, per-artifact certificates. We study this in the context of the Ethereum Virtual Machine (EVM), a low-level virtual machine that executes smart contracts on the Ethereum blockchain. We build EquiVM, a foundational framework in Lean comprising an executable EVM semantics and a specification language that characterizes the intended behavior of smart contracts, but commits to no source language or compilation toolchain. In EquiVM, refinement is stated for deployed bytecode of arbitrary provenance, interaction with unknown code is part of the semantics, and each proof is a replayable, machine-checked certificate. No previous technique achieves this combination. Using frontier commercial LLMs, twenty-three real-world contracts are proved end to end with minimal human guidance, among them most of the MakerDAO stablecoin system, at up to a hundred million tokens and a hundred hours of proof time per contract. We conclude that foundational mechanized proofs can now be bought at the price of tokens, and that this shift can reshape how verification frameworks are architected.","abs_url":"https://arxiv.org/abs/2607.26306","pdf_url":"https://arxiv.org/pdf/2607.26306v1","primary_category":"cs.PL","published":"2026-07-28","tags":["verification","ai-agents","theorem-proving-ml"],"lean_note":"EquiVM, a Lean 4 framework with executable EVM semantics and Sol- specification language, states and proves refinement of deployed bytecode via LLM agents.","repo":"https://github.com/OpenZeppelin/openzeppelin-contracts","thumb":"thumbs/2607.26306.jpg"},{"id":"2607.26230","title":"Formally certifying number field invariants","authors":["Alain Chavarri Villarello","Sander R. Dahmen"],"abstract":"Number fields, which generalize the rational numbers, are fundamental objects in number theory. Many of their key arithmetic properties are captured by invariants whose computation is among the central tasks of computational algebraic number theory and a focus of several computer algebra systems and databases. In this paper, we describe a Lean 4 formalization for certifying several of these number field invariants. Building on previous work on certifying rings of integers, we extend this certification approach to further invariants including the signature, the unit group modulo $p$-th powers, and, ultimately, the class group. We also improve discriminant certification, allowing verifications for higher-degree number fields infeasible in previous work. We introduce structures based on representations of algebraic objects suited to computation, including reusable ones for certifying ideal arithmetic. Along the way, we formalize several underlying mathematical results, for instance on real closed fields and pseudo-remainder sequences, which are of independent interest. We apply our framework to verify hundreds of entries of the $\\textit{L-functions and modular forms database}$ (LMFDB) concerning the discriminant, signature, class number, and class group structure of various number fields. To this end, we wrote a SageMath script that computes the certificates and outputs Lean proofs of the corresponding statements.","abs_url":"https://arxiv.org/abs/2607.26230","pdf_url":"https://arxiv.org/pdf/2607.26230v1","primary_category":"cs.LO","published":"2026-07-28","tags":["formalization","verification","number-theory","mathlib"],"lean_note":"Lean 4 formalization certifies number field invariants (discriminant, signature, class group) using externally-computed certificates checked against Mathlib definitions.","repo":"https://github.com/artie2000/real_closed_field","thumb":"thumbs/2607.26230.jpg"},{"id":"2607.25628","title":"Kernel-Checked Exclusions for the Erdős-Selfridge Odd Covering Problem: Any Odd Covering of $\\mathbb{Z}$ Has lcm Exceeding 10000","authors":["Ibrahim Mian","Shayaan Siddique"],"abstract":"The Erdős-Selfridge odd covering problem (Erdős problem #7) asks whether a covering system of $\\mathbb{Z}$ exists whose moduli are all odd, distinct, and greater than 1. The problem is open. We present a Lean 4 formalization, checked end to end by the proof kernel, of the exclusion: any covering of $\\mathbb{Z}$ by finitely many congruence classes with distinct odd moduli > 1 has lcm of the moduli exceeding 10000. The proof composes a formalized density argument (a covering by divisors of $N$ exceeding 1 forces $2N \\le σ_1(N)$, so the lcm is abundant or perfect), a kernel-checked abundancy floor (no odd $N < 945$ qualifies), a family of Chinese-Remainder capacity certificates -- decidable per-$N$ arithmetic inequalities each refuting every covering with distinct moduli > 1 dividing that $N$ -- for all 23 odd abundant numbers below $10^4$, and a kernel-checked enumeration establishing that those 23 are the only odd non-deficient candidates. The result is transported to the official StrictCoveringSystem $\\mathbb{Z}$ formulation of Erdős #7 in google-deepmind/formal-conjectures, with a bidirectional periodicity bridge between coverings of $\\mathbb{Z}$ and finite checks over $\\mathbb{Z}/N\\mathbb{Z}$ suitable for consuming future SAT-style search output. All 63 published theorems depend on exactly propext, Classical.choice, and Quot.sound: no sorry, no native_decide, no solver in the trusted base. The mathematical content is known -- the density argument is folklore, and far larger uncertified classifications of covering numbers exist -- so the contribution is epistemic rather than mathematical: these exclusions are theorems of the Lean kernel, with an axiom gate enforced mechanically in continuous integration.","abs_url":"https://arxiv.org/abs/2607.25628","pdf_url":"https://arxiv.org/pdf/2607.25628v1","primary_category":"cs.LO","published":"2026-07-28","tags":["formalization","verification","number-theory"],"lean_note":"Lean 4 over Mathlib formalizes, with kernel-checked proofs, that any odd covering system of Z has lcm exceeding 10000.","repo":"https://github.com/ibrahimmian36/centurion","thumb":"thumbs/2607.25628.jpg"},{"id":"2607.25262","title":"Type Safety via Hoare Logic with Separation and Pure Types","authors":["Wenhua Li","Darius Foo","Quang Trung Ta","Wei-Ngan Chin"],"abstract":"Type safety has traditionally rested on carefully crafted type systems, under the motto \"well-typed programs cannot go wrong\". Modern demands push type systems past this basic guarantee: toward memory safety (e.g., Rust), stronger data-structure invariants (e.g., GADTs), and broader typability (e.g., MLstruct). The motto absorbs each such property by enlarging the set of states deemed \"wrong\", but collapses them into one binary verdict: heap ownership, flow-sensitive changes to a variable's type, and the gap between a recoverable and a fatal error are relational, stateful facts about intermediate states that one verdict cannot tell apart. Worse, each demand typically brings its own extension, making it hard to say what each guarantees or how they combine. Floyd-Hoare logic supplies a unified foundation. We present a framework for type-safety verification built from four ingredients: (i) case specifications for path-sensitive typing; (ii) separation types, inspired by separation logic, for flow-sensitive type mutation and must-aliasing; (iii) a disciplined distinction between Err (runtime error values our types track) and Abrt (compile-time errors), yielding the refined motto well-typed programs must never abort; and (iv) type predicates for data-structure invariants. Since all four are ordinary types in one Boolean algebra rather than separate extensions, the framework subsumes both GADTs and liquid types within one type logic, spanning weak specifications that tolerate Err to strong ones that eliminate it. Subtyping reduces to one decidable emptiness test, so a single lightweight procedure serves the whole framework with no SMT oracle in its trusted base. We formalise the Hoare rules and prove soundness in a machine-checked Lean mechanisation; by proof reflection it yields a self-certifying type-checker, evaluated on a benchmark suite.","abs_url":"https://arxiv.org/abs/2607.25262","pdf_url":"https://arxiv.org/pdf/2607.25262v1","primary_category":"cs.PL","published":"2026-07-28","tags":["verification","formalization","logic","type-theory"],"lean_note":"Meta-theory of the TypeHL type-safety framework is mechanised in Lean 4, yielding a self-certifying reflective type-checker.","thumb":"thumbs/2607.25262.jpg"},{"id":"2607.24504","title":"Classifying Capabilities (Extended Version)","authors":["Cao Nguyen Pham","Oliver Bračevac","Yichen Xu","Yaoyu Zhao","Martin Odersky"],"abstract":"Capture checking in Scala 3 enables lightweight and practical effect and resource tracking by recording capabilities in types. However, the system offers no way to reason about kinds of capabilities. Natural constraints such as \"retaining only the control-flow capabilities of this closure\" or \"excluding all thread-local capabilities from this argument\" become inexpressible. Both arise in the Scala 3 standard library: \"Try\" re-throws caught exceptions, so it retains only the control-flow capabilities of its body, and \"Future\" must not capture thread-local resources. The inability to state these constraints has kept parts of the library outside capture checking. We introduce capability classifiers: a tree-structured, user-extensible hierarchy of tags that classify capabilities by their semantic role. Projections filter capture sets by classifier, supporting both inclusion (\"c.only[C]\") and exclusion (\"c.except[C]\"). The tree structure enables decidable disjointness reasoning: classifiers on separate branches are guaranteed to be disjoint regardless of unknown extensions elsewhere in the hierarchy. We formalize classifiers as an extension of System Capless, a core calculus for capture checking, introducing a classifier kind algebra based on intersection, union, and subtraction of classifier subtrees. We extend the operational semantics to model exception interception and establish type safety, effect safety, and handler coverage via a big-step proof, fully mechanized in Lean 4. Classifiers are implemented in the Scala 3 capture checker, and we demonstrate their use on standard library types and real-world effect exclusion patterns.","abs_url":"https://arxiv.org/abs/2607.24504","pdf_url":"https://arxiv.org/pdf/2607.24504v1","primary_category":"cs.PL","published":"2026-07-27","tags":["verification","type-theory","formalization"],"lean_note":"Type safety, effect safety, and handler coverage for capability classifiers proven via a mechanized big-step proof in Lean 4.","thumb":"thumbs/2607.24504.jpg"},{"id":"2607.24483","title":"The exact solution of Bellman's lost-in-a-forest problem for the golden gnomon","authors":["Alexander Temerev","Alessio Doria"],"abstract":"We solve Bellman's lost-in-a-forest problem for the golden gnomon $G$, the isosceles triangle with equal sides $1$ and apex angle $108^\\circ$: the shortest curve guaranteed to reach the boundary of $G$ from an unknown starting position and heading is a symmetric seven-piece path of segments, circular shoulders, and tangents, of exactly determined length $C=1.282676025459\\ldots$. To our knowledge, this is the first proved exact optimum for an isosceles triangle whose base angle is below $45^\\circ$. The curve's parameters come from one isolated quartic root, and $C$ is transcendental. Equivalently, $C^{-1}G$ is the smallest homothetic golden-gnomon cover of all unit arcs. The proof introduces a balanced support calibration: one weighted family of escape inequalities, built on the linear relation among the triangle's three normals, exactly saturated by the candidate, through eighteen exact support windows, and confronting every shorter competitor at once. Aggregation along the normal fan compresses the calibration to a finite zero-sum family of supported vectors; summation by parts then bounds its total by path length whenever the running suffix balance, the ledger, stays in the unit disk. A local two-gap surgery and cyclic bitonicity force a shortest hypothetical counterexample into exactly the temporal order the ledger tolerates. Lean 4 verifies the two finite algebraic certificate families and the reusable discrete ledger identities and bounds.","abs_url":"https://arxiv.org/abs/2607.24483","pdf_url":"https://arxiv.org/pdf/2607.24483v1","primary_category":"math.MG","published":"2026-07-27","tags":["formalization","verification","geometry"],"lean_note":"Lean 4 verifies the two finite algebraic certificate families and the reusable discrete ledger identities and bounds underlying the exact optimum.","repo":"https://github.com/atemerev/gnomon","thumb":"thumbs/2607.24483.jpg"},{"id":"2607.23828","title":"Powers of the Vandermonde determinant are eventually non-SNP","authors":["Thien Le","Melanie Weber"],"abstract":"We prove a conjecture of Monical, Tokcan, and Yong that every fixed positive power of the Vandermonde determinant is non-SNP in all sufficiently many variables, where a polynomial is non-SNP if there is a lattice point in its Newton polytope that does not appear with nonzero coefficient. This means our result proves that for every even power $k\\geq4$, there is always such a missing lattice monomial in large enough dimensions. The odd case follows from alternation, and the quadratic case was previously known. For every even power $k\\geq4$, we exhibit an explicit lattice point in the Newton polytope of $a_{δ_k}^k$ whose coefficient vanishes. The vanishing is obtained from a Dyson constant-term identity, proved using the finite-variable Jack scalar product and Macdonald's specialization formula. The key even-power construction and proof strategy arose from prompting with OpenAI Codex (GPT Sol 5.6 Extra High), a large language model; the complete transcript appears in the appendix. The authors subsequently checked and organized the argument. The accompanying Lean formalization is available at https://github.com/steven-le-thien/vandermonde-snp.","abs_url":"https://arxiv.org/abs/2607.23828","pdf_url":"https://arxiv.org/pdf/2607.23828v1","primary_category":"math.CO","published":"2026-07-26","tags":["formalization","combinatorics"],"lean_note":"An accompanying Lean formalization verifies the combinatorial proof that fixed powers of the Vandermonde determinant are eventually non-SNP.","repo":"https://github.com/steven-le-thien/vandermonde-snp","thumb":"thumbs/2607.23828.jpg"},{"id":"2607.23715","title":"Formally Verified Synthesizable Floating-Point Data Types in ARCH HDL","authors":["Shuqing Zhao"],"abstract":"We report the design and end-to-end verification of first-class IEEE-754 binary32 (FP32) and bfloat16 (BF16) arithmetic for ARCH, a hardware description language intended to be generated by language models. Every operator - comparisons, conversions, add, sub, mul, and fused multiply-add (FMA) - is described once against a single bit-vector IR and rendered three ways from one source: synthesizable SystemVerilog, an SMT-LIB model, and a Lean 4 proof model. The three artifacts cannot drift apart structurally, and the residual per-node printer correspondence is machine-checked: a Yosys-to-SMT miter proves the emitted SystemVerilog equivalent to the SMT model for all 24 operators. Verification splits at the solver-tractability frontier: multiplier-free operators (comparisons, add/sub over all 2^64 inputs, conversions, and all binary BF16 arithmetic) are proved exhaustively equivalent to the SMT-LIB FloatingPoint theory; the SAT-hard multiplier-bearing operators (FP32 mul and FMA) are proved correctly rounded in Lean, sorry-free, against a value-level round-to-nearest-even specification over exact dyadic values. Physical characterization exposed the FMA as the timing outlier: its exact-wide 470-bit datapath does not pipeline in our flow. We reimplemented it as a bounded 98-bit guard/round/sticky datapath that pipelines to 268 MHz on Nangate45, and proved, in Lean and over all 2^96 inputs, that it is bit-identical to the exact-wide reference, so it inherits the reference's proven correct rounding. The equivalence is tractable precisely because the shared multiplier appears on both sides and cancels: neither a SAT solver nor the proof ever solves a multiplier equivalence. (The BF16 FMA is deliberately an FP32-accumulating fusion, characterized as exactly that.) All machine-checked claims are pinned to a tagged open-source release.","abs_url":"https://arxiv.org/abs/2607.23715","pdf_url":"https://arxiv.org/pdf/2607.23715v1","primary_category":"cs.CL","published":"2026-07-26","tags":["verification"],"lean_note":"IEEE-754 FP32/BF16 multiplier-bearing operators (mul, FMA) are proved correctly rounded and bit-exact in sorry-free Lean 4.","repo":"https://github.com/arch-hdl-lang/arch-com","thumb":"thumbs/2607.23715.jpg"},{"id":"2607.23500","title":"Formalizing Flag Algebras in Lean","authors":["Gyeongwon Jeong","Seonghun Park","Jihoon Hyun","Sang-il Oum","Hongseok Yang"],"abstract":"Razborov's flag algebra method is a powerful tool for proving asymptotic inequalities in extremal graph theory, often reducing the task to finding a finite certificate by semidefinite programming. We present a machine-checked formalization of the method for finite simple graphs, together with a certificate-to-proof compiler that turns externally generated certificate data into algebraic proofs checked by Lean. The formalization covers the foundations of the method: partially labeled graphs, their densities in large graphs, the quotient algebra of density expressions, graph-limit semantics through positive homomorphisms, and the downward operators used to average out labels. The compiler treats the external semidefinite programming output as candidate data rather than trusted input: Lean independently computes the required density and multiplication facts, verifies positive semidefiniteness exactly over $\\mathbb{Q}$, and carries out the algebraic normalization steps of flag-algebra proofs. Our case studies yield formal proofs of seven Turán-type upper bounds, including Mantel's theorem and the Erdős pentagon theorem, a $C_4$-density bound for triangle-free graphs, and edge-density bounds for $K_4$-free, $K_5$-free, and $C_5$-free graphs. Independently of the compiler, we formalize the matching constructions that complete the exact Turán densities of Mantel's theorem and the Erdős pentagon theorem, and prove two inequalities of Goodman. Our constrained semantics also prompted a meta-theoretic comparison of two ways of imposing graph constraints: building a hereditary constraint into the flag algebra from the start, or testing inequalities afterward on constrained graph limits with labels chosen at random. We state the resulting root-plantability criterion characterizing when the two approaches agree; a forthcoming paper will present the complete account.","abs_url":"https://arxiv.org/abs/2607.23500","pdf_url":"https://arxiv.org/pdf/2607.23500v1","primary_category":"cs.LO","published":"2026-07-26","tags":["formalization","verification","combinatorics"],"lean_note":"Formalizes Razborov's flag algebra method in Lean with a certificate-to-proof compiler verifying semidefinite programming output over ℚ.","thumb":"thumbs/2607.23500.jpg"},{"id":"2607.23427","title":"A note on Lata\\la's argument in SK model","authors":["Seiichiro Kusuoka","Shuta Nakajima"],"abstract":"In this note, we consider the Sherrington--Kirkpatrick model with deterministic external field. Let $q=q(β,h)$ denote the solution of the replica-symmetric self-consistency equation \\[ q=\\mathbb E\\tanh^2\\!\\left(h+β\\sqrt q\\,Z\\right), \\qquad Z\\sim N(0,1), \\] where $β$ and $h$ are inverse temperature and external field, respectively. By refining Lata\\la' s argument, previously limited to \\(β< \\frac{1}{2}\\), and using the Kearns--Saul inequality, we prove overlap concentration and convergence of the free energy to the replica symmetric formula with error \\(O(N^{-1})\\) whenever \\[ β^2\\frac{q}{{\\rm arctanh}q}<1. \\] Note that for any $β<1$ and $h\\in \\mathbb R$, the condition above is satisfied. Moreover, for every nonzero $h$, this region contains a nonempty interval with $β>1$.","abs_url":"https://arxiv.org/abs/2607.23427","pdf_url":"https://arxiv.org/pdf/2607.23427v1","primary_category":"math.PR","published":"2026-07-26","tags":["formalization","probability"],"lean_note":"A Lean 4 formalization was developed (with AI assistance) to support the extended replica-symmetry argument for the SK model.","repo":"https://github.com/njimaMath/research_public","thumb":"thumbs/2607.23427.jpg"},{"id":"2607.23390","title":"When Can Depth Replace Precision? A Resource Theory of Quantized Neural Computation","authors":["Mojtaba Soltanalian"],"abstract":"When can additional low-bit residual computation replace missing numerical precision for a fixed input-output map? We model a quantized residual system over a fixed horizon as a pure schedule selecting fields from a declared low-bit operation library, and use relaxed controls to characterize its infinite-depth limit. The distance from the target to the closed relaxed reachable set is the exact structural floor: no increase in depth can remove it for that library. Pure schedules approach the relaxed class at rate $O(D^{-1})$ under bounded-variation time dependence and $O(D^{-\\vartheta}+D^{-1})$ under Holder dependence of exponent $\\vartheta$. Execution arithmetic can reverse this conclusion: full-state write-back introduces a $Dρ_z$ penalty and can freeze residual updates, whereas increment error feedback replaces this growth by a bounded carry term and obeys an exact common-lattice conservation law. A fixed-teacher converse makes this rate sharp: for coherent depth-$L$ first-order high-precision comparators, accuracy matching requires $D=Θ(L)$. Learned codebooks add a metadata resource, while state-dependent routing introduces hybrid event conditions. Verified primal and dual bounds yield feasible, impossible, or unresolved decisions before training. Companion software implements the workflow, and Lean 4 machine-checks the exact discrete core. Depth replaces precision only relative to a declared library, horizon, execution semantics, and routing model.","abs_url":"https://arxiv.org/abs/2607.23390","pdf_url":"https://arxiv.org/pdf/2607.23390v1","primary_category":"cs.LG","published":"2026-07-25","tags":["verification","ml-theory"],"lean_note":"Lean 4 machine-checks the exact discrete core of the quantized residual reachability and depth-precision framework.","thumb":"thumbs/2607.23390.jpg"},{"id":"2607.22972","title":"Learned Interventions in Lean 4 grind","authors":["Evan Wang","Simon Chess","Sophie Szeto","Theodore Meek"],"abstract":"Lean 4's grind tactic combines congruence closure, E-matching, and case-splitting into a single automated solver, and like any such solver, it relies on hand-tuned heuristics to decide what to instantiate and where to case-split. These heuristics are tempting targets for learning, but there is a catch: because grind's search is non-monotone, a learned heuristic that helps one proof can break another, and an always-on replacement usually nets out near zero. We avoid this by invoking a learned intervention only after stock grind has already failed: a failure-triggered cascade that, by construction, cannot lose a proof grind already had. We apply it to two of grind's internal decisions. A cost-aware E-matching filter solves slightly more problems and runs about 5% faster. A lookahead step proves five theorems it otherwise times out on. We also report the negative result that motivated the design: across four feature-based models, statically predicting the correct case split is no better than random, because whether a split explodes is a runtime property that the features do not capture. Our results suggest that learning within theorem-proving tactics is most effective as a mechanism for deciding when and how to spend bounded search, backed by a reliable symbolic fallback.","abs_url":"https://arxiv.org/abs/2607.22972","pdf_url":"https://arxiv.org/pdf/2607.22972v2","primary_category":"cs.LG","published":"2026-07-25","tags":["tactic-automation","theorem-proving-ml","tooling"],"lean_note":"Learned interventions are integrated into Lean 4's grind tactic to filter e-matching instances and select case splits via failure-triggered lookahead.","thumb":"thumbs/2607.22972.jpg"},{"id":"2607.22524","title":"Machine-Checked Arithmetic Bit Complexity of the Kannan-Bachem Smith Normal Form in Lean 4","authors":["Junye Ji"],"abstract":"We formalize in Lean 4 the Kannan-Bachem Smith normal form algorithm for nonsingular square integer matrices. The program returns $S,U,U^{-1},V,V^{-1}$ and proves $UAV=S$, $U^{-1}SV^{-1}=A$, four inverse identities, the Smith divisibility conditions, and equality of $S$ with a canonical reference matrix. Stabilization terminates because each recursive pass strictly decreases the binary size of the active pivot; the outer algorithm recurses on the lower-right block. The computation also emits a flat trace of designated sign-magnitude arithmetic leaves. Branch conditions, quotients, Bezout data, and matrix entries are taken from the recorded primitive runs. Composite phases form their traces by concatenating the charge lists returned by the executed children. Verified self-delimiting codecs define the input and output sizes. Coefficient and work recurrences, closed by a kernel-checked polynomial-envelope calculus, give fixed polynomial bounds for both trace cost and the encoded length of the five output matrices. The theorem concerns these arithmetic primitives; structural operations and compiled Lean runtime are outside the model.","abs_url":"https://arxiv.org/abs/2607.22524","pdf_url":"https://arxiv.org/pdf/2607.22524v1","primary_category":"cs.LO","published":"2026-07-24","tags":["formalization","verification","algebra","number-theory"],"lean_note":"Formalizes the Kannan-Bachem Smith normal form algorithm in Lean 4 over Mathlib, proving correctness and polynomial arithmetic bit-complexity bounds.","thumb":"thumbs/2607.22524.jpg"},{"id":"2607.22511","title":"CausalForge: A Formally Grounded, Self-Improving Agentic Framework for Automated Research in Causal Inference","authors":["Jiyuan Tan","Vasilis Syrgkanis"],"abstract":"Automating theoretical research is constrained not only by the generation of candidate results, but also by their reliable evaluation. A common approach is to close the research loop with a large language model (LLM) reviewer. However, such reviewers remain empirically unreliable: they may accept fabricated papers and detect them at rates close to chance (Bad Scientist, 2025). We present CausalForge, a framework for automated theoretical research in causal inference grounded in the Lean proof assistant. CausalForge combines Causalean, a foundational Lean library for causal inference containing 7,035 machine-checked declarations developed with language-model assistance under human design and review, with CausalSmith, a self-improving agentic pipeline that selects research topics, proposes results, formalizes statements, constructs proofs, and presents the resulting artifacts for human inspection. Because a machine-checked proof establishes only that a formal statement follows from its assumptions, not that the statement faithfully captures the intended scientific claim, the pipeline augments kernel verification with a statement audit that compares each formal theorem against the informal claim it is intended to express. We evaluate the system using artifacts produced by completed autonomous research runs. The source code, formal library, and run records are available at https://github.com/Jiyuan-Tan/CausalForge.","abs_url":"https://arxiv.org/abs/2607.22511","pdf_url":"https://arxiv.org/pdf/2607.22511v1","primary_category":"stat.ML","published":"2026-07-24","tags":["formalization","ai-agents","autoformalization","economics"],"lean_note":"Introduces Causalean, a Lean 4 library of 7,035 machine-checked causal-inference declarations, plus an agentic pipeline that formalizes and proves self-proposed results in Lean.","repo":"https://github.com/Jiyuan-Tan/CausalForge","thumb":"thumbs/2607.22511.jpg"},{"id":"2607.21533","title":"Benchmarking Agents for Proving Theorems in Quantum Algorithms and Quantum Information","authors":["Lei Zhang","Yusheng Zhao","Yimeng Cao","Ranyiliu Chen","Mingrui Jing","Jizhe Lai","Ziao Tang","Jingu Xie","Hongshun Yao","Xuanqiang Zhao","Guocheng Zhen","Chengkai Zhu","Xin Wang"],"abstract":"Formal verification is becoming increasingly practical for quantum computing, yet the ability of AI agents to construct machine-checkable proofs in this domain remains unmeasured. We introduce Lean-QuantumAlg-Bench and Lean-QIT-Bench, two Lean 4 benchmarks containing 36 and 40 theorem-completion tasks for quantum algorithms and quantum information theory, respectively. Every task compiles in a fixed environment and is evaluated by deterministic proof checking and targeted semantic review, with difficulty weights assigned before model execution. We evaluate four models-GPT-5.5, Kimi K3, DeepSeek V4-Pro, and MiniMax M3-within a common theorem-proving framework under two settings: a task-only baseline and library-augmented deduction (LAD), which additionally provides access to a verified domain library. The highest difficulty-weighted scores are 60.4 out of 100 on the quantum-algorithm benchmark and 59.6 out of 100 on the quantum-information benchmark. LAD improves both score and completion rate in all eight model-benchmark comparisons, with gains of up to 15.9 points, providing evidence that verified libraries can strengthen domain-specific proof agents. The results reveal recurring weaknesses of agentic proving in areas such as quantum simulation, quantum learning, quantum information measures, and entanglement theory. Monetary and wall-clock costs per score point also vary substantially across models, highlighting important capability-efficiency trade-offs. We expect these benchmarks to establish a reproducible baseline for developing more capable and reliable proof agents, and to pave the way toward self-evolving AI scientists for advancing quantum information science.","abs_url":"https://arxiv.org/abs/2607.21533","pdf_url":"https://arxiv.org/pdf/2607.21533v1","primary_category":"quant-ph","published":"2026-07-23","tags":["benchmark","ai-agents","theorem-proving-ml","quantum"],"lean_note":"Introduces two Lean 4 benchmarks (36 and 40 theorem-completion tasks) for quantum algorithms and quantum information, evaluating AI proof agents via Lean compilation.","repo":"https://github.com/QudeLeap/Lean-QuantumAlg-Bench","thumb":"thumbs/2607.21533.jpg"},{"id":"2607.20816","title":"Cofinite Zeros of High Derivatives","authors":["Eric Hou"],"abstract":"We construct a nonzero transcendental entire function such that every nonempty open subset of the complex plane contains a zero of every sufficiently high derivative; equivalently, the union of the zero sets along every infinite increasing sequence of derivative orders is dense. The construction is probabilistic and uses a bounded-coefficient Fock series. A saddle estimate, a one-coordinate small-ball bound, and Jensen's formula give summable outer probabilities for zero-free disks. The resulting function satisfies the explicit growth bound $|f(z)|\\leq\\sqrt2\\exp(|z|^2)$ and therefore also supplies a counterexample to a 1973 theorem of Boas and Reddy as printed. A machine-checked Lean 4 formalization verifies the existence theorem, the growth bound, and their supporting lemmas.","abs_url":"https://arxiv.org/abs/2607.20816","pdf_url":"https://arxiv.org/pdf/2607.20816v1","primary_category":"math.CV","published":"2026-07-23","tags":["formalization","analysis"],"lean_note":"A Lean 4 formalization (Mathlib v4.28.0) verifies the existence theorem, the explicit growth bound, and their supporting lemmas.","repo":"https://github.com/erichou1/cofinite-derivative-zeros","thumb":"thumbs/2607.20816.jpg"},{"id":"2607.20418","title":"Pure-DP Statistical Query Release at the Conjectured Square-Root Rate","authors":["Jack Fitzsimons"],"abstract":"Nikolov and Ullman asked whether k statistical queries on a universe of size T can be released under pure differential privacy with expected worst-coordinate error at the square-root rate suggested by known lower bounds. We prove their conjectured upper bound. For every database size n and privacy parameter $\\varepsilon>0$, there is an $\\varepsilon$-differentially private mechanism with expected error $O(\\min\\{1,\\sqrt{\\log(2T)\\log(2k)/(\\varepsilon n)}\\})$. This matches the lower-bound dependence in the standard high-dimensional regimes where those bounds apply; the shifted logarithms and outer minimum make the upper bound valid without additional parameter assumptions. The construction starts from a selection-only private multiplicative weights transcript, then replaces its probability mass function by a distance-penalized likelihood envelope. To prove that the modification preserves accuracy, a likelihood-level Maurey argument upper-bounds each Hamming-ball maximum by a small family of auxiliary PMW laws. Renyi moment bounds control nearby balls, a direct mixture bound controls distant balls, and grouping radii at the privacy scale prevents an additional $1/\\varepsilon$ factor in the error. The mechanism is information-theoretic. A companion Lean 4 development machine-checks the finite construction, pure privacy after deterministic decoding, and the displayed all-regimes upper bound.","abs_url":"https://arxiv.org/abs/2607.20418","pdf_url":"https://arxiv.org/pdf/2607.20418v1","primary_category":"cs.DS","published":"2026-07-22","tags":["verification","cryptography"],"lean_note":"A Lean 4 development machine-checks the finite mechanism construction, its pure differential privacy after deterministic decoding, and the claimed error upper bound.","thumb":"thumbs/2607.20418.jpg"},{"id":"2607.20186","title":"From Dag-Like Proofs to Boolean Circuits in Lean","authors":["Lorenzo Saraiva","Edward Hermann Haeusler"],"abstract":"In this article, we present a method for encoding Dag-Like Derivability Structures (DLDS), obtained via horizontal compression of Natural Deduction proofs in purely implicational minimal logic, as Boolean circuits. These DLDS compress Natural Deduction tree-like proofs into directed acyclic graphs, preserving logical correctness while reducing redundancy. We formally define the circuit construction process and establish its pointwise correctness, showing that, for any fixed path assignment, the resulting Boolean circuit agrees with the intended dependency-propagation semantics. A Lean formalization establishes machine-checked guarantees for the circuit evaluator and includes a restricted bridge for the uncompressed simple-tree fragment, connecting valid DLDS instances in that fragment to genuine circuit acceptance of their extracted paths under the route and discharge conditions formalized in Lean. This approach opens new perspectives for automated theorem proving and formal certification.","abs_url":"https://arxiv.org/abs/2607.20186","pdf_url":"https://arxiv.org/pdf/2607.20186v1","primary_category":"cs.LO","published":"2026-07-22","tags":["formalization","verification","logic"],"lean_note":"Formalizes in Lean a Boolean circuit evaluator for Dag-Like Derivability Structures, proving pointwise correctness and a simple-tree fragment bridge.","thumb":"thumbs/2607.20186.jpg"},{"id":"2607.19727","title":"Foundations of Machine-Checked Control Theory in Lean","authors":["Moritz Doll","Iman Shames"],"abstract":"We introduce an open-source library for machine-checked control theory in the interactive proof assistant Lean to lay foundations for the verification of cyber-physical systems. To this end, as representative theorems, we present formalizations of Lyapunov stability theory and the small-gain theorem. First, the machinery employed for formalizing Lyapunov stability, i.e., neighborhood filters, allows stating a Lyapunov theorem that covers both points and sets and applies to continuous, discrete, and hybrid systems. Second, the small-gain theorem is proved via stating input-output systems as relations without the usual well-posedness assumption. The Lean formalization of each of these theorems is then presented. We conclude by discussing the library architecture and mentioning some of the other system theoretic results that are formalized in the library along with future plans.","abs_url":"https://arxiv.org/abs/2607.19727","pdf_url":"https://arxiv.org/pdf/2607.19727v1","primary_category":"math.OC","published":"2026-07-22","tags":["formalization","verification","analysis"],"lean_note":"An open-source Lean/Mathlib library formalizes control theory, including Lyapunov stability and the small-gain theorem for cyber-physical system verification.","thumb":"thumbs/2607.19727.jpg"},{"id":"2607.19263","title":"The Limits of Quantum Computers for Power Flow","authors":["Cameron Khanpour","Samuel Talkington"],"abstract":"This letter proves realistic grid properties limit the applicability of quantum computers for power flow. Grids that split into two large regions meeting at only a few buses, common in transmission networks, force the pseudo condition number of the DC susceptance matrix to grow polynomially in the network size, and long chains of lines bridging such regions force quadratic growth, making recent empirical observations rigorous. The bounds also hold with overwhelming probability for arbitrary bounded random line susceptances. Combined with query and tomography lower bounds, this precludes end-to-end quantum advantage for DC power flow at every readout level, and these obstructions persist through AC power flow, optimal power flow, and unit commitment. All proofs are formally verified with accompanying Lean 4 source code.","abs_url":"https://arxiv.org/abs/2607.19263","pdf_url":"https://arxiv.org/pdf/2607.19263v1","primary_category":"quant-ph","published":"2026-07-21","tags":["formalization","verification","quantum"],"lean_note":"All numbered theorems and lemmas bounding the pseudo condition number of DC power flow matrices are formally verified in Lean 4.","repo":"https://github.com/eigenergy/quantum-power-flow-limits","thumb":"thumbs/2607.19263.jpg"},{"id":"2607.18921","title":"Circuit Claims Depend on What Is Extracted and How It Is Compared","authors":["Yang Sheng","Jie Fu"],"abstract":"Circuit extraction identifies a small set of model components whose presence preserves a target behavior under ablation, and the resulting circuit is often read as the mechanism behind that behavior. We argue that this reading is under-determined: preserving behavior does not single out one circuit, because the claim it supports depends on which circuit is reported and how two circuits are compared. We make this concrete in a synthetic Lean tactic-prediction benchmark -- predicting the next step of a proof -- where fixed proof rules with randomized surface form let differences between extracted circuits be attributed to these choices rather than to the task. Across dense and weight-sparse checkpoints (most weights constrained to zero) of the same transformer, evaluated on atomic (single-rule) and compositional (multi-rule) proofs, we vary which extracted object is reported (a compact prediction-preserving circuit, a broader graph that also keeps surrounding read, write, and routing structure, or the smallest subgraph meeting a post-ablation loss threshold), and whether each attention head's query and key are represented jointly or separately. Exact component-to-component edge overlap is low and sensitive to these choices, at times dropping to a random baseline, while two coarser summaries stay stable: the set of selected attention heads, and the circuit-size ranking of conditions that differ in which supervised checkpoint initializes reinforcement learning (RL). The largest accuracy gains from RL on compositional proofs come with the most structure beyond the atomic circuits. A circuit-level claim is therefore well defined only once one states which circuit is reported, the pruning threshold used to extract it, and the level at which circuits are compared. We distill these requirements into a reporting practice for circuit-extraction studies.","abs_url":"https://arxiv.org/abs/2607.18921","pdf_url":"https://arxiv.org/pdf/2607.18921v1","primary_category":"cs.LG","published":"2026-07-21","tags":["theorem-proving-ml","benchmark","ml-theory","logic"],"lean_note":"Uses a synthetic Lean tactic-prediction benchmark (predicting next proof tactic) to study circuit-extraction stability in transformers.","repo":"https://github.com/Stepuuu/circuit-extraction-stability","thumb":"thumbs/2607.18921.jpg"},{"id":"2607.18455","title":"Extended Abstract: From Pattern Unification Towards Pattern Matching Unification","authors":["David Richter","Timon Böhler"],"abstract":"We revisit the role of higher-order unification in dependently typed languages and identify a fundamental limitation of existing pattern-based fragments: their inability to synthesize functions defined by case analysis. Even simple and ubiquitous constraints arising from type inference, particularly from use of induction principles, fall outside the expressive power of Miller patterns and their modern extensions. We observe that such constraints naturally correspond to definitions by dependent pattern matching. Motivated by this correspondence, we propose integrating dependent pattern matching into the unification process. We present a prototype implementation of a small dependently typed language that collects delayed unification constraints and resolves them via a pattern matching compiler. Our approach successfully infers solutions that are rejected by current systems such as Rocq and Lean, suggesting a new direction for unification that unifies type inference and pattern matching compilation.","abs_url":"https://arxiv.org/abs/2607.18455","pdf_url":"https://arxiv.org/pdf/2607.18455v1","primary_category":"cs.PL","published":"2026-07-20","tags":["tooling","type-theory"],"lean_note":"A prototype dependently typed language integrating pattern matching into unification was implemented in under 3000 lines of Lean.","repo":"https://github.com/AndrasKovacs/elaboration-zoo","thumb":"thumbs/2607.18455.jpg"},{"id":"2607.18139","title":"1-out-of-5 Maximin-Share Allocations Always Exist for Four Agents","authors":["Christoph Schwerdtfeger"],"abstract":"For four agents with nonnegative additive valuations, a complete 1-out-of-5 maximin-share allocation always exists, improving the previous 1-out-of-6 guarantee. Together with known exact-MMS counterexamples, this completely characterizes the four-agent case: the guarantee holds exactly for $d\\geq5$. The main technical contribution is a balanced-residual partition lemma: removing rejected bundles with one of the four highest-ranked goods apiece leaves a remainder that still admits the required number of unit-valued balanced bundles. In its central $2+2$ case, three unit bundles repair two pairs of colliding high-valued goods. The theorem is machine-checked in Lean 4.","abs_url":"https://arxiv.org/abs/2607.18139","pdf_url":"https://arxiv.org/pdf/2607.18139v1","primary_category":"econ.TH","published":"2026-07-20","tags":["formalization","economics"],"lean_note":"The four-agent 1-out-of-5 maximin-share existence theorem is machine-checked in Lean 4 against Mathlib, sorry-free.","thumb":"thumbs/2607.18139.jpg"},{"id":"2607.17477","title":"On Some Problems from the Kourovka Notebook","authors":["Wouter van Doorn","Elias Judin","Pietro Monticone","Daniel Morrison"],"abstract":"The Kourovka Notebook is a long-running collection of open problems in group theory. In this paper we present solutions to eight of its problems. We construct a group with exactly two maximal locally soluble normal subgroups and show that, for every $1 \\le k\\le n!$, there is a group containing $n$ distinct elements whose $n!$ ordered products take exactly $k$ distinct values. We also give examples showing that group order together with the statistic $\\sum_g\\varphi(\\lvert g\\rvert)$ does not determine simplicity, and we construct a surjective non-injective Rota-Baxter operator on a non-abelian group. Further, we determine the group generated by the class transpositions of moduli at most $k$, prove that every power graph of a finite group that is a cograph is chordal, show that the right-relatively convex subgroups of a right-orderable group need not form a sublattice of its subgroup lattice, and disprove a proposed rank inequality for certain $p$-group extensions. All of these solutions were autonomously discovered and formally verified in Lean by Aristotle, a formal reasoning agent developed by Harmonic.","abs_url":"https://arxiv.org/abs/2607.17477","pdf_url":"https://arxiv.org/pdf/2607.17477v1","primary_category":"math.GR","published":"2026-07-20","tags":["formalization","ai-agents","algebra"],"lean_note":"Eight open Kourovka Notebook group-theory problems were autonomously solved and formally verified in Lean 4 by Harmonic's Aristotle agent.","repo":"https://github.com/pitmonticone/Kourovka","thumb":"thumbs/2607.17477.jpg"},{"id":"2607.17352","title":"Self-Modifying Lean Proof Agents with Verifier-Grounded Benchmark Coevolution","authors":["Yuqing Li","Zeguan Wu","Yu Gan","Junyu Liu"],"abstract":"Designing effective Lean proof agents is a central challenge in formal mathematical reasoning. Beyond building stronger provers, recent work emphasizes the workflow around Lean: how an agent decomposes proof obligations, uses tools and compiler feedback, diagnoses failures, repairs proofs, and maintains structured proof context. Motivated by code-level self-evolving agents, we study whether such workflows can be evolved rather than hand-designed. We present a self-evolving Lean proof agent in which a small fixed, trusted runtime wraps a fully mutable workspace: the proof workflow, prompts, and tools. Unlike most self-evolving systems, which optimize against a fixed external benchmark, our system coevolves the agent and its benchmark. Between generations, the highest-scoring agent (the champion) revises the active task distribution through a mastery-throttled curriculum update that introduces harder proof obligations only after the current level is mastered, and a single-anchor recalibration re-runs the champion on the updated benchmark to keep scores comparable as difficulty rises. All evolution stays inside a Lean-grounded verification loop: however the agent rewrites itself, a success counts only when its behavior yields Lean-verified proofs under a trusted snapshot, and each attempt must emit a machine-readable, Lean-grounded proof context whose representation may evolve but whose groundedness is enforced. We run the coevolving trajectory and a fixed-benchmark baseline for 15 active generations and compare them on a held-out miniF2F test split. The best coevolving agent reaches a 45.1% held-out solve rate, versus 12.7% for the seed and 32.0% for the best fixed-benchmark agent, showing that verifier-grounded self-evolution can improve Lean proof workflows under a coevolving benchmark.","abs_url":"https://arxiv.org/abs/2607.17352","pdf_url":"https://arxiv.org/pdf/2607.17352v1","primary_category":"cs.AI","published":"2026-07-19","tags":["ai-agents","theorem-proving-ml","benchmark"],"lean_note":"A self-evolving proof agent rewrites its workflow while a fixed Lean verifier grounds solve verdicts, evaluated on miniF2F and PutnamBench.","thumb":"thumbs/2607.17352.jpg"},{"id":"2607.16997","title":"PriorProof: A Point-in-Time Measure of Technique Novelty for Formal Proofs","authors":["Neel Somani"],"abstract":"Mathematicians distinguish proofs that explain, simplify, or introduce a nonstandard route, but these judgments are difficult to operationalize. We study a deliberately narrower construct: time-relative proof-route nonstandardness in formal mathematics. For a Lean theorem, PriorProof extracts the dependency footprint of its elaborated proof term and scores the weighted surprisal of that footprint under a retrieval-conditioned, hierarchically smoothed prior built only from an earlier quarterly snapshot of Mathlib. The method requires no hand-built technique ontology and no human labels: statement retrieval is learned from proof-derived contrastive pairs, while the scored object is read mechanically from proof terms. In a blinded topology study, 100 presentations collapse to 76 distinct underlying pairs: 12 canonical contrasts shown three times for consistency screening and 64 distinct stratified pairs. Against the majority of three retained domain raters, PriorProof agrees on 53/76 pairs (69.7%, Wilson 95% CI 58.7-78.9%), including 11/12 canonical pairs (91.7%, 64.6-98.5%) and 42/64 stratified pairs (65.6%, 53.4-76.1%). Score-gap quartiles are nonmonotone after repeat collapse; the endpoints are 12/19 (63.2%, 41.0-80.9%) in the smallest-gap bin and 16/19 (84.2%, 62.4-94.5%) in the largest, supporting an endpoint-calibration tendency rather than a resolved staircase. The best language-model condition agrees on 60/76 pairs (78.9%, 68.5-86.6%); on paired outcomes, PriorProof alone is correct on 8 pairs and the model alone on 15 (exact two-sided McNemar p = 0.210), so the difference is not established at this sample size. We therefore present PriorProof not as a replacement for expert or model judgment, but as a decomposable, time-anchored signal whose score gap provides an interpretable reliability indicator.","abs_url":"https://arxiv.org/abs/2607.16997","pdf_url":"https://arxiv.org/pdf/2607.16997v1","primary_category":"cs.AI","published":"2026-07-18","tags":["theorem-proving-ml","mathlib","topology","meta-study"],"lean_note":"Extracts dependency footprints from elaborated Lean 4 proof terms in Mathlib and scores proof-route nonstandardness against earlier quarterly snapshots.","repo":"https://github.com/neelsomani/priorproof","thumb":"thumbs/2607.16997.jpg"},{"id":"2607.16855","title":"Rtl2lean: Automated RTL-to-Lean Translation with Hierarchical Theorem Generation and Lemma Reuse","authors":["Hongqin Lyu","Junxing Dong","Yonghao Wang","Zhiteng Chao","Tiancheng Wang","Huawei Li"],"abstract":"Formal verification with interactive theorem provers can provide strong correctness guarantees for register transfer level designs, but applying it to existing SystemVerilog code requires substantial manual effort in semantic modeling and proof construction. This paper presents Rtl2lean, a framework that automatically translates RTL designs into executable Lean 4 models and builds a hierarchical theorem library for subsequent verification. The generated model represents hardware execution as a pure state transition function, while a four layer theorem framework captures combinational semantics, sequential updates, single cycle behavior, and reachability and invariants. When a high level property cannot be discharged by the existing theorem base, an LLM based proving loop proposes intermediate lemmas from the current proof context and Lean feedback. Only lemmas accepted by the Lean kernel are added to the reusable lemma pool. Experiments on six SystemVerilog designs generate 403 theorems, all of which are successfully checked by Lean. Among 358 foundational lemmas, 287 are available for automatic reuse, yielding a reusable lemma ratio of 80.2 percent. The results demonstrate that Rtl2lean can construct machine checked RTL proof libraries with low checking overhead and substantial cross property lemma reuse.","abs_url":"https://arxiv.org/abs/2607.16855","pdf_url":"https://arxiv.org/pdf/2607.16855v1","primary_category":"cs.AR","published":"2026-07-18","tags":["verification","tooling","autoformalization","theorem-proving-ml"],"lean_note":"Translates SystemVerilog RTL into executable Lean 4 state-transition models and builds hierarchical theorem libraries with LLM-assisted, kernel-checked lemma reuse.","thumb":"thumbs/2607.16855.jpg"},{"id":"2607.18313","title":"A Proof of Bala's Congruence Conjecture for A028342","authors":["Ahaan Kallat"],"abstract":"Let $a(n)$ be the sequence A028342 in the On-Line Encyclopedia of Integer Sequences (OEIS), defined by the exponential generating function $\\sum_{n\\ge0} a(n)x^n/n! = \\prod_{i\\ge1}(1-x^i)^{-1/i}$. Equivalently, $a(n)$ counts permutations of an $n$-element labeled set in which every cycle is assigned one divisor of its length, where a cycle of length $m$ has $d(m)$ choices, $d(m)$ being the number of positive divisors of $m$. We prove a family of congruences for $a$, conjectured by Peter Bala. They state that $k \\mid a(n+k)+a(n)$ for odd $k$, that $k \\mid a(n+k)-a(n)$ for $k\\equiv 0,2,6 \\pmod 8$, and that $k \\mid 2(a(n+k)-a(n))$ for $k\\equiv 4\\pmod 8$. The proof first establishes a product congruence $a(n+k)\\equiv a(n)a(k)\\pmod k$, and then computes $a(p^r)\\bmod p^r$ for each prime power by counting the colored permutations fixed by a subgroup of order $p$.","abs_url":"https://arxiv.org/abs/2607.18313","pdf_url":"https://arxiv.org/pdf/2607.18313v1","primary_category":"math.CO","published":"2026-07-17","tags":["formalization","number-theory","combinatorics"],"lean_note":"Theorem 1.1 and all supporting propositions and lemmas about congruences for OEIS sequence A028342 were formally verified in Lean 4 with Mathlib.","repo":"https://github.com/ahaankallat/bala-a028342-lean","thumb":"thumbs/2607.18313.jpg"},{"id":"2607.16533","title":"Oblivious Probabilistic Outcome Logic: Verifying Probabilistic Programs with an Oblivious Adversary","authors":["Hanxi Chen","Noam Zilberstein","Andrew C. Myers","Alexandra Silva"],"abstract":"In the context of probabilistic programs, an oblivious adversary resolves nondeterminism without seeing the outcomes of random draws. Obliviousness is a common assumption in online algorithms and distributed protocols, but the complex interaction between random draws and adversarial choices makes it challenging to reason about correctness. While there has been significant progress toward reasoning about programs that combine randomization with nondeterminism, most of the work has focused on the adaptive model, whose omniscient view of program state is too powerful to establish correctness for certain classes of programs. We introduce Oblivious Probabilistic Outcome Logic (opOL), a new logic for reasoning about probabilistic programs with nondeterminism controlled by an oblivious adversary. Building on Outcome Logic and Probabilistic Separation Logic, opOL models adversarial choice as a resource and uses probabilistic independence to ensure that random outcomes are hidden from the adversary. The opOL proof system provides expressive and compositional rules for case analysis on both random and nondeterministic outcomes, and for proving almost-sure termination. Expressivity is tested through several case studies, including a paging algorithm and a leader election protocol. The opOL metatheory and case studies are mechanized in Lean 4.","abs_url":"https://arxiv.org/abs/2607.16533","pdf_url":"https://arxiv.org/pdf/2607.16533v1","primary_category":"cs.PL","published":"2026-07-17","tags":["verification","logic"],"lean_note":"The opOL metatheory and case studies are mechanized in Lean 4.","thumb":"thumbs/2607.16533.jpg"},{"id":"2607.16171","title":"A Globally Asymptotically Stable Planar Homogeneous Polynomial Vector Field With No Polynomial Lyapunov Function","authors":["Jun Liu","Maxwell Fitzsimmons"],"abstract":"We disprove the conjecture that every globally asymptotically stable homogeneous polynomial vector field admits a homogeneous polynomial Lyapunov function. The counterexample is a planar homogeneous cubic polynomial vector field with integer coefficients. It admits no positive definite homogeneous polynomial with nonpositive Lie derivative and, more strongly, no real-analytic Lyapunov function even locally. Nevertheless, it has an explicit degree-two homogeneous Lyapunov function that is radially unbounded, continuously differentiable everywhere, and smooth away from the origin. We also provide a machine-checked Lean 4 formalization of the main result.","abs_url":"https://arxiv.org/abs/2607.16171","pdf_url":"https://arxiv.org/pdf/2607.16171v1","primary_category":"math.DS","published":"2026-07-17","tags":["formalization","analysis"],"lean_note":"Provides a machine-checked Lean 4 formalization of the main result disproving the homogeneous polynomial Lyapunov converse conjecture.","repo":"https://github.com/j49liu/homogeneous-lyapunov-counterexample-lean","thumb":"thumbs/2607.16171.jpg"},{"id":"2607.15174","title":"Mech: Mechanised Choreographic Programming","authors":["Xueying Qin","Marco Peressotti","Fabrizio Montesi"],"abstract":"Choreographic programming (CP) is a programming paradigm for the correct-by-construction development of concurrent and distributed systems: programmers write the intended overall behaviour of a system from a global perspective in a choreography, which is then automatically compiled into communicating endpoint programs by a procedure known as endpoint projection (EPP). The central promise is that the projected endpoint programs, when executed together, are behaviourally equivalent to the source choreography. Fulfilling this promise becomes delicate for expressive CP languages. Existing mechanisations of CP treat only restricted fragments, while textbook and general purpose language implementations with rich features leave crucial interactions informal. In particular, general branching in knowledge of choice, general recursion, and nondeterministic choice in choreographies have not yet been integrated in a machine-checked theory. We present Mech, a new mechanisation of CP in Lean 4 that captures these features. There are two central technical challenges in our development of Mech. First, the sketched semantics from the literature does not correctly capture how nondeterministic choice interacts with concurrency. We therefore formulate new semantics that align nondeterministic choreographic executions with the behaviours of projected endpoint programs. Second, managing all these features in proofs is complex. We address this by uncovering new algebraic laws for choreographies, the operators used in their semantics, EPP, and their combinations. Using our development, we prove completeness and soundness of EPP and derive communication safety and deadlock-freedom for projected networks, yielding the most extensive mechanised theory of CP to date.","abs_url":"https://arxiv.org/abs/2607.15174","pdf_url":"https://arxiv.org/pdf/2607.15174v1","primary_category":"cs.PL","published":"2026-07-16","tags":["verification","formalization","logic"],"lean_note":"Mechanises choreographic programming in Lean 4, proving soundness and completeness of endpoint projection with general recursion and nondeterminism.","thumb":"thumbs/2607.15174.jpg"},{"id":"2607.14699","title":"Exact Computation of Non-Gaussian Mismatch Penalties in Wiener-Hermite Cross-Correlation Identification","authors":["Serhii Zabolotnii"],"abstract":"Wiener-Hermite cross-correlation identification represents a polynomial response in the Hermite basis. Under Gaussian excitation the basis is orthogonal and a diagonal rule recovers it exactly; under non-Gaussian excitation the same basis is kept, but its Gram matrix gains off-diagonal terms and the diagonal rule is no longer the population projection. We give the exact finite-order excess $L^2(P)$ risk of this mismatch: a moment quadratic form from two Hankel-Cholesky factorizations and one diagonal solve, at $O(s^3)$ cost from moments to order $2s$. Closed cumulant forms at orders three and four expose which non-Gaussian features drive it; symmetry protects the Gaussian basis only through order two. A bootstrap decides, from data, whether a matched basis is worth building; on a Wiener-Hammerstein benchmark it separates a near-Gaussian channel (penalty $\\approx 10^{-4}$) from a skewed output (penalty $0.05$). The computation is a weighted-$L^2$ projection whose core normal-system correspondence is machine-checked in Lean 4.","abs_url":"https://arxiv.org/abs/2607.14699","pdf_url":"https://arxiv.org/pdf/2607.14699v1","primary_category":"stat.ME","published":"2026-07-16","tags":["verification","probability"],"lean_note":"The core normal-system correspondence underlying the weighted-L2 projection penalty formula is machine-checked in Lean 4.","repo":"https://github.com/SZabolotnii/Ku-Projection-Framework-code-supplement","thumb":"thumbs/2607.14699.jpg"},{"id":"2607.14582","title":"MathCoPilot: An Interactive System for Human-AI Symbiotic Paradigm of Mathematical Research","authors":["Junjie Zhang","Jiayu Liu","Wenbin Liu","Zhenya Huang","Doudou Wang","Yan Jiang","Leiye Xu","Tao Xiong","Wen Huang","Qi Liu","Guoping Hu","Enhong Chen","Mengping Zhang","Xiangdong Ye"],"abstract":"Existing LLM-based theorem provers have achieved impressive results on formal mathematics benchmarks, yet they remain confined to acting as autonomous agents that prove a stated proposition. In this paper, we propose MathCoPilot, a human-in-the-loop system that embodies a new human--AI symbiotic paradigm for mathematical research, in which the mathematician steers the high-level mathematical direction while AI agents carry out the detailed formalization and proof work under continuous human guidance. MathCoPilot unifies three core capabilities: (1) an interactive workbench where the mathematician and AI agents collaborate through a living proof blueprint that decomposes a proof into navigable steps the human can directly inspect, direct, and refine; (2) automated proving skill orchestration with adaptive knowledge base search and Lean-integrated iterative verification; and (3) topic-driven paper retrieval and automated formalization into a verified Lean knowledge base. Using MathCoPilot, we systematically compare four state-of-the-art LLMs, including Gemini~3.1~Pro, GPT-5.4, and Claude~Opus~4.7, on a FormalMATH subset and on two real PDE theorems requiring deep domain expertise, evaluating their ability to produce verified Lean~4 proofs and to identify errors in deliberately incorrect proofs. Our results show that while current models can handle undergraduate-level problems with high success rates under favorable autoformalization conditions, substantial challenges remain for domain-specific theorems requiring genuine mathematical understanding.","abs_url":"https://arxiv.org/abs/2607.14582","pdf_url":"https://arxiv.org/pdf/2607.14582v1","primary_category":"cs.AI","published":"2026-07-16","tags":["ai-agents","theorem-proving-ml","autoformalization","tooling"],"lean_note":"MathCoPilot is a human-in-the-loop system where AI agents generate and iteratively verify Lean 4 proofs against Mathlib under mathematician guidance.","thumb":"thumbs/2607.14582.jpg"},{"id":"2607.14420","title":"A Fast Quantitative Analyzer for NetKAT","authors":["Thomas Lu","Qiancheng Fu","Kevin Batz","Oliver Bøving","Tiago Ferreira","Mark Moeller","Nate Foster","Alexandra Silva"],"abstract":"When designing a network, engineers must navigate trade-offs (e.g., one topology offers more aggregate bandwidth, another lower latency or better resilience) that demand reasoning about quantitative properties. We present a fast analyzer for quantitative network properties based on weighted NetKAT (wNetKAT), a domain-specific language that provides a semantic foundation for quantitative reasoning by modeling network behavior using weights drawn from a semiring. At the core of our development is the design of a symbolic data structure -- weighted symbolic packet programs (wSPPs) -- that compactly represent the semantics of weighted policies, for which a direct implementation would be intractable. We show how to compute all policy constructs symbolically; unsurprisingly, the crux is Kleene star, for which we design a tailored algorithm. We further develop trace-carrying Pareto semirings, which compute multi-objective frontiers together with the network paths that realize them. We formalize the development in Lean and provide an optimized Rust implementation. Being parametric on a semiring, our implementation covers both classical and quantitative analyses: we show that it is competitive with KATch, a heavily optimized Boolean-reachability verifier, and orders of magnitude faster than McNetKAT and Storm on probabilistic analyses. A case study comparing Fat-tree and Jellyfish data-center topologies shows the framework supports multi-objective design-time analysis.","abs_url":"https://arxiv.org/abs/2607.14420","pdf_url":"https://arxiv.org/pdf/2607.14420v1","primary_category":"cs.PL","published":"2026-07-15","tags":["verification","formalization"],"lean_note":"The weighted NetKAT symbolic analysis framework, including weighted symbolic packet programs and semiring semantics, is formalized in Lean.","thumb":"thumbs/2607.14420.jpg"},{"id":"2607.14087","title":"Stochastic Domination of Gaussian Maxima: A Resolution of the Weak Simplex Conjecture","authors":["Abhijeet Mulgund"],"abstract":"We prove a stochastic comparison for Gaussian maxima. Let $R$ be an $m\\times m$ correlation matrix satisfying $R-\\mathbf{1} \\mathbf{1}^{\\mathsf T}/m\\succeq0$, let $X\\sim\\mathcal{N}(0,R)$, and let $Z_1,\\ldots,Z_m$ be independent standard Gaussian random variables. Then $\\max_{1\\leq i\\leq m}X_i \\leq_{\\mathrm{st}} \\max_{1\\leq i\\leq m}Z_i$, or equivalently, $\\mathbb{P}\\{X_i\\leq c\\text{ for every }i\\}\\geqΦ(c)^m$ for every $c\\in\\mathbb{R}$. This comparison resolves the Weak Simplex Conjecture: among $d+1$ equiprobable equal-energy signals in $\\mathbb{R}^d$ transmitted over an additive white Gaussian noise channel, the regular simplex maximizes the probability of correct maximum-likelihood decoding at every signal-to-noise ratio. It also proves the inequality asserted by the Simplex Mean Width Conjecture and gives an exact formula for the largest number of equiprobable messages that can be sent at prescribed energy and error probability by a deterministic no-feedback AWGN code under a per-codeword energy constraint. The proof combines a Gaussian product inequality for log-concave functions with an adaptive tilting argument that makes the inequality applicable to the one-sided threshold events defining the maximum. A lean formalization of this argument is available at https://github.com/abhmul/weak-simplex-conjecture-lean.","abs_url":"https://arxiv.org/abs/2607.14087","pdf_url":"https://arxiv.org/pdf/2607.14087v2","primary_category":"math.PR","published":"2026-07-15","tags":["formalization","probability"],"lean_note":"A Lean formalization of the adaptive tilting and Gaussian product inequality argument proving the stochastic domination is provided.","repo":"https://github.com/abhmul/weak-simplex-conjecture-lean","thumb":"thumbs/2607.14087.jpg"},{"id":"2607.14082","title":"Building Shor's Algorithm in Lean: An Agentic Formalization of Quantum Attacks on RSA-2048 and P-256","authors":["Lei Zhang","Yusheng Zhao","Hongshun Yao","Xin Wang"],"abstract":"Large language models are increasingly assisting with demanding formal theorem-proving tasks, particularly when grounded in machine-checked libraries such as Lean. Agentic systems further amplify this process by searching, reusing, and extending existing formal developments to uncover new discoveries. In quantum computing, Shor's algorithm and its variants present such a demanding case for Lean formalization. In this work, we formalize this algorithm family in Lean through agentic formalization: software agents analyze sources, write Lean code and repair proofs, with human review of the scientific claims and machine checking of the resulting formal proofs. Our formalization develops the mathematical foundations for analyzing quantum attacks in two cryptographic settings: a 2048-bit modulus in the RSA-2048 and the standardized elliptic curve over a 256-bit prime field (P-256). To support these analyses, the formalization ranges from quantum algorithms for order finding to reversible quantum circuits for modular and elliptic-curve arithmetic. Based on [Quantum 5, 433] and [ASIACRYPT 2017, 241--270], we formalize the logical resource estimates for RSA-2048 and P-256, respectively, and provide additional estimates of classical operations. We expect the results pave the way for broader machine-checked quantum cryptanalysis and represent a step toward AI-assisted design and verification of quantum algorithms.","abs_url":"https://arxiv.org/abs/2607.14082","pdf_url":"https://arxiv.org/pdf/2607.14082v1","primary_category":"quant-ph","published":"2026-07-15","tags":["formalization","ai-agents","quantum","cryptography"],"lean_note":"Formalizes Shor's algorithm family in Lean via agentic LLM systems, verifying quantum resource estimates for attacking RSA-2048 and P-256.","thumb":"thumbs/2607.14082.jpg"},{"id":"2607.13921","title":"Generative Compilation: On-the-Fly Compiler Feedback as AI Generates Code","authors":["Niels Mündler-Sasahara","Hristo Venev","Dawn Song","Martin Vechev","Jingxuan He"],"abstract":"Languages with rich static semantics, such as Rust, provide stronger guarantees for AI-generated code, but their strictness makes generation more difficult. Off-the-shelf compilers can provide useful feedback post-generation, but does not guide intermediate generation steps, such as those during autoregressive LLM decoding. Constrained decoding intervenes earlier by rejecting invalid tokens during sampling, but requires white-box model access and costly reimplementation for semantic constraints. We introduce generative compilation, the first approach to obtaining compiler feedback on partial programs during generation. The core technical device is a sealor: a lightweight, mostly syntax-guided transformation that converts partial programs into complete ones that standard compilers can diagnose. It is designed such that possible-to-complete partial programs are never rejected, while preserving enough code context to catch genuine dead ends early. We construct such a sealor on a core Rust-like calculus and prove that it satisfies these properties, all mechanized in Lean. We extend it to the first partial-program checker for real Rust. We evaluate our method on challenging repository-level Rust coding tasks, across both frontier black-box and open-weight models. We show that generative compilation reduces non-compiling outputs and improves functional correctness, relative to standard post-generation feedback. It does so by detecting a broad range of errors close to their source and early during generation, thereby reducing errors cascades and enabling focused diagnostics. More broadly, generative compilation is a step toward making compilers a first-class citizen of AI-assisted programming active during generation, rather than a separate post-generation check.","abs_url":"https://arxiv.org/abs/2607.13921","pdf_url":"https://arxiv.org/pdf/2607.13921v2","primary_category":"cs.PL","published":"2026-07-15","tags":["verification","formalization"],"lean_note":"A sealor transforming partial Rust-like programs into complete ones is defined on a core calculus and its correctness properties are mechanized in Lean.","thumb":"thumbs/2607.13921.jpg"},{"id":"2607.13662","title":"Definitional Inversion, Without Normalisation","authors":["Mario Carneiro","Thierry Coquand","Adrien Frabetti Mathieu","Meven Lennon-Bertrand","Paul-André Melliès","Stephanie Weirich"],"abstract":"We contribute a new proof technique, based on domain theory, to prove key meta-theoretic properties of dependent type systems: definitional inversion properties, i.e. injectivity and no-confusion of type constructors. This proof technique is independent of normalisation, and indeed applies even for the \"type-in-type\" rule of Martin-Löf's original type theory. Our proof is the first to establish injectivity of type constructors for such a system in the presence of $η$ laws. More generally, the technique is motivated by, and intended for, the metatheory of systems such as Idris, Lean, or dependent Haskell, whose underlying type theory is known to be non-normalising, as well as projects such as MetaRocq or Lean4Lean, where Gödel's second incompleteness theorem means we cannot show normalisation of the object logic in itself. We showcase the method on a small type theory, then explain how it extends to more ambitious extensions.","abs_url":"https://arxiv.org/abs/2607.13662","pdf_url":"https://arxiv.org/pdf/2607.13662v1","primary_category":"cs.LO","published":"2026-07-15","tags":["formalization","type-theory","logic"],"lean_note":"Meta-theoretic proof of definitional inversion for dependent type theory is mechanised in Lean (alongside Agda and Rocq), targeting the Lean4Lean project.","thumb":"thumbs/2607.13662.jpg"},{"id":"2607.13612","title":"The SIGReg Objective as Variational Free Energy: A Theoretical Active-Inference Account of JEPA World Models","authors":["Fabio Arnez","Alexandra Gomez-Villa"],"abstract":"Joint-Embedding Predictive Architectures (JEPAs) are the dominant design for latent world models, yet they are usually justified by empirical performance rather than a normative principle. We show that the choice of anti-collapse regulariser determines whether a JEPA's training objective, a prediction loss plus a weighted embedding regulariser, is a valid Active Inference (AIF) variational free energy. We organise four non-contrastive regularisers (VICReg, LogDet, PairDist, and SIGReg) into an entropy-estimator hierarchy indexed by a prior-miscalibration gap, and show that the gap's sign, whether the estimator bounds the latent entropy from above or below, decides whether the AIF surprise bound survives: VICReg and LogDet are unsafe upper bounds, PairDist a safe lower bound, and SIGReg eliminates the gap. We then prove a correspondence theorem: under the standard constant-noise encoder model and successful SIGReg enforcement (isotropic-Gaussian embeddings), the gap vanishes, the objective becomes an exact information bottleneck, the surprise bound is preserved, and the latent goal cost becomes an exact proxy for AIF pragmatic value, whereas VICReg leaves an irreducible second-order anisotropy term. We extend the correspondence to multi-step expected free energy, ensemble epistemic value, and a learned-policy regime, and we identify the one AIF term no current JEPA world model computes: the state-epistemic value, a future-state coverage signal. The predictions differ in kind, not degree, and are stated here as theoretical consequences left for empirical test in separate work; full proofs are in Appendix A, and the algebraic core of every result is machine-verified in Lean 4 (Appendix D).","abs_url":"https://arxiv.org/abs/2607.13612","pdf_url":"https://arxiv.org/pdf/2607.13612v1","primary_category":"cs.LG","published":"2026-07-15","tags":["verification","ml-theory"],"lean_note":"The algebraic core of every theoretical result (Gaussian bridge, gap decomposition, correspondence theorem) is machine-verified in Lean 4.","repo":"https://github.com/FabioArnez/sigreg-vfe-correspondence-lean","thumb":"thumbs/2607.13612.jpg"},{"id":"2607.13531","title":"Ripple: An Open, AI-Formalized Lean 4 Framework for Computing with CRNs","authors":["Ho-Lin Chen","Xiang Huang"],"abstract":"We present Ripple, an open, AI-formalized Lean 4 framework for the mathematics of computing real numbers with chemical reaction networks (CRNs). Ripple formalizes the full ladder of models -- the GPAC / CRN continuum and the CRN-computable reals, the large-population-protocol (LPP) compilation pipeline, and a continuous-time Markov chain (CTMC) layer bridged to the deterministic mean-field limit by three machine-checked versions of Kurtz's theorem, and two Turing-completeness results -- the Bournez-Graça-Pouly GPAC Turing-completeness construction and the Soloveichik-Cook-Winfree-Bruck stochastic-CRN universality theorem. The development is reliable (its core constructions are verified to depend on exactly the three Mathlib foundational axioms, with no sorry); it exposed genuine, fixable gaps in published proofs (the approximate-majority convergence argument and the LPP main theorem); and it proves new results -- a fully machine-checked construction of Apéry's constant ζ(3) as a CRN-computable number via its holonomic generating function, the same recipe turning the modular 1/π series of Ramanujan into a sharp open problem. The formalization was carried out predominantly by AI agents using only publicly available models, so the workflow is reproducible.","abs_url":"https://arxiv.org/abs/2607.13531","pdf_url":"https://arxiv.org/pdf/2607.13531v1","primary_category":"cs.LO","published":"2026-07-15","tags":["formalization","verification","ai-agents","number-theory"],"lean_note":"Formalizes in Lean 4 the mathematics of computing reals with chemical reaction networks, including CRN-computability, Kurtz's theorem, and two Turing-completeness results.","repo":"https://github.com/zinan-huang/Ripple","thumb":"thumbs/2607.13531.jpg"},{"id":"2607.13225","title":"Graph Puzzles III.1: A Proof of Sabidussi's Compatibility Conjecture","authors":["Nikolay Ulyanov"],"abstract":"We prove Sabidussi's compatibility conjecture. Let $G$ be a finite connected multigraph in which every vertex has even degree and the minimum degree is at least four, and let $T$ be a closed trail that traverses every edge exactly once. The edges of $G$ can be partitioned into circuits (connected 2-regular subgraphs) so that no circuit contains the two edges used consecutively anywhere in $T$. In fact, the edges can be four-coloured so that every such pair receives two different colours and the subgraph formed by the edges of each colour has even degree at every vertex. Formalization in Lean 4 is also available in the author's github.","abs_url":"https://arxiv.org/abs/2607.13225","pdf_url":"https://arxiv.org/pdf/2607.13225v1","primary_category":"math.CO","published":"2026-07-14","tags":["formalization","combinatorics"],"lean_note":"A proof of Sabidussi's compatibility conjecture is accompanied by a Lean 4 formalization available on the author's GitHub.","repo":"https://github.com/gexahedron/sabidussi-lean","thumb":"thumbs/2607.13225.jpg"},{"id":"2607.13165","title":"Interchange graphs of (0,1)-matrices are maximally Hamiltonian","authors":["Jeffrey S. Baggett","Huiya Yan"],"abstract":"For integer vectors R,S let A(R,S) denote the class of (0,1)-matrices with row sum vector R and column sum vector S. Its interchange graph G(R,S) has A(R,S) as its vertex set, two matrices being adjacent when they differ by a single 2 x 2 interchange. Brualdi conjectured that G(R,S) is Hamiltonian for every R,S. We prove the stronger statement that G(R,S) is maximally Hamiltonian: Hamilton-laceable when bipartite, and Hamilton-connected when not. The proof is a structural induction on the number of matrices in the class, organized by the structure theory of interchange graphs. Deleting inactive lines and splitting invariant positions expresses any class as a Cartesian product, reducing the argument to the prime factors. The bipartite classes are products of complete transposition graphs; we settle them together, without induction, by proving they are paired 2-disjoint-path-coverable and hence Hamilton-laceable, using a recent theorem of Coleman, Fischberg, Gong, Harrington and Wong on paired disjoint path covers. The non-bipartite classes divide into three cases: products assembled from smaller factors, a base of Johnson graphs and small classes, and the large prime classes, treated by a pivot-and-fiber construction whose line quotients are matroid base-exchange graphs. The complete argument has been machine-checked in the Lean 4 proof assistant from first principles together with seven cited results of the literature; the disjoint-path-cover results it imports are themselves proved within the formalization.","abs_url":"https://arxiv.org/abs/2607.13165","pdf_url":"https://arxiv.org/pdf/2607.13165v2","primary_category":"math.CO","published":"2026-07-14","tags":["formalization","combinatorics"],"lean_note":"Proves that interchange graphs of (0,1)-matrices are maximally Hamiltonian, with the complete argument machine-checked from first principles in Lean 4.","thumb":"thumbs/2607.13165.jpg"},{"id":"2607.13159","title":"Beyond Mock Modularity: Elliptic Corrections for Higher Dyson Ranks","authors":["Claudia Alfes","Ken Ono","Ashvin Swaminathan"],"abstract":"When $m = 1$, the Dyson rank generating function is a classical bridge between partition theory, Ramanujan's mock theta functions, and the theory of harmonic Maass forms and nonholomorphic Jacobi forms. The rank is a statistic on partitions, and the higher Dyson systems, for $m \\geq 2$, are a natural multivariable refinement of it, combining $m$ graded rank contributions. Unlike the classical case, these higher systems are not expected to fit the mock-modular framework, which raises the question of what analytic structure governs them. We show that their root-of-unity specializations carry a hidden elliptic structure. A finite $q$-difference recurrence produces an explicit polynomial obstruction to the expected index $m$ elliptic transformation law, and because the obstruction is finite, its partial fractions canonically determine finitely many Appell--Lerch correction terms that remove it. The corrected functions satisfy a twisted index $m$ elliptic law; a natural translation removes the twist, and their holomorphic finite parts admit finite theta decompositions. Thus, the natural analogue of Dyson's mock-modular phenomenon at higher $m$ is not mock modularity but a finite theta decomposition governed by an index $m$ elliptic transformation law. These results grew out of a human--AI collaboration, and the key new formulas were formalized and machine-verified in Lean/Mathlib by AxiomProver.","abs_url":"https://arxiv.org/abs/2607.13159","pdf_url":"https://arxiv.org/pdf/2607.13159v1","primary_category":"math.NT","published":"2026-07-14","tags":["formalization","number-theory","combinatorics"],"lean_note":"Key new q-series formulas for higher Dyson rank corrections were formalized and machine-verified in Lean/Mathlib by AxiomProver.","repo":"https://github.com/AxiomMath/HigherDyson","thumb":"thumbs/2607.13159.jpg"},{"id":"2607.12981","title":"An Agentic Formalization for Certified Quantum Neural Network Design","authors":["Mingrui Jing","Lei Zhang","Yusheng Zhao","Hongshun Yao","Xin Wang"],"abstract":"A central model in quantum machine learning is the quantum neural network (QNN), whose design requires balancing expressivity and trainability. Technically, expressivity is studied through circuit-function analysis, such as quantum signal processing, while trainability is analyzed using dynamical-Lie-algebra (DLA) methods. To support certified QNN design, we formalize these major components of QNN theory in a connected lean 4 development checked by a proof kernel, where every analytic input is either proved or exposed as a named hypothesis. On the expressivity side, we prove exact if-and-only-if characterizations of single-qubit QNNs, a resource-counted quantum phase processing theorem, and an overparameterization ceiling that bounds the quantum Fisher information rank by the DLA dimension. On the trainability side, we derive the direct-sum loss-variance law through a de-circularized second-moment interface. A parameterized Casimir-uniqueness engine discharges the required inputs for fully controllable, orthogonal, and matchgate circuit families, while single-qubit and product-Clifford ensembles close the two-design assumptions directly. A capstone theorem pairs the conditional variance law with exact loss reconstruction in DLA coordinates. The development record identifies eight corrections and clarifications that were not explicit in the informal arguments. We expect this work to provide a machine-checkable foundation for QNN theory and a step toward AI-assisted or automated design of quantum machine learning algorithms.","abs_url":"https://arxiv.org/abs/2607.12981","pdf_url":"https://arxiv.org/pdf/2607.12981v1","primary_category":"quant-ph","published":"2026-07-14","tags":["formalization","quantum","mathlib"],"lean_note":"Formalizes quantum neural network expressivity and trainability theory in Lean 4 with Mathlib, kernel-checked with tracked axioms.","repo":"https://github.com/QudeLeap/Lean-QuantumAlg","thumb":"thumbs/2607.12981.jpg"},{"id":"2607.12873","title":"Record compositions of alternating permutations and noncommutative symmetric functions","authors":["Evan Chen","Ken Ono","Michal Mogielnicki"],"abstract":"Amdeberhan, Shareshian, and Stanley recently proved that a function $\\varphi$ arising in the theory of partition Eisenstein series counts the alternating permutations of $\\{1,\\dots,2n\\}$ with a given `record' partition, and they asked whether there is a similar theory for record compositions, suggesting a role for noncommutative symmetric functions. Here we solve their open problem by showing that the number of alternating permutations of $\\{1,\\dots,2n\\}$ with record composition $(α_1,\\dots,α_\\ell)$ is \\[ \\prod_{j=1}^{\\ell}\\binom{2s_j-1}{2α_j-1}E_{2α_j-1}, \\] where $s_j=α_1+\\dots+α_j$, $E_k$ is an Euler number, and the record composition of $w=a_1a_2\\dots a_{2n}$ (so $a_1>a_2<a_3>\\dotsb$) lists the factor lengths obtained by cutting $a_1a_3\\dots a_{2n-1}$ before each left-to-right maximum other than the first. These numbers are the coefficients of a natural lift of the degree-$n$ sprout symmetric function with seed $\\sec(\\sqrt{t}\\,)$ to noncommutative symmetric functions, expanded in products of noncommutative power sums of the first kind. An analogous refinement holds for every sprout sequence whose seed is given by the exponential formula. AxiomProver autonomously produced and verified the results in this paper in Lean.","abs_url":"https://arxiv.org/abs/2607.12873","pdf_url":"https://arxiv.org/pdf/2607.12873v1","primary_category":"math.CO","published":"2026-07-14","tags":["formalization","ai-agents","combinatorics"],"lean_note":"AxiomProver autonomously formalized and verified the paper's three main combinatorial theorems in Lean 4.31.0.","repo":"https://github.com/AxiomMath/record-compositions","thumb":"thumbs/2607.12873.jpg"},{"id":"2607.12650","title":"Evidence-Grounded Verified Agentic Reasoning: A Path Toward Eliminating LLM Hallucination in Empirical Inference via Tool-Attested Kernel Proofs","authors":["Junyu Ren"],"abstract":"Tool access alone does not make LLM empirical reasoning governable: accepted outputs need not descend from attested evidence, and accepted deductions need not hold up under formal scrutiny. We present EG-VAR (Evidence-Grounded Verified Agentic Reasoning), a Lean 4-based tool-calling architecture in which the Lean kernel is the sole minter of Verified claims via tool-attestation axioms and declared source lifts. Every verified output structurally descends from an attested tool call (Thm. 3.1) and a kernel-checked chain of valid inference (Thm. 3.2); residual outputs are honest Abstain with a replayable audit trail. On a subcollection of TableBench numerical reasoning (n=120), EG-VAR attains 120/120 versus a 95% same-tool baseline; on counterfactual stress tests (5 domains x 2 models), EG-VAR stays 100% source-faithful while same-tool drops to 80-90% (no-tool 50-80%). With the LLM as deployment-time formalizer, residual semantic-formalization error is 3.3% on Sonnet and 1.7% on Opus. We position EG-VAR as a technical-governance interface for high-stakes empirical claims: a formal sidecar makes the target proposition, source scope, evidence boundary, proof obligation, and abstention condition auditable, eliminating unsupported Verified outputs today while turning formalization errors, lift and source-authority disputes, ambiguities, and abstentions into explicit audit targets. Over time, typed sidecars in datasets, APIs, public records, and AI-generated documents can amortize this formalization burden into reusable infrastructure.","abs_url":"https://arxiv.org/abs/2607.12650","pdf_url":"https://arxiv.org/pdf/2607.12650v1","primary_category":"cs.LG","published":"2026-07-14","tags":["verification","ai-agents","theorem-proving-ml"],"lean_note":"Uses the Lean 4 kernel as sole minter of Verified empirical claims, type-checking LLM-proposed proof tactics under tool-attestation axioms.","repo":"https://github.com/7pocheR/eg-var","thumb":"thumbs/2607.12650.jpg"},{"id":"2607.12226","title":"Foundational Constraint Solving for Expressive Refinement Typing","authors":["Jam Kabeer Ali Khan","Petros Markopoulos","Nico Lehmann","Ranjit Jhala"],"abstract":"SMT-based program verifiers are hamstrung by two problems: expressiveness, because predictable verification restricts to the boundaries of SMT decidability, and trust, because the solver is a large, unverified artifact whose soundness bugs may quietly compromise every tool built on it. We present FLEX, a foundational Constrained Horn Clause (CHC) solver implemented in LEAN, that reduces the trusted base to the kernel alone, and allows using LEAN's entire proof ecosystem to verify low-level systems code, via three contributions. First, FLEX encodes CHCs as plain LEAN propositions where the Horn variables are existentially bound predicates, and shows how to implement CHC solvers as tactics (meta-programs) that compute kernel checkable proofs of the CHC propositions. Second, we show how to implement two verified CHC generators in LEAN: a Floyd-Hoare style generator for an imperative language, and a refinement-type-based generator for a functional calculus, which can be composed with the solving tactics to yield the first end-to-end foundational CHC-based verifiers. Finally, we show how FLEX allows us to leapfrog the expressiveness limitations of SMT by unleashing LEAN's entire ecosystem of proof machinery to prove arbitrary functional correctness properties of various low-level Rust libraries using the FLUX refinement type checker, and demonstrate the viability of FLEX as a trustworthy CHC backend, by showing it automatically discharges 95.7% of the CHCs from FLUX's benchmark suite.","abs_url":"https://arxiv.org/abs/2607.12226","pdf_url":"https://arxiv.org/pdf/2607.12226v1","primary_category":"cs.PL","published":"2026-07-14","tags":["verification","tactic-automation","tooling"],"lean_note":"FLEX implements a foundational Constrained Horn Clause solver in Lean, encoding CHCs as propositions and discharging them via certifying tactics with kernel-checked proofs.","repo":"https://github.com/jam-khan/Flex","thumb":"thumbs/2607.12226.jpg"},{"id":"2607.11648","title":"Superlinear complexity of the $(3/2)^n$ steering word","authors":["Ralf Stephan"],"abstract":"Write $(3/2)^n = m_n + \\eps_n$ with $m_n$ the nearest integer and $\\eps_n\\in[-\\tfrac12,\\tfrac12)$, and let $T=(t_n)$, $t_n=2m_{n+1}-3m_n$, be the resulting \\emph{steering word}: the step-by-step record of the map $x\\mapsto\\tfrac32 x$ on the orbit of $1$, coded by nearest-integer rounding. Using results by Corvaja--Zannier and Nair--Kumar--Rout we prove that the subword complexity $\\pT(k)$ of $T$ is superlinear, $\\pT(k)/k\\to\\infty$. The argument is completely formalized in Lean-4, depending only on the Subspace Theorem.","abs_url":"https://arxiv.org/abs/2607.11648","pdf_url":"https://arxiv.org/pdf/2607.11648v2","primary_category":"math.NT","published":"2026-07-13","tags":["formalization","number-theory"],"lean_note":"The proof that the (3/2)^n steering word has superlinear subword complexity is fully formalized in Lean 4, depending only on the Subspace Theorem.","repo":"https://github.com/rwst/Superlinear-Complexity","thumb":"thumbs/2607.11648.jpg"},{"id":"2607.11376","title":"Correlated and uncorrelated long--time asymptotics of type D ASEP","authors":["Jeffrey Kuan"],"abstract":"The type D ASEP is an asymmetric two--species interacting particle system on $\\Z$, in which two separately conserved species hop, bind into a composite ``bound pair'', and split. The model, along with its reversible measures and orthogonal polynomial duality, was constructed using the representation theory of $U_q(\\so_{2n})$. The reversible measures and orthogonal polynomial duality are each a product of two copies of the single-species ASEP reversible measures and orthogonal polynomial duality. In this paper, we study the long-time asymptotics of the type D ASEP. In the fixed--$q$ regime, using an exact current--decoupling identity, we prove that the asymptotic hydrodynamic limit and Tracy--Widom fluctuations decouple, as predicted from the duality. In the weak--asymmetry (Edwards--Wilkinson) regime, when $q=1-c/N^2$, we prove that the two density fluctuation fields \\underline{decouple}: each converges to a linear stochastic heat equation, with no cross--coupling in either the drift or the noise, the limiting noises having vanishing cross--correlation. More surprisingly, we then prove that the two limiting normal random variables are \\underline{correlated} with a seemingly new correlation function. The correlation is exactly equal to $(1-e^{-4c})/(4c)$, with the positive parts of the normal random variables having correlations expressed by the Bessel--Struve function. This paper, with the exception of the abstract and introduction, was written entirely by Claude Opus 4.8 and Fable 5. The proofs were then formalized in Lean, using Aristotle by Harmonic AI. The human author of this paper verified the proofs manually.","abs_url":"https://arxiv.org/abs/2607.11376","pdf_url":"https://arxiv.org/pdf/2607.11376v1","primary_category":"math-ph","published":"2026-07-13","tags":["formalization","probability"],"lean_note":"Proofs of the type D ASEP long-time asymptotics were formalized in Lean using Aristotle by Harmonic AI.","thumb":"thumbs/2607.11376.jpg"},{"id":"2607.11258","title":"TreeThink: A Modular Tree Search Library for Mathematical Reasoning with LLMs","authors":["Burak S. Akbudak","Zeynel A. Uluşan","Can S. Erer","Gözde Gül Şahin"],"abstract":"Tree search algorithms enable systematic exploration of the proof space in neural theorem proving. Existing LLM tree search libraries primarily target natural language reasoning and do not provide native integration with formal verifiers, while theorem proving systems often rely on task-specific search implementations. We introduce TreeThink, an open-source Python library for modular, fully asynchronous tree search in neural theorem proving. It integrates established tree search methods with vLLM-based inference pipelines and diverse node evaluation techniques, ranging from lightweight heuristics to neural evaluators. We support Lean~4, Rocq, and Isabelle/HOL alongside natural language. It connects directly to each language's Read-Eval-Print Loop (REPL) server for real-time verification and proof state extraction. We evaluate TreeThink on miniF2F and MATH500, demonstrating cross-language formal proof search, natural language reasoning support, and up to 6.3$\\times$ wall-clock speedup from asynchronous execution. Source code is released under the MIT license at https://github.com/GGLAB-KU/treethink , and the library is accessible as a downloadable package at https://pypi.org/project/treethink/ .","abs_url":"https://arxiv.org/abs/2607.11258","pdf_url":"https://arxiv.org/pdf/2607.11258v1","primary_category":"cs.CL","published":"2026-07-13","tags":["tooling","theorem-proving-ml","benchmark"],"lean_note":"Modular tree-search library integrating Lean 4 via REPL for neural theorem proving, evaluated on miniF2F alongside Rocq and Isabelle.","repo":"https://github.com/GGLAB-KU/treethink","thumb":"thumbs/2607.11258.jpg"},{"id":"2607.10314","title":"Mechanised operational semantics of Rowhammer","authors":["Martin Berger","Amir Naseredini"],"abstract":"Rowhammer is a hardware vulnerability in dynamic random-access memory (DRAM) in which repeated accesses to aggressor rows can induce bit-flips in victim rows. This phenomenon violates a core assumption of conventional programming language semantics: reading or writing one memory location does not modify others. Despite the security importance of this phenomenon, there is no formal framework connecting Rowhammer faults with program behaviour. We present a probabilistic small-step operational semantics for an idealised imperative language subject to Rowhammer-style faults. The semantics abstracts from DRAM internals and semiconductor physics. A general probabilistic fault model parameterises the semantics, representing Rowhammer-style faults by assigning probabilities to bit-flips during read or write operations. The resulting distributions are propagated through programs using the standard monadic structure of probabilistic computation. As a case study, we formalise a well-known defence that places program variables sufficiently far apart in physical memory that an access to one variable cannot disturb another. We prove a distribution-independent semantic collapse theorem: for every finite execution, including prefixes of terminating and non-terminating executions, the protected projection of the probabilistic Rowhammer semantics is the Dirac distribution of the corresponding Rowhammer-free execution. We develop an observation-parametric account of secure information flow. Non-interference is expressed as a hyperproperty comparing the distributions of low observations from low-equivalent initial memories. Consequently, physical separation preserves non-interference for every admissible fault model, while every Rowhammer non-interference violation reflects a violation already present in the Rowhammer-free semantics. The development is fully mechanised in Lean using mathlib.","abs_url":"https://arxiv.org/abs/2607.10314","pdf_url":"https://arxiv.org/pdf/2607.10314v1","primary_category":"cs.PL","published":"2026-07-11","tags":["formalization","verification","logic"],"lean_note":"Fully mechanises a probabilistic small-step operational semantics of Rowhammer faults and information-flow security in Lean using Mathlib.","thumb":"thumbs/2607.10314.jpg"},{"id":"2607.10216","title":"Formalizing Abstract Simplicial Complexes & Stellar Subdivisions in Lean","authors":["Garett Cunningham","Daniel Zach","Stefan Friedl"],"abstract":"The theory of simplicial complexes is a cornerstone of topology, offering a sophisticated tool for computing invariants. We present a formalization of abstract simplicial complexes and stellar subdivisions in the Lean proof assistant. We adopt a purely combinatorial framework in order to provide a cohesive foundation for studying the theory of stellar subdivisions as seen in many contexts of combinatorial topology. In particular, we provide formalizations of morphisms between abstract simplicial complexes; several crucial constructions and operations on complexes, such as links and joins; and perform a comprehensive study of how stellar subdivisions interact with these operations. We state and prove a number of identities commonly used in the study of triangulated manifolds, such as deriving equivalences between links in an abstract simplicial complex $K$ and in a stellar subdivision $σ_s K$, including results with no references in the standard literature. To our knowledge, this is the first formalization of stellar subdivisions in any proof assistant.","abs_url":"https://arxiv.org/abs/2607.10216","pdf_url":"https://arxiv.org/pdf/2607.10216v1","primary_category":"cs.LO","published":"2026-07-11","tags":["formalization","topology","combinatorics"],"lean_note":"Formalizes abstract simplicial complexes, links, joins, and stellar subdivisions in Lean, proving identities used in combinatorial topology.","repo":"https://github.com/not-gary/pachner","thumb":"thumbs/2607.10216.jpg"},{"id":"2607.10084","title":"An Overlap Construction for Relative Linear Extension Ratios","authors":["Maseeh Ghodsi"],"abstract":"Chan and Pak introduced the relative linear extension ratio $ρ(P,x)=e(P)/e(P-x)$, where $e(P)$ is the number of linear extensions of a finite poset $P$, and let $ν(c,d)$ be the least number of elements of a poset that realizes $ρ(P,x)=d/c$. They proved that $ν(c,d)\\le d/c+O(\\log d\\log\\log d)$ for $d\\ge 3c$, and asked whether the hypothesis $d\\ge 3c$ can be relaxed to $d\\ge(1+\\varepsilon)c$ or removed. We prove the fixed-gap form of this question: for every fixed $\\varepsilon>0$, $ν(c,d)\\le \\frac{d}{c}+O_{\\varepsilon}(\\log d\\log\\log d)$ whenever $d\\ge(1+\\varepsilon)c$, and the implied constant is absolute once $d\\ge 2c$. The new ingredient is a one-element overlap construction: if $x$ is minimal in $P$ and $y$ is minimal in $Q$, then there is a poset $R$ with $|R|=|P|+|Q|-1$ and an element $z$ such that $ρ(R,z)=ρ(P,x)+ρ(Q,y)-1$. Together with the continued-fraction construction of Chan and Pak and Rukavishnikova's tail bound for sums of partial quotients, this removes the factor $3$ in their range. We also show that the fixed-gap hypothesis is essentially optimal for this construction. In the range $1 < d/c < 2$, with $h=d-c$, the size bound the construction can certify is at least $\\lfloor c/h\\rfloor$, so the method reaches the stated error term only when $h$ is at least of order $c/(\\log c\\log\\log c)$. The remaining obstruction to removing the hypothesis is a short-interval problem for sums of partial quotients, which we describe. The deductive part of the argument has been checked with the Lean proof assistant.","abs_url":"https://arxiv.org/abs/2607.10084","pdf_url":"https://arxiv.org/pdf/2607.10084v1","primary_category":"math.CO","published":"2026-07-11","tags":["formalization","combinatorics"],"lean_note":"The deductive core—continuant floor bound, residue-counting steps, and case analysis of the main theorem—is machine-checked in Lean 4 using its standard library.","thumb":"thumbs/2607.10084.jpg"},{"id":"2607.14137","title":"Untrusted Authors, Trusted Answers: A Calculus of Fidelity-Graded Translations","authors":["Christoph Kirsch"],"abstract":"Verified translation has two well-studied extremes: prove the translator once (certified compilation), or validate each run of one translator (translation validation). Both treat a single translation in isolation. We study translations as a graph -- many source languages, several reasoning targets, multiple independently built routes -- where the honest answer to \"is this translation correct?\" differs from edge to edge. We present a calculus of fidelity-graded translations: pairs of languages close commuting squares that are checkable per program and compose by pasting; declared fidelity grades compose by weakest link, are re-established per run by inline checking, and are exceeded by agreement between independently derived routes; and an end-to-end theorem isolates a fundamental asymmetry -- witness-carrying answers are self-certifying at the source, while universal answers are where grades, branches, and certificates earn their cost. The compositional core is mechanized in Lean 4. The calculus is implemented in hurdy-gurdy, a platform of 13 languages and 13 pairs around two reasoning hubs, built as a two-directional experiment in LLM-generated correctness: independent LLM agents wrote every pair, largely unsupervised, with the architecture's cross-checks as the only semantic gate, and the platform's intended player is itself an LLM. All code, and most of this paper, is LLM-generated; the human contribution is the architecture. The same gate is the intended growth model: hurdy-gurdy scales in language support through pairs contributed by anyone -- with LLMs, with agents, or by hand -- admitted by architecture, not authorship. We report conjoined coverage, branch agreement, a compliance-derived benchmark with machine-derived ground truth, witness replay, certified unreachability, and escape-rate experiments for the gate.","abs_url":"https://arxiv.org/abs/2607.14137","pdf_url":"https://arxiv.org/pdf/2607.14137v1","primary_category":"cs.PL","published":"2026-07-10","tags":["formalization","verification","logic"],"lean_note":"The compositional core of the fidelity-graded translation calculus, including pasting and route telescope theorems, is mechanized in Lean 4 with an axiom audit.","repo":"https://github.com/cksystemsgroup/hurdy-gurdy","thumb":"thumbs/2607.14137.jpg"},{"id":"2607.09632","title":"Lean-QIT: Towards a Formal Infrastructure for Quantum Information Theory","authors":["Chengkai Zhu","Ziao Tang","Guocheng Zhen","Yimeng Cao","Yusheng Zhao","Ranyiliu Chen","Xuanqiang Zhao","Lei Zhang","Xin Wang"],"abstract":"Quantum information theory (QIT) characterizes the capabilities and fundamental limits of quantum information processing, underpinning quantum communication, computation, and error correction. Formalizing its coding theorems requires connecting finite-block protocols, analytic inequalities, and asymptotic limits within a unified machine-checked framework. Existing developments, however, lack a reusable operational layer that defines codes, error criteria, achievable rates, and capacities independently of their information-theoretic characterizations. In this work, we present LeanQIT, a Lean 4 library for finite-dimensional QIT. It provides composable, kernel-checked interfaces for quantum states and channels, source and channel codes, finite-block performance criteria, hypothesis testing, one-shot quantities, and asymptotic rate constructions. Using this infrastructure, we formalize Schumacher's quantum source-coding theorem, the Holevo--Schumacher--Westmoreland classical-capacity theorem, and the entanglement-assisted classical-capacity theorem together with its strong converse. By separating operational definitions from analytic characterizations and exposing reusable achievability, converse, and asymptotic components, Lean-QIT provides a machine-readable foundation for formal QIT and a compositional knowledge substrate for emerging AI-assisted formalization, automated proof search, and agentic reasoning in quantum information and computation.","abs_url":"https://arxiv.org/abs/2607.09632","pdf_url":"https://arxiv.org/pdf/2607.09632v1","primary_category":"quant-ph","published":"2026-07-10","tags":["formalization","verification","quantum"],"lean_note":"Presents Lean-QIT, a Lean 4 library formalizing finite-dimensional quantum information theory, including Schumacher, HSW, and entanglement-assisted capacity theorems.","repo":"https://github.com/QuAIR/Lean-QIT","thumb":"thumbs/2607.09632.jpg"},{"id":"2607.09383","title":"System Capybara: Tracking Capabilities for Separation and Freshness (Extended Version)","authors":["Yichen Xu","Oliver Bračevac","Cao Nguyen Pham","Yaoyu Zhao","Martin Odersky"],"abstract":"Substructural type systems give strong static control over aliasing. Examples include uniqueness, separation, and borrowing. How can such control be brought to established languages whose programming models rely on higher-order abstraction, unrestricted aliasing, and pervasive sharing? We study this problem in the context of Scala. We show how to retrofit these guarantees selectively instead of globally: ordinary code keeps Scala's usual aliasing discipline, while stronger guarantees can be enforced where they matter. Our starting point is Scala's capture checking, whose treatment of capabilities is inspired by the object-capability tradition: capabilities are ordinary values, and capture sets record, in a value's type, which capabilities the value may use. We develop System Capybara, which adds a selective alias-control layer to this mechanism. By tracking separation, consumption, freshness, and read-only access for capabilities, Capybara recovers key reasoning principles from substructural and ownership-based disciplines without global invariants. We give a type-preserving translation from the surface calculus Capybara to CoreCapybara, a core calculus extending System Capless, the earlier foundation for capture checking. The translation uses quantifiers for capture polymorphism and freshness, and constraint-indexed modal types for separation. We prove a semantic soundness result for the core calculus in Lean 4 and derive type safety, memory safety (no use-after-free or double-free), immutability of read-only computations, and data-race freedom for well-typed programs. Finally, we implement Scala 3's new separation checker, which brings higher-order separation reasoning about effects, capabilities, and resources to ordinary Scala, including fearless concurrency.","abs_url":"https://arxiv.org/abs/2607.09383","pdf_url":"https://arxiv.org/pdf/2607.09383v1","primary_category":"cs.PL","published":"2026-07-10","tags":["verification","type-theory","logic"],"lean_note":"The core calculus CoreCapybara's semantic soundness (type, memory safety, immutability) is mechanized in Lean 4.","thumb":"thumbs/2607.09383.jpg"},{"id":"2607.09217","title":"OpenProver: Agentic and Interactive Theorem Proving with Lean 4","authors":["Matěj Kripner","Milan Straka"],"abstract":"In this system paper, we present OpenProver, an open-source system for LLM-driven automated theorem proving (ATP) with integrated Lean 4 formal verification. OpenProver integrates a Planner-Worker-Verifier architecture inspired by recent ATP agentic systems such as Aletheia. A Planner agent maintains a compact Whiteboard scratchpad and an unbounded Repository of intermediate findings, and decomposes mathematical work into parallel Workers. OpenProver is fully open-source, offers reproducible evaluation through automatic formal verification of generated proofs, and provides an interactive terminal interface for human-guided proof search. In interactive mode, OpenProver allows the human operator to monitor and steer the proof search process, motivated by the established human-AI synergy in interactive code generation. To showcase the potential for quantitative ablation experiments enabled by automatic formal verification, we evaluate OpenProver on ProofNet and compare it with a simple baseline. OpenProver is publicly available at https://github.com/kripner/OpenProver.","abs_url":"https://arxiv.org/abs/2607.09217","pdf_url":"https://arxiv.org/pdf/2607.09217v1","primary_category":"cs.AI","published":"2026-07-10","tags":["ai-agents","theorem-proving-ml","tooling"],"lean_note":"OpenProver is an agentic LLM theorem-proving system integrating Lean 4 formal verification, evaluated on ProofNet formal theorems.","repo":"https://github.com/kripner/OpenProver","thumb":"thumbs/2607.09217.jpg"},{"id":"2607.09072","title":"Agentic Proof and Property-Based Testing via Property-Templates in Data-Intensive Computing","authors":["Seongmin Lee","Yaoxuan Wu","Miryung Kim"],"abstract":"As the cost of code generation becomes cheaper with AI, the new bottleneck in software engineering has shifted to intent specification and validation. Overcoming this durability crisis of AI-driven coding requires more than traditional fuzzing: each candidate property must be proven correct over a model and shown to hold on the real implementation, making formal proof and systematic property-based testing (PBT) complementary. However, validating properties this way at scale requires solving two subproblems: verifying candidate properties and operationalizing PBT without AI hallucination. We hypothesize that recurring property patterns, cast as property templates--abstract, parameterized forms with holes--address both at once. This paper investigates recurring property patterns in Apache Spark. In data-intensive scalable computing systems, correctness properties arise from the principles of data partition, computation decomposition, and dataflow computation. For instance, aggregation decomposition relates a global function executed on the entire dataset to a local function followed by a recombiner. We design an agentic, dual-track validation framework that uses property templates to formally verify correctness in the Lean 4 theorem prover and instantiate PBT templates as executable PySpark tests. Our evaluation shows that property templates increase agentic proof engineering success by up to 2.6x (1.6x on average) and reduce proof hallucinations by 59%. Template-guided PBT synthesis reduces intent misalignments from 22 to 1 and cuts synthesis cost by up to 5.7x (3.8x on average). Template-guided synthesis further exceeds a state-of-the-art Spark fuzzer and approaches unguided LLM-based PBT on code coverage. Finally, comparing the two tracks is informative: when a proof succeeds yet a PBT finds a counterexample, the mismatch identifies a gap between the formal model and implementation.","abs_url":"https://arxiv.org/abs/2607.09072","pdf_url":"https://arxiv.org/pdf/2607.09072v1","primary_category":"cs.SE","published":"2026-07-10","tags":["verification","ai-agents","theorem-proving-ml"],"lean_note":"Agentic framework uses parameterized Lean 4 proof templates to machine-check correctness properties of Apache Spark operations.","repo":"https://github.com/apache/datafusion-comet","thumb":"thumbs/2607.09072.jpg"},{"id":"2607.09014","title":"New bounds for double covers of the discrete box {0,1,2}^d","authors":["Patrick White"],"abstract":"A proper sub-box of $A=\\{0,1,2\\}^d$ is a product $S_1\\times\\dots\\times S_d$ with each $\\varnothing\\neq S_i\\subsetneq\\{0,1,2\\}$. A double cover is a finite multiset of proper sub-boxes covering every point of $A$ exactly twice; write $f(d)$ for the minimum size of a double cover. Leader, Milicevic and Tan asked whether $f(d)\\ge 2^d$ for all $d$ (Question 4.1 of the PatternBoost paper of Charton-Ellenberg-Wagner-Williamson), analogous to the Alon-Bohman-Holzman-Kleitman partition bound $2^d$. No better than the trivial volume bound was previously known, for any $d\\ge 2$. We prove the first nontrivial lower bounds. A modular refinement of the parity argument gives $f(d)\\ge 2^{d+1}/(d+1)$; a slicing argument gives $f(4)\\ge 19$, $f(5)\\ge 33$, both above $2^d$, resolving the question for $d=4,5$ -- the first cases beyond the trivially known $d\\le 3$. A finer \"line rigidity\" argument yields $f(6)\\ge 60$, breaking the profile-statistic barrier (capped at $57$, shown here). This is formally verified in Lean 4: $f(6)\\ge 60$ is machine-checked on the three standard Mathlib axioms alone. On the upper-bound side, a dimension-lifting construction $f(r+3)\\le 6\\cdot 2^r+3f(r)$ gives $f(6)\\le 81$ (improving the known $82$) and $f(d)\\le(\\tfrac65+o(1))2^d$ asymptotically; a refinement improves the constant to $\\tfrac87$. This makes partial progress on PatternBoost's problem of reducing their constant $1.28$, and refutes the closed-form guess $f(d)=5\\cdot 2^{d-2}+1$ from $d=7$ on. Together, $60\\le f(6)\\le 81$. Finally we isolate the construction-side obstruction -- an \"S+c=2^j+1\" phenomenon, every skeleton sitting exactly one box past the partition bound -- and show it is of a piece with the Leader-Milicevic-Tan question itself.","abs_url":"https://arxiv.org/abs/2607.09014","pdf_url":"https://arxiv.org/pdf/2607.09014v1","primary_category":"math.CO","published":"2026-07-10","tags":["formalization","combinatorics"],"lean_note":"The lower bound f(6)>=60 is machine-checked in Lean 4, relying only on the three standard Mathlib axioms.","thumb":"thumbs/2607.09014.jpg"},{"id":"2607.09793","title":"A counterexample to a subadditivity conjecture of Cohen for Sophie Germain cyclic numbers","authors":["Josué Alexander Ibarra"],"abstract":"An integer $n \\ge 1$ is cyclic if $\\gcd(n,\\varphi(n))=1$ (equivalently, if every group of order $n$ is cyclic), and Sophie Germain cyclic if both $n$ and $2n+1$ are cyclic. Let $C_σ(N)$ count the Sophie Germain cyclic integers in $[1,N]$. Cohen conjectured that $C_σ$ is subadditive, $C_σ(m+n) \\le C_σ(m)+C_σ(n)$ for all $1 \\le m \\le n$ (his Conjecture 66), having checked $m,n \\le 10^6$ without finding a counterexample. We give one: at $m=31$, $n=3928$, $C_σ(3959)=697 > 696 = C_σ(31)+C_σ(3928)$. The argument is short, and is verified by the Lean 4 kernel.","abs_url":"https://arxiv.org/abs/2607.09793","pdf_url":"https://arxiv.org/pdf/2607.09793v1","primary_category":"math.NT","published":"2026-07-09","tags":["formalization","number-theory"],"lean_note":"A counterexample to Cohen's subadditivity conjecture for Sophie Germain cyclic numbers is formalized and kernel-verified in Lean 4 over Mathlib.","thumb":"thumbs/2607.09793.jpg"},{"id":"2607.08986","title":"A Formalization of the Mean-Field Derivation of the Vlasov Equation: AI-Assisted Lean Formalization as a Strategy Game","authors":["Joseph K. Miller"],"abstract":"We formalize a research result in the Lean 4 proof assistant by having a mathematician direct an AI system, and frame the activity as a formalization game. The objective is to turn a LaTeX document into Lean. The game is won when the development compiles, contains no sorry, and a machine check shows the target theorems rest on Lean's foundational axioms alone. Reuse is a second check, by a definition we introduce: whether the development yields a self-contained layer of general mathematics the wider library could absorb. The case study is a complete, axiom-clean formalization of well-posedness for the nonlinear Vlasov equation via Dobrushin's mean-field route -- existence, uniqueness, the stability estimate and mean-field limit, and a short-window superposition principle (weak solutions are Lagrangian). The human's role was to direct, not to write proofs: to scope the definitions, steer the decompositions, and triage the library's gaps; the AI agent executed. The formalization certifies the proof of each statement as written; whether the written statement is the intended theorem stays the mathematician's judgment. The optimal-transport machinery that fell out of the build (in particular, properties of the Wasserstein-1 metric and the Kantorovich-Rubinstein duality theorem) separates into a self-contained layer that compiles against Mathlib alone: about a sixth of the development (49 of 299 declarations), behind a 22-declaration interface with no reverse dependency. The headline theorems ran in about a week, the full development in about a month. We report the quantitative claims as observations of one game, not as general laws. The game's rules name no particular system, so the methodological framing is meant to outlast the tools of any one run.","abs_url":"https://arxiv.org/abs/2607.08986","pdf_url":"https://arxiv.org/pdf/2607.08986v1","primary_category":"cs.AI","published":"2026-07-09","tags":["formalization","ai-agents","analysis","mathlib"],"lean_note":"Formalizes Dobrushin's mean-field derivation of the Vlasov equation in Lean 4 via an AI-directed workflow, exporting a Mathlib-absorbable optimal-transport layer.","repo":"https://github.com/Hydrodynamical/Vlasov_Meanfield_Formalization","thumb":"thumbs/2607.08986.jpg"},{"id":"2607.08692","title":"From Rules to Nash Equilibria: A Lean 4 Case Study in Game-Theoretic Analysis of a Competitive Trading Card Game","authors":["Arthur F. Ramos","Tulio Soria"],"abstract":"We present a metagame analysis of the competitive Pokemon Trading Card Game, machine-checked in Lean 4 over real tournament data. The headline game-theoretic results, including Nash equilibrium, replicator dynamics, and the matrix-level type-bridge computation, rely on native_decide, which trusts Lean's compiler rather than its kernel; the trust boundary is made explicit. The artifact spans approximately 31,900 lines, 87 files, and 2,627 theorems, of which roughly 200 directly verify empirical claims, with no sorry, admit, or custom axioms. Analyzing Trainer Hill data from January to February 2026 for events with at least 50 players, over 14 archetypes and their full pairwise matchup matrix, we prove a popularity paradox: the most played deck, Dragapult, with 15.5% metagame share, has only 46.7% expected win rate, while Grimmsnarl, with 5.1% share, achieves 52.7%. A machine-checked Nash equilibrium of the raw game assigns Dragapult 0% weight; exhaustive enumeration over all nonempty support subsets confirms a unique symmetric Nash equilibrium of the constant-sum symmetrization with seven-deck support. Against this equilibrium mix, Dragapult falls 40.4 permil below the game value. Single-step replicator dynamics indicate downward fitness pressure on Dragapult, upward pressure on Grimmsnarl, and strongest extinction pressure on Alakazam. A 10,000-iteration sensitivity analysis confirms qualitative stability, with core support decks appearing in more than 96% of resampled equilibria. The primary contribution is methodological: a reproducible case study showing how formal verification can turn qualitative metagame narratives into machine-checkable, re-runnable strategic science.","abs_url":"https://arxiv.org/abs/2607.08692","pdf_url":"https://arxiv.org/pdf/2607.08692v1","primary_category":"cs.GT","published":"2026-07-09","tags":["formalization","verification","economics"],"lean_note":"Game-theoretic metagame analysis of the Pokemon TCG formalized in Lean 4, using native_decide over exact rationals to verify Nash equilibria and replicator dynamics.","thumb":"thumbs/2607.08692.jpg"},{"id":"2607.08656","title":"Cantor measures with odd base do not admit Fourier frames","authors":["Jaume de Dios Pont","Lukas Liehr","Mitchell A. Taylor"],"abstract":"We prove that the Cantor measure with base $b$ does not admit a Fourier frame whenever $b > 1$ is an odd integer. In particular, this answers a question of Strichartz on the existence of a Fourier frame for the middle third Cantor measure. A formalization of our main result in Lean 4 is also provided.","abs_url":"https://arxiv.org/abs/2607.08656","pdf_url":"https://arxiv.org/pdf/2607.08656v1","primary_category":"math.FA","published":"2026-07-09","tags":["formalization","analysis"],"lean_note":"The main theorem that odd-base Cantor measures admit no Fourier frame is formalized and verified in Lean 4 using Mathlib.","repo":"https://github.com/jaumededios/Cantor_Measure_Frames","thumb":"thumbs/2607.08656.jpg"},{"id":"2607.08366","title":"Minimum modulus for the unique multiset-sum problem","authors":["José A. R. Fonollosa"],"abstract":"Fix n >= 2. A set A = {a_0 < a_1 < ... < a_{n-1}} of n residues in Z_N is \"valid mod N\" if the all-ones multiset is the only size-n multiset drawn from A whose sum is p := sum_i a_i (mod N). For the super-increasing set A = {2^k - 1 : 0 <= k <= n-1} we determine the least valid modulus exactly: N_min(n) = 2^n - 2^floor(log_2 n) for all n >= 2. Both directions of the proof are elementary, resting on a sharp minimal-digit-sum estimate for representations by binary coins, and the full theorem has been machine-checked in Lean 4/Mathlib for all n (https://github.com/jarfo/min-modulus). We conjecture that no size-n residue set admits a smaller valid modulus. This validity condition is exactly what makes the permanent of an n x n matrix equal to a single coefficient of a row-product polynomial modulo x^N - 1, extractable by a size-N discrete Fourier (or number-theoretic) transform; the theorem thus identifies the smallest transform, N ~ 2^n, for which this evaluation is exact. That application -- and the resulting common framework for the classical formulas of Ryser and Glynn and this transform -- is developed in a companion paper (arXiv:2607.09949).","abs_url":"https://arxiv.org/abs/2607.08366","pdf_url":"https://arxiv.org/pdf/2607.08366v2","primary_category":"math.CO","published":"2026-07-09","tags":["formalization","combinatorics","number-theory"],"lean_note":"The full Main Theorem determining the least valid modulus for the unique multiset-sum problem is formalized and kernel-checked in Lean 4/Mathlib.","repo":"https://github.com/jarfo/min-modulus","thumb":"thumbs/2607.08366.jpg"},{"id":"2607.07857","title":"Multi-agent Autoformalization of Tensor Network Theory","authors":["Sirui Lu","Erickson Tjoa","J. Ignacio Cirac"],"abstract":"We build a team of specialized large language-model agents and present an agent-driven workflow for research-level formalization in theoretical physics, with the autoformalization of the fundamental theorem of matrix-product states as a demonstration. The agents, coordinated through a structured mathematical blueprint and periodic human review, orchestrated and executed the full formalization autonomously. For some statements, the agents were able to explore new proof routes that are not part of the standard literature. Along the way the agents produced extensive tensor-network and quantum-information libraries not previously available in Mathlib, Lean's mathematical library. As a physical application, the formalization also extends towards symmetry-protected topological phases in one dimension. We find that the main bottleneck in large-scale autoformalization is enforcing mathematical intent and we provide a detailed study of the full process and various subtleties involved. We release the codebase as the library \\href{https://github.com/LionSR/TNLean}{TNLean}, together with a \\nChapters{}-chapter \\href{https://lionsr.github.io/TNLean/blueprint/}{blueprint} of the formalization effort.","abs_url":"https://arxiv.org/abs/2607.07857","pdf_url":"https://arxiv.org/pdf/2607.07857v1","primary_category":"quant-ph","published":"2026-07-08","tags":["autoformalization","ai-agents","formalization","quantum"],"lean_note":"Multi-agent LLM workflow autoformalizes the fundamental theorem of matrix-product states in Lean 4, producing new tensor-network and quantum-information Mathlib libraries.","repo":"https://github.com/LionSR/TNLean","thumb":"thumbs/2607.07857.jpg"},{"id":"2607.06944","title":"$p$-adic Hahn series with sparse support","authors":["Shanwen Wang","Yijun Yuan"],"abstract":"Let $p$ be a prime number. We introduce a sparseness condition on the supports of $p$-adic Hahn series, and prove that this condition implies transcendence over $\\breve{\\mathbf Q}_p$, the completed maximal unramified extension of $\\mathbf{Q}_p$. As an application, we prove the order-type conjecture of $\\mathbf{Q}_p$-algebraic $p$-adic Hahn series with bounded support under the condition that the support has only finitely many accumulation points. All results in this paper have been fully formalized in the Lean theorem prover (v 4.31.0), building over Mathlib.","abs_url":"https://arxiv.org/abs/2607.06944","pdf_url":"https://arxiv.org/pdf/2607.06944v1","primary_category":"math.NT","published":"2026-07-08","tags":["formalization","number-theory"],"lean_note":"Fully formalizes transcendence results for sparse-support p-adic Hahn series in Lean 4 over Mathlib.","thumb":"thumbs/2607.06944.jpg"},{"id":"2607.06693","title":"Stable Phase Retrieval for Spans of Independent Random Variables","authors":["Pedro Abdalla","Jaume de Dios Pont","João P. G. Ramos","Mitchell A. Taylor"],"abstract":"We prove that, after $L^2$ normalization, stable phase retrieval holds over the $L^2$-spans of independent real-valued centered random variables if and only if all but possibly one coordinate satisfies a uniform two-sided $L^1$ bound. This provides a complete characterization of stable phase retrieval for such subspaces, building upon the pioneering work of Calderbank--Daubechies--Freeman--Freeman and confirming the conjectured characterization communicated to us by those authors. We provide two different proofs of this fact, both based on a decomposition of the $\\ell^2$-coefficients of each random variable. The first is a compactness proof, which makes use of the infinite divisibility of limit laws of tail sums. The second is a quantitative proof, which substitutes the compactness step with an explicit dichotomy based on anticoncentration estimates of Sperner type. This latter proof was partially LLM generated based on the ideas in the first proof and a considerable amount of guidance by the authors. An autoformalization of our main result in Lean 4 is also provided, following the ideas in the quantitative proof.","abs_url":"https://arxiv.org/abs/2607.06693","pdf_url":"https://arxiv.org/pdf/2607.06693v1","primary_category":"math.FA","published":"2026-07-07","tags":["formalization","autoformalization","analysis","probability"],"lean_note":"The main stable phase retrieval characterization is autoformalized and machine-checked in Lean 4 using Mathlib, with the proof partly translated by LLMs.","repo":"https://github.com/jaumededios/RandomVarSPR","thumb":"thumbs/2607.06693.jpg"},{"id":"2607.06627","title":"On the polynomial values represented by quadratic forms","authors":["Bogdan Grechuk","Jamal Agbanwa"],"abstract":"Many Diophantine equations can be reduced to the question of whether, for a given non-degenerate quadratic form $F$ and a univariate polynomial $P$ with integer coefficients, $P(x)$ can be represented by $F$ for infinitely many values of $x$. We develop a method for answering this question for certain cubic and quartic polynomials $P$, as well as for certain polynomials of the form $P(x)=R(Q(x))$, where $R(t)$ and $Q(x)$ are polynomials of degree $3$ and $2$, respectively. Applying this method with $F(y,z)=y^2+z^2$, $R(t)=t^3-4$ and $Q(x)=x^2$, we conclude that $x^6-4$ is a sum of two squares infinitely often. In turn, this implies that the equation $y^2+x^3y+z^2+1=0$ has infinitely many integer solutions. Prior to this work, it was the shortest equation for which it was open whether its integer solution set is finite or infinite. We conclude with a list of the new shortest equations whose finiteness problem remains open. All main results of this paper has been formalized in Lean using Aristotle.","abs_url":"https://arxiv.org/abs/2607.06627","pdf_url":"https://arxiv.org/pdf/2607.06627v1","primary_category":"math.GM","published":"2026-07-07","tags":["formalization","number-theory"],"lean_note":"All main results, including that x^6-4 is a sum of two squares infinitely often, were formalized in Lean using Aristotle.","repo":"https://github.com/JAgbanwa/PolynomialDiophantineEquations","thumb":"thumbs/2607.06627.jpg"},{"id":"2607.06379","title":"Axioms for physical reasoning: codifying the Seiberg--Witten solution in Lean","authors":["Michael R. Douglas"],"abstract":"Mathematicians have embraced interactive theorem provers with growing enthusiasm -- building large shared libraries and machine-checking a string of landmark results. Theoretical physics is different: most of its results are not theorems but justified by arguments the community trusts without a rigorous proof. For many -- the one we treat here among them -- no rigorous proof is within reach. For 4d Yang--Mills theory, deriving exact rigorous results from first principles would first require constructing the interacting theory nonperturbatively, which is a sizable piece of one of the Clay Millennium prize problems. We argue here that an interactive theorem prover can be used to verify some non-rigorous physics arguments. The method is to postulate a short list of explicit, named physical postulates, which imply the physical results by virtue of a machine-checkable proof. The trust that remains then rests on that short, inspectable list, and the prover can report, for any downstream result, exactly which assumptions it used. We carry this out for the Seiberg--Witten solution of ${N}=2$ $SU(2)$ super-Yang--Mills -- the genus-one case -- formalized in Lean 4; the higher-genus $SU(N)$ generalization is developed in the same repository as an axiomatized skeleton and left to future work. We describe what is proved, what is assumed, how the assumptions are checked -- external review and an independent numerical oracle -- and why this discipline is a sound standard for validating AI-generated results in theoretical physics. What we offer is a discipline, reviewable on its own terms: a reader may take the Seiberg--Witten mathematics on trust and still assess the formalization method.","abs_url":"https://arxiv.org/abs/2607.06379","pdf_url":"https://arxiv.org/pdf/2607.06379v1","primary_category":"hep-th","published":"2026-07-07","tags":["formalization","verification","physics"],"lean_note":"Formalizes the Seiberg–Witten solution of N=2 SU(2) super-Yang–Mills in Lean 4 by isolating named physical postulates and machine-checking their consequences.","repo":"https://github.com/mrdouglasny/seiberg-witten","thumb":"thumbs/2607.06379.jpg"},{"id":"2607.06341","title":"Harnessing Code Agents for Automatic Software Verification","authors":["Shuangxiang Kan","Shuanglong Kan","Sebastian Ertel"],"abstract":"Formal verification offers the strongest guarantee of software correctness, but it does not scale: the proofs demanded by interactive theorem provers such as Coq require enormous expert effort. Large language models (LLMs) promise to generate these proofs automatically, yet existing approaches wire a fixed, human-designed proof strategy into the system and constrain the model to follow it (retrieving premises and predicting tactics one step at a time, or splitting goals by divide-and-conquer), and still prove only a fraction of their target theorems. We show that imposing such a strategy is unnecessary and limiting. Handing the whole lemma to a general LLM code agent (for example, Claude Code), free to choose its own approach, and wrapping it in a verification harness is both simpler and more effective, achieving full coverage: every targeted lemma proved, with no failures and no Coq expert intervention. The agent writes the proofs under feedback and hard constraints from the harness that keep each one sound (accepted only when the prover's kernel closes it), complete (no obligation left unproved or silently dropped), and terminating (no divergent tactics). We evaluate this harness plus code agent along three dimensions. (1) Core logic: on Iris, the state-of-the-art separation logic for concurrent and memory-manipulating programs, Aria proves all 4,257 lemmas of the four core modules and the 217 lemmas verifying Rust's standard libraries built on it, fully automatically. (2) Comparison with prior LLM provers: on reglang, where prior provers manage barely one in eight, Aria proves all 318. (3) Generality: on iris-lean, the unfinished Lean 4 port of Iris, it proves 72 not-yet-ported lemmas, showing the approach is not specific to Coq. A state-of-the-art model (Claude Opus 4.7) can write proofs for verified software development fully and automatically.","abs_url":"https://arxiv.org/abs/2607.06341","pdf_url":"https://arxiv.org/pdf/2607.06341v1","primary_category":"cs.FL","published":"2026-07-07","tags":["verification","ai-agents","logic"],"lean_note":"A code-agent verification harness (Aria) proves 72 not-yet-ported lemmas in iris-lean, the in-progress Lean 4 port of Iris, checked by the Lean kernel.","repo":"https://github.com/coq-community/reglang","thumb":"thumbs/2607.06341.jpg"},{"id":"2607.06089","title":"Closed-form fractional radial links for elliptical Mahalanobis discriminant analysis","authors":["Serhii Zabolotnii"],"abstract":"We study binary classification under shared-generator elliptical class-conditional distributions. The log-likelihood ratio is an additive function of the two squared Mahalanobis radii, with radial link $\\varphi=\\log g$; QDA is recovered only when this link is affine. We derive the Bayes radial-link family from the within-class radius law and estimate it by a finite fractional-power stochastic-polynomial projection instead of tuning a generic spline. The link is identifiable from the radius law, the plug-in estimator is $\\sqrt{n}$-consistent and asymptotically normal under finite-moment regularity conditions, and the induced classifier is asymptotically Bayes-optimal in an iterated sieve limit. The structural bridge, GAM membership, and identity-link/affine-generator dichotomy are verified in Lean 4 without unproven placeholders. Against the global Mahalanobis-GAM of Ghosh et al. (2025), reimplemented with mgcv REML splines at equal input budget, the derived link is never significantly worse on three UCI benchmarks and is decisively better on breast_cancer ($[+0.009,+0.021]$ global, $[+0.109,+0.136]$ global+local). Across six real financial series under temporal-dependence-robust validation, it is never significantly worse than the fitted GAM and is significantly better on three of five heavy-tailed series plus the light-tailed control. Relative to QDA, it improves the heaviest-tailed series (oil $[+0.024,+0.070]$, S&P 500 $[+0.038,+0.126]$, JPY/USD $[+0.009,+0.047]$) and ties elsewhere. A closed-form rate simulation corroborates the $\\sqrt{n}$ rate and the predicted excess-risk dichotomy between QDA's approximation-limited floor and the derived link's vanishing excess risk. The contribution is no significant loss relative to a tuned global GAM without spline smoothing-parameter selection, plus improved accuracy over QDA where generator curvature matters.","abs_url":"https://arxiv.org/abs/2607.06089","pdf_url":"https://arxiv.org/pdf/2607.06089v1","primary_category":"math.ST","published":"2026-07-07","tags":["formalization","verification","ml-theory"],"lean_note":"Lean 4 certifies the structural algebra: the elliptical log-likelihood-ratio radial-link bridge, GAM membership, and the affine-link collapse to QDA.","repo":"https://github.com/SZabolotnii/DSGE-MV_Elliptical_LLR-code-supplement","thumb":"thumbs/2607.06089.jpg"},{"id":"2607.05987","title":"Formalizing Scarf, Brouwer, and Nash in Lean","authors":["Yuwei Lyu","Kai Li"],"abstract":"We formalize in Lean 4 a complete combinatorial route from Scarf's theorem to Brouwer's fixed point theorem and to the existence of mixed Nash equilibria in finite games. The development follows Ivanov's indexed-order formulation of Scarf's theorem, formalizes the room--door incidence structure and parity argument, instantiates the theorem on finite grids of the standard simplex, and carries out the compactness and continuity argument needed to obtain a fixed point. We then extend the result to finite products of simplices by an explicit embedding--projection construction and use this product theorem to prove mixed Nash equilibrium existence via the Nash map. As a secondary by-product, we derive BrouwerBench, a preliminary 80-item Lean-grounded benchmark for probing proof-structure understanding within this single formal development.","abs_url":"https://arxiv.org/abs/2607.05987","pdf_url":"https://arxiv.org/pdf/2607.05987v1","primary_category":"cs.LO","published":"2026-07-07","tags":["formalization","benchmark","logic","economics"],"lean_note":"Formalizes in Lean 4 a combinatorial route from Scarf's theorem to Brouwer's fixed point theorem and mixed Nash equilibrium existence, using Mathlib.","repo":"https://github.com/math-xmum/Brouwer","thumb":"thumbs/2607.05987.jpg"},{"id":"2607.05874","title":"A Unified Framework for Formalizing Matrix Decomposition Proofs","authors":["Wanli Ma","Zichen Wang","Zaiwen Wen"],"abstract":"Existence proofs for many matrix decompositions share a recursive routine: a local transformation prepares the matrix, a slice is selected, a recursive solution is obtained, and the result is lifted and transported back. Formalizing this routine uniformly in dependent type theory is difficult because recursive subproblems may change index types, and reconstruction must preserve structural predicates across block embeddings and reindexings. We develop a Lean~4 framework that separates decomposition schemas, transformations, reduction strategies, measures, lifting, transport, and subtype induction. The framework uses general index types, packages square and rectangular matrices in universe types, and provides a decomposition driver that assembles strategy data into subtype-induction instances. It has been instantiated across PLU, LU, LDL/Cholesky, QR variants, Gauss rank normal form, Hessenberg reductions, Schur variants, normal spectral decomposition, SVD, bidiagonalization, tridiagonalization, UTV, Smith normal form, rational canonical form, and Jordan-type forms at varying levels of statement strength. Across these instances, repeated decomposition proofs are best treated not as separate tasks but as instances of a more general inductive statement whose interface records a certified proof path compatible with the chosen decomposition statement.","abs_url":"https://arxiv.org/abs/2607.05874","pdf_url":"https://arxiv.org/pdf/2607.05874v1","primary_category":"math.NA","published":"2026-07-07","tags":["formalization","algebra","mathlib"],"lean_note":"A Lean 4 framework abstracts the recursive proof pattern shared by many matrix decompositions into reusable schema, transport, and subtype-induction machinery.","repo":"https://github.com/wl-ma/MatDecompFormal","thumb":"thumbs/2607.05874.jpg"},{"id":"2607.05739","title":"Integer values of $\\tan(\\arctan 1+\\arctan 2+\\cdots+\\arctan n)$ are rare","authors":["Ken Ono"],"abstract":"For $n\\ge1$, we let $$x_n:=\\tan\\bigl(\\sum_{k=1}^{n}\\arctan k\\bigr).$$ In 2008, Amdeberhan, Medina, and Moll conjectured that $x_n\\not \\in \\mathbb{Z}$ for every $n\\ge5$. This was known for a set of positive integers of density $\\tfrac{120}{817}\\approx0.1469$. We prove that an integer value $x_n=m$ satisfies $|m|\\ge e^{(1/2+o(1))\\,n\\log n}$, which we use to deduce that $$\\#\\{\\,1\\leq n\\le N:x_n\\in\\mathbb{Z}\\,\\}=O(\\log N). $$ In particular, the conjecture holds for a density-one set of $n$. The results in this note were formalized in Lean/Mathlib and produced autonomously by AxiomProver from natural-language statements.","abs_url":"https://arxiv.org/abs/2607.05739","pdf_url":"https://arxiv.org/pdf/2607.05739v1","primary_category":"math.NT","published":"2026-07-07","tags":["formalization","autoformalization","number-theory"],"lean_note":"Number-theoretic results bounding integer values of arctangent sums were formalized in Lean/Mathlib and produced autonomously by AxiomProver from natural-language statements.","repo":"https://github.com/AxiomMath/TanArctan","thumb":"thumbs/2607.05739.jpg"},{"id":"2607.05492","title":"Lean-Quantum: Toward AI-Assisted Formalization of Quantum Information","authors":["Kazumi Kasaura","Kei Tsukamoto","Kento Mori","Risa Mizuno","Takahiro Namatame","Yuta Oriike","Masaya Taniguchi","Sho Sonoda","Hayata Yamasaki"],"abstract":"Quantum information theory is built on entropic quantities; among them, the sandwiched Rényi relative entropy is a fundamental divergence with various applications, and its data processing inequality (DPI) under quantum channels is a cornerstone result. In this work, we present a Lean 4 library for quantum information, designed as a reusable formal infrastructure for theoretical analysis. As a central demonstration of the library, we formalize the DPI for the sandwiched Rényi relative entropy for positive semidefinite operators on finite-dimensional quantum systems. The library provides a basis-independent operator-theoretic framework for finite-dimensional quantum mechanics compatible with the standard mathematical library Mathlib, including reusable interfaces for finite-dimensional systems, states, channels, tensor products, partial traces, Choi operators, Kraus representations, and Stinespring representations. It also builds infrastructure for noncommutative trace inequalities, including operator monotonicity and convexity via the real continuous functional calculus, block-operator positivity, Hilbert-Schmidt operator spaces, Jensen's operator inequality, generalized perspectives, operator power means, and Lieb-Ando trace inequalities. On top of this framework, we formalize entropy-specific ingredients for the DPI: variational formulas for the sandwiched quasi-entropy via Young and reverse-Young inequalities, tensor-product compatibility of real powers, and Haar measures on unitary groups. Together, these components yield a Lean formalization of the DPI, give strong subadditivity as a corollary, and provide the last missing component needed to complete the Lean formalization of the generalized quantum Stein's lemma. More broadly, the development provides machine-checkable foundations for future formalized and AI-assisted research in quantum information theory.","abs_url":"https://arxiv.org/abs/2607.05492","pdf_url":"https://arxiv.org/pdf/2607.05492v1","primary_category":"quant-ph","published":"2026-07-06","tags":["formalization","verification","quantum","analysis"],"lean_note":"Presents a Lean 4 library for finite-dimensional quantum information and formalizes the data processing inequality for the sandwiched Rényi relative entropy.","repo":"https://github.com/Hayata-Yamasaki-Group/lean-quantum","thumb":"thumbs/2607.05492.jpg"},{"id":"2607.04534","title":"Mechanism-level routing failure in LLMs over Lean-verified algebraic structures","authors":["Manuel Israel Cázares","Wenlin Zhang","Haobo Ma"],"abstract":"We present an empirical study of structural routing failure in large language models (LLMs) over a formally verified algebraic corpus. The task requires selecting the correct proof-mechanism label from a fixed closed template set for compact mathematical objects drawn from the FiberRing formalization in Lean 4, where each item is anchored to a Lean-verified artifact and assigned a label from the corresponding certificate family. Our central finding is a mechanism-level routing ceiling: under blind conditions, gpt-oss-120b achieves 80.3% template accuracy on 22 FiberRing items (n=66; temperature=0, seed=0), while Llama 3.3 70B reaches 68.2%. Exposing a mechanism-bearing Lean verdict/witness cue (Condition A2) raises accuracy to 90.9% and 81.8% -- gaps of +10.6 and +13.6 pp termed cue-induced routing uplift. The dominant failure is a CRT-to-ring-equivalence misroute: gpt-oss-120b misroutes 7 of 12 CRT items (58.3%) blind, zero under A2. A cross-model dissociation in Llama is notable: verdict accuracy is identical in both conditions (95.5%), while template accuracy improves 13.6 pp -- confirming that truth inference and proof-mechanism classification are separable capacities. A cross-corpus extension (Set B; 6 POM/CollisionKernel items, 72 evaluations) provides a small cross-module check: CRT-granularity compression reappears with different labels, and an inverse cross-model dissociation emerges. These findings extend the router hypothesis (Cazares 2026) to formal algebraic structures. The full pipeline, manifest, and results are at https://github.com/bytepro-ai/fiber-routing-eval.","abs_url":"https://arxiv.org/abs/2607.04534","pdf_url":"https://arxiv.org/pdf/2607.04534v1","primary_category":"cs.CL","published":"2026-07-05","tags":["theorem-proving-ml","benchmark","algebra"],"lean_note":"Evaluates LLM mechanism-classification routing over the FiberRing Lean 4 algebraic corpus, using Lean-verified artifacts as anchored ground-truth labels.","repo":"https://github.com/bytepro-ai/fiber-routing-eval","thumb":"thumbs/2607.04534.jpg"},{"id":"2607.04077","title":"Anomalous Partial Quotients in the Continued Fraction of $\\sqrt{ζ(3)-S_N}$","authors":["David Victor Feldman"],"abstract":"Let $S_N = \\sum_{j=1}^N j^{-3}$ and $R_N = ζ(3) - S_N$. The simple continued fraction of $\\sqrt{R_N}$ has partial quotients of generic size $O(N)$. We prove that at the sequence of indices $N_k = (Q_{2k+1}-1)/2$, where $Q_{2k+1}$ are companion Pell numbers, the continued fraction begins \\[ \\sqrt{R_{N_k}} = \\bigl[0;\\; M_k-1,\\; 1,\\; 6M_k^3+12M_k-2,\\; 1,\\; \\ldots\\,\\bigr], \\] with $M_k = P_{2k+1}$ (Pell numbers), and the third partial quotient grows cubically while generic ones are linear. We determine all partial quotients through the fifth: \\begin{align*} \\PQ_0 &= M_k - 1, & \\PQ_2 &= 6M_k^3 + 12M_k - 2, & \\PQ_4 &= \\Bigl\\lfloor\\frac{10M_k - 261}{261}\\Bigr\\rfloor, \\PQ_1 &= 1, & \\PQ_3 &= 1, & \\PQ_5 &= \\Bigl\\lfloor\\frac{261}{r_k}\\Bigr\\rfloor + ε_k, \\end{align*} where $r_k = (10M_k) \\bmod 261$ satisfies the recurrence $r_{k+1} \\equiv 6r_k - r_{k-1} \\pmod{261}$, and $ε_k = -1$ at the $k$ with $r_k \\mid 261$ (the two residue classes $k \\equiv 57, 62 \\pmod{60}$), and $ε_k = 0$ otherwise. All six formulas follow from the Euler--Maclaurin expansion of $1/\\sqrt{R_{N_k}}$, carried to sufficient precision, combined with the Pell identity $Q_{2k+1}^2 - 2M_k^2 = -1$. The delicate first step, $\\PQ_0 = M_k - 1$, is proved by rationalizing the irrational factor $\\sqrt{2}$ in the Euler--Maclaurin expansion; we complement this proof with a heuristic derivation via Gosper's bihomographic continued-fraction algorithm that exposes the underlying mechanism. All claimed results have been formalized in LEAN with the aid of Aristotle.","abs_url":"https://arxiv.org/abs/2607.04077","pdf_url":"https://arxiv.org/pdf/2607.04077v1","primary_category":"math.NT","published":"2026-07-05","tags":["formalization","number-theory"],"lean_note":"All claimed results on continued-fraction partial quotients of √(ζ(3)−S_N) at Pell indices were formalized in Lean with the aid of Aristotle.","thumb":"thumbs/2607.04077.jpg"},{"id":"2607.03963","title":"Neuro-Symbolic Reasoning for Vulnerability Detection","authors":["Yanjie Zhao","Hongjie Chen","Li Lu","Zhou Yang","Xiao Cheng","Haoyu Wang"],"abstract":"Ask a large language model (LLM) whether a pointer dereference is safe, and it can often produce a plausible justification for ``yes''. The difficulty is that a fluent justification is not a proof. This gap is precisely where automated vulnerability detection lives: deciding, for a given operation in source code, whether a memory safety defect such as a null dereference, use-after-free, or double free can actually occur. We trace the unreliability of LLM-based vulnerability detection to a mechanism, the premature discharge of safety obligations, and argue that the remedy is not better prompting but a separation of roles: the component that interprets the code must not also be the one that decides a safety obligation is met. In this paper, we present LeanGuard, a neuro-symbolic framework that assigns each act to the side equipped for it. On the neural side, an LLM serves strictly as a semantic filter over candidate facts extracted from the abstract syntax tree (AST): it prunes spurious facts and keeps the real ones, but never discharges an obligation or decides the verdict on its own. On the symbolic side, the surviving facts are compiled into a verification model in Lean 4 (a formal proof assistant whose kernel accepts a conclusion only when it is formally proved), where every dangerous operation must be matched by a guard that provably covers it in scope; absent such a guard, the obligation stays open rather than being argued away. Because a function rarely arrives with full context, this symbolic model is necessarily partial: an unproved obligation is not yet a defect. An evidence-aware adjudicator therefore weighs the symbolic and neural verdicts by the quality of each. We instantiate the framework on five CWE classes to ask how far this division of labor can be pushed.","abs_url":"https://arxiv.org/abs/2607.03963","pdf_url":"https://arxiv.org/pdf/2607.03963v1","primary_category":"cs.SE","published":"2026-07-04","tags":["verification","ai-agents"],"lean_note":"Extracted code facts are compiled into a Lean 4 verification model whose kernel checks whether safety obligations are provably discharged.","thumb":"thumbs/2607.03963.jpg"},{"id":"2607.03406","title":"LeanDY: Type-Based and Trace-Based Symbolic Protocol Verification in Lean","authors":["Simon Jeanteur","Lorenzo Veronese","Magdalena Solitro","Matteo Maffei"],"abstract":"Computer-aided formal verification is a widely used approach for the symbolic analysis of cryptographic protocols. However, many modern protocols rely on features that remain challenging for existing techniques. In particular, reasoning about state, time-dependent behavior, inductively defined data structures, unbounded executions, and conditional secrecy requires a level of expressiveness that is difficult to reconcile with effective automation. As a result, protocol verification has largely followed two disjoint paths: fully automated methods with limited expressiveness, or interactive proofs in general-purpose theorem provers that offer flexibility but only limited, non-specialized automation. We present an orthogonal approach that bridges this gap by combining compositional type-based reasoning with trace-based reasoning, enabling modular verification of stateful and unbounded protocols. Guided by the language-and-automation co-design (LAC) principle, our approach delivers protocol-specific automation while retaining high expressiveness. We implement this framework as the LeanDY library for the Lean proof assistant, building on and extending the design of DY*, and combining protocol-specific automation with interactive proofs. Our framework supports, in a unified setting, a broad class of functional and security requirements, including secrecy and authentication for stateful protocols, as well as recursive conditional secrecy for protocols using XOR. We formalize SegWit-style blockchain primitives in LeanDY and demonstrate its expressiveness by carrying out an in-depth formalization of payment channels on top of this blockchain model, verifying punishment mechanisms and properties that depend on chain liveness.","abs_url":"https://arxiv.org/abs/2607.03406","pdf_url":"https://arxiv.org/pdf/2607.03406v2","primary_category":"cs.CR","published":"2026-07-03","tags":["verification","cryptography","tooling"],"lean_note":"Presents LeanDY, a Lean library combining type-based and trace-based symbolic verification of cryptographic protocols, applied to blockchain payment channels.","repo":"https://github.com/SecPriv/leandy","thumb":"thumbs/2607.03406.jpg"},{"id":"2607.01734","title":"Reformalization of the Jordan Curve Theorem","authors":["Simon Guilloud","Sankalp Gambhir","Samuel Chassot"],"abstract":"We present a case study in reformalization, a variant of autoformalization in which the input proof is not natural language but a formal development in a different proof assistant. Concretely, we report three reformalizations of the Jordan Curve Theorem: from Mizar to Lean, from HOL Light to Lean, and from HOL Light to Agda. We analyse the results and identify pipeline design choices that matter for practical reformalization tasks.","abs_url":"https://arxiv.org/abs/2607.01734","pdf_url":"https://arxiv.org/pdf/2607.01734v1","primary_category":"cs.AI","published":"2026-07-02","tags":["autoformalization","ai-agents","formalization","topology"],"lean_note":"LLM-based agentic reformalization ports the Jordan Curve Theorem into Lean from Mizar and HOL Light sources, using Lean's checker as arbiter.","repo":"https://github.com/epfl-lara/jordan-curve-theorem","thumb":"thumbs/2607.01734.jpg"},{"id":"2607.01544","title":"Formalized $q$-series: The Rogers-Ramanujan Identities and Beyond","authors":["Kenny Lau","Seewoo Lee","Ken Ono"],"abstract":"The theory of $q$-series and basic hypergeometric series plays a crucial role at the intersection of combinatorics, number theory, and representation theory. From the classical partition identities of Euler and Jacobi to modern developments in class field theory, vertex operator algebras, and the Monstrous Moonshine conjecture, $q$-series provide the analytic framework for a wide range of profound applications. In this paper, we discuss the formalization of this theory in the Lean proof assistant, a process that requires careful design of scalable and versatile structures to reconcile formal algebraic identities with analytic convergence properties. We address these foundational challenges by focusing on the construction of $q$-Pochhammer symbols, $q$-binomial coefficients, Bailey's Lemma and similar primitives. To demonstrate the utility of this work, we provide fully verified proofs of the Jacobi Triple Product formula and the celebrated Rogers-Ramanujan identities, which serve as both historical and technical benchmarks for the field. This work establishes a rigorous computational foundation for the future formalization of mock theta functions, modular forms, and the diverse algebraic structures that underpin their applications across mathematics and physics.","abs_url":"https://arxiv.org/abs/2607.01544","pdf_url":"https://arxiv.org/pdf/2607.01544v1","primary_category":"math.NT","published":"2026-07-01","tags":["formalization","number-theory","combinatorics","mathlib"],"lean_note":"Formalizes q-series theory in Lean/Mathlib, verifying the Jacobi Triple Product, Bailey's Lemma, and the Rogers-Ramanujan identities.","repo":"https://github.com/wwylele/PentagonalNumberTheorem","thumb":"thumbs/2607.01544.jpg"},{"id":"2607.00815","title":"LRAT-Catcher: Importing SAT Solver Certificates into Lean4 by Reflection","authors":["Stefan Szeider"],"abstract":"SAT solvers settle combinatorial problems beyond the reach of interactive theorem provers and produce LRAT certificates for independent verification. We present LRAT-Catcher, a standalone, general-purpose tool that imports a DIMACS formula together with an LRAT certificate into Lean 4 as a theorem. LRAT-Catcher runs the formally verified LRAT checker from Lean core as compiled native code via reflection. This scales to instances where Mathlib's explicit proof-term import exhausts memory. LRAT-Catcher also composes cube-and-conquer solving runs entirely inside Lean. Per-cube refutations are combined with a cover-completeness certificate, itself an LRAT proof, into a single unsatisfiability theorem. Verified encodings connect CNF-level results to the original combinatorial problems. We evaluate the tool against Mathlib's proof-term import and the external checker cake_lpr on establishing the Schur number S(4) = 44 and the Ramsey number R(4,4) = 18 as Lean theorems.","abs_url":"https://arxiv.org/abs/2607.00815","pdf_url":"https://arxiv.org/pdf/2607.00815v1","primary_category":"cs.LO","published":"2026-07-01","tags":["tooling","verification","logic","combinatorics"],"lean_note":"LRAT-Catcher imports DIMACS formulas and LRAT SAT certificates into Lean 4 as theorems via native reflection using Lean core's verified checker.","repo":"https://github.com/leansolving/lrat-catcher","thumb":"thumbs/2607.00815.jpg"},{"id":"2607.00563","title":"Certificate-Carrying Transformation of Event-Driven Block Programs","authors":["Yuan Si","Jialu Zhang"],"abstract":"Block-based end-user languages such as Scratch run tens of millions of programs. Existing tools establish behavior preservation through program analysis and testing without a checked guarantee. We turn optimization into certificate-carrying source-to-source rewriting. An untrusted optimizer proposes a rewrite; a trusted, fail-closed checker accepts it only after recomputing every side condition that the rewrite's behavior preservation depends on under an explicit observation lens. The checker is the sole authority: given a correct checker and a small, explicitly stated set of model-to-VM assumptions, an optimizer bug cannot mint an unsound acceptance. The observation lens is a parameter, and the central soundness argument is a cooperative-frame refinement theorem: a write overwritten before any thread observes it, within a window in which no thread yields, can be removed. We mechanize this theorem in Lean and show that one parametric statement covers two concrete rewrite families instantiated to variable state and renderer state. We build a checker for six rewrite families and evaluate it on 300 real Scratch projects. The checker accepts a behavior-preserving rewrite on 94.3% of projects (283 of 300); certification costs under one tenth of a second per project; and a cross-family adversarial campaign of 4,278 perturbed rewrites produces zero false accepts. An audit found eight false accepts the per-family test suites missed; each is now rejected. An ablation that strips the semantic side conditions, leaving analysis and testing alone, ships rewrites the virtual machine confirms change behavior; the full checker rejects every one. The result shows how to provide behavior-preservation guarantees for a concurrent, event-driven, end-user language. The checker recomputes every required condition instead of trusting optimizer claims, keeping the trusted base small.","abs_url":"https://arxiv.org/abs/2607.00563","pdf_url":"https://arxiv.org/pdf/2607.00563v1","primary_category":"cs.PL","published":"2026-07-01","tags":["verification","formalization","logic"],"lean_note":"Mechanizes the cooperative-frame refinement theorem for dead-store elimination in Lean 4, with two decision procedures as verified-executable Lean functions.","thumb":"thumbs/2607.00563.jpg"},{"id":"2607.00091","title":"A Priori Integral Persistent Excitation in Conservative Polynomial ODEs with Higher-Order Interactions","authors":["Aleksandr Semenov","Alexander Fradkov"],"abstract":"The paper proposes an approach for verifying integral persistent excitation, which is important in problems of parameter identification and adaptive control in nonlinear dynamical systems. The approach works for conservative polynomial ODEs a priori without knowledge of the parameters. Rigorous proofs of the corresponding theorems are provided. An example of a nonlinear dynamical system with higher-order interactions and the application of the proposed method to it are analyzed. The proof of the main result is formalized in the Lean formal verification language.","abs_url":"https://arxiv.org/abs/2607.00091","pdf_url":"https://arxiv.org/pdf/2607.00091v1","primary_category":"math.OC","published":"2026-06-30","tags":["formalization","analysis"],"lean_note":"The proof of the main theorem on absence of trajectory sticking to hyperplanes and affine integral persistent excitation is formalized in Lean.","repo":"https://github.com/Sashkasem/IPE_proof_lean","thumb":"thumbs/2607.00091.jpg"},{"id":"2606.31134","title":"Beyond the Library: An Agentic Framework for Autoformalizing Research Mathematics","authors":["Arshia Soltani Moakhar","Iman Gholami","Max Springer","Mahdi JafariRaviz","MohammadTaghi Hajiaghayi"],"abstract":"While Large Language Models (LLMs) have demonstrated exceptional capabilities in mathematical reasoning, they frequently produce subtle errors that evade human detection. Formal mathematical languages like Lean 4 offer mechanical proof checking, strongly motivating the need for autoformalization: the automatic translation of natural language mathematics into verifiable code. Recent trends indicate that general-purpose LLMs, heavily optimized for standard programming, now outperform smaller models explicitly fine-tuned for Lean. Leveraging this shift, we introduce an agentic autoformalization framework powered by general coding LLMs. At the core of our system is an orchestrator that manages a multi-agent pipeline tailored for research-level mathematics. Because cutting-edge research frequently relies on concepts outside the scope of existing libraries like Mathlib, our system dynamically extends necessary type definitions and validates them via a novel Auxiliary Lemma technique before formalizing the primary theorems. We applied our approach to PutnamBench, producing machine-checked Lean proofs for a random sample of 32 problems. Furthermore, we evaluate our system on five papers from the ACM Symposium on Theory of Computing (STOC) spanning combinatorics, communication complexity, mechanism design, and learning theory, successfully formalizing their main theorems and validating the generated formalizations with human experts; for all five we also formalize the proofs alongside the statements, and notably two of them are proved with no axioms beyond Lean's kernel. All of our formalizations are available at https://beyondthelibrary.github.io/formal_arxiv .","abs_url":"https://arxiv.org/abs/2606.31134","pdf_url":"https://arxiv.org/pdf/2606.31134v2","primary_category":"cs.AI","published":"2026-06-30","tags":["autoformalization","ai-agents","theorem-proving-ml","benchmark"],"lean_note":"An agentic multi-agent framework autoformalizes research mathematics into Lean 4, extending type definitions beyond Mathlib and validating via auxiliary lemmas.","thumb":"thumbs/2606.31134.jpg"},{"id":"2606.31002","title":"Beyond Compilation: Evaluating Faithful Natural-Language-to-Lean Statement Formalization","authors":["Ke Zhang","Patricio Gallardo Candela","Sudhir Murthy","Yi Xie","Zhi Wang","Maziar Raissi"],"abstract":"Theorem-proving benchmarks evaluate proof search against fixed formal statements, but natural-language-to-Lean formalization must generate the formal statement itself. In this setting, compilation is only a validity check: a Lean declaration may type-check while omitting hypotheses, changing domains, or expressing a vacuous claim. We study faithful statement formalization as both an evaluation problem and a bottleneck-attribution problem. On a 400-entry graduate-level benchmark spanning real analysis, complex analysis, topology, and algebra, our protocol combines Lean compilation, cross-model semantic judging, and human expert calibration. The resulting picture is different from compile-rate evaluation: a full tool-augmented agent reaches 89.5% compilation but only 60.5% consensus faithfulness, exposing a 29.0-point compile-pass but consensus-unfaithful gap. Targeted human audits support the metric as a conservative decision boundary: across available case-level audits, 96.0% of consensus-positive outputs are human-confirmed faithful, while 82.4% of compile-pass consensus-negative outputs are human-confirmed semantic failures. Under this metric, existing one-shot formalizer models and prover-oriented Lean models remain low, suggesting that formal validity, proof-oriented Lean competence, and faithful statement generation should be reported separately. We then use a full $2^3$ factorial design to decompose three recurring interventions in formalization pipelines: parametric expert drafting, Mathlib/context search, and Lean elaboration feedback. Elaboration feedback is the largest validity intervention, but it also exposes a larger compile-pass semantic-failure bucket; search mainly improves grounding and selectivity; and fine-tuned drafting is largely substitutable in this tool stack once feedback and grounding are available.","abs_url":"https://arxiv.org/abs/2606.31002","pdf_url":"https://arxiv.org/pdf/2606.31002v1","primary_category":"cs.AI","published":"2026-06-30","tags":["autoformalization","benchmark","mathlib"],"lean_note":"Introduces a 400-entry benchmark and protocol for evaluating faithful natural-language-to-Lean statement formalization beyond mere compilation.","thumb":"thumbs/2606.31002.jpg"},{"id":"2606.30868","title":"Vector alignment in matrix Lie groups","authors":["Congzhou M Sha"],"abstract":"The difference in gauge between two observers of the same physical system can be thought of as a group element acting on their common vector representations. Recovering that group element from a finite, noisy list of paired observations may be of use in both theory and experiment. The Kabsch and Horn algorithms efficiently align point clouds in $\\mathbb R^3$, reconciling rotated frames of reference in Galilean relativity (i.e. $SO(3)$). In a previous work, we proposed an alternative Lie algebra method which extends to the Lorentz group $SO(3,1)_+$, and putatively to all Lie groups. In this work, we report the explicit formulae for applying the Lie algebra method to the classical matrix Lie groups (general linear $GL(n)$, special linear $SL(n)$, special orthogonal $SO(n)$, unitary $U(n)$, indefinite special orthogonal $SO(p,q)$, symplectic $Sp(n)$, spin $Spin(n)$, special Euclidean $SE(n)$) over both the real and complex fields. The four steps (pseudoinverse, matrix logarithm, projection onto the Lie algebra, matrix exponential) are exact in the noiseless case. The only group-dependent step is the projection, which we show produces the unique least squares-optimal element of the Lie algebra whenever its image lies in $\\mathfrak g$ and its residual is orthogonal to $\\mathfrak g$. Additionally, the Lie algebra method is optimal only to leading order for noisy data, so we refine it with a Newton-style correction. This correction matches the Lie algebra method in the noiseless case and direct least squares optimization in the noisy case, with performance between that of the Lie algebra method without correction and naive least squares optimization. The projections, their optimality, and the identity underlying the correction are formally proven in Lean~4.31.0 (with Mathlib 4.31.0), and numerical experiments are benchmarked in Julia.","abs_url":"https://arxiv.org/abs/2606.30868","pdf_url":"https://arxiv.org/pdf/2606.30868v1","primary_category":"math.NA","published":"2026-06-29","tags":["formalization","verification","algebra"],"lean_note":"Formally verifies in Lean 4 with Mathlib the Lie algebra projection formulae, their optimality, and the real-embedding exponential identity.","repo":"https://github.com/JuliaDiff/ReverseDiff.jl","thumb":"thumbs/2606.30868.jpg"},{"id":"2606.30782","title":"A Lean 4 Formalization of Scott's \\emph{Continuous Lattices} (1972)","authors":["Lars Warren Ericson"],"abstract":"We present a complete machine-checked formalization of Dana Scott's landmark 1972 paper \\emph{Continuous Lattices} \\textbf{[Sco72]}, carried out in Lean 4 against mathlib and including the March 1972 Milner correction in \\textbf{[Sco72]} (pp.~135--136). Scott's paper develops a model for \\(λ\\)-calculus from a topological starting point. He defines \\emph{injective} \\(T_0\\)-spaces -- those with a strong extension property for continuous maps -- and shows that they are exactly the \\emph{continuous lattices}: complete lattices whose Scott topology is determined by the order via the way-below relation (\\(\\ll\\)). On this foundation he studies projections, retractions, products, function spaces, and inverse limits. The capstone (Theorem 4.4) constructs an inverse limit \\(D_\\infty\\) of function-space approximants and proves \\(D_\\infty \\cong [D_\\infty \\to D_\\infty]\\), yielding a purely mathematical model for Church's untyped \\(λ\\)-calculus. Our development formalizes \\textbf{43 numbered results} from Scott's Sections 1--4 (Propositions, Corollaries, Lemmas, and Theorems), each as a sorry-free Lean theorem, together with supporting infrastructure (step functions, the \\(\\Uparrow a\\) basis of Scott opens, Milner's coarser-than-Scott hypothesis, the function-space tower, and the \\(i_\\infty\\)/\\(j_\\infty\\) pair). The formalization is \\textbf{classical} (uses \\texttt{Classical.choice} transitively) and follows Scott's proof dependency order. Where the Lean proof required choices not visible in the original -- or where dead ends were encountered -- we record detailed notes in Section 5. All proofs check with the standard footprint \\(\\texttt{[propext, Classical.choice, Quot.sound]}\\).","abs_url":"https://arxiv.org/abs/2606.30782","pdf_url":"https://arxiv.org/pdf/2606.30782v1","primary_category":"cs.LO","published":"2026-06-29","tags":["formalization","logic","topology"],"lean_note":"Complete sorry-free Lean 4 formalization against Mathlib of Scott's 1972 Continuous Lattices, including 43 numbered results and the D∞ λ-calculus model.","thumb":"thumbs/2606.30782.jpg"},{"id":"2606.29687","title":"A Machine-Verified Proof of a Quantum-Optimization Conjecture","authors":["Uri Kol","Maor Ben-Shahar","Kfir Sulimany","Dirk Englund"],"abstract":"We report a machine-verified resolution of a problem open for over a decade in quantum optimization: the Farhi, Goldstone and Gutmann (FGG) conjecture that depth-$p$ Quantum Approximate Optimization Algorithm (QAOA) on the ring of disagrees attains approximation ratio $(2p+1)/(2p+2)$ exactly. We found the proof using a large language model, Claude Fable 5, and verified its correctness end-to-end by the Lean 4 proof assistant. Our methodology includes several ingredients: building on a substantial Lean library of quantum information, we formalized the QAOA components and the known parts of the problem, and reduced the conjecture to a single open mathematical statement. The model was then handed the library and our agentic toolkit, and tasked with closing that gap by constructing a proof in Lean. The resulting process is a feedback loop between the model's natural-language reasoning and Lean's mechanical verification, which converged to a machine-verified proof. Human verification is required only for the structural scaffolding - that the formal statement faithfully encodes the intended claim - while the proof itself is supplied by the model and certified mechanically by Lean. The proof is nevertheless striking - the model uncovered a hidden dynamical symmetry of the problem and exploited it, borrowing tools and machinery from an adjacent field to turn a hard existence problem into an explicit construction. This work paves the way for resolving open conjectures in quantum information science and beyond.","abs_url":"https://arxiv.org/abs/2606.29687","pdf_url":"https://arxiv.org/pdf/2606.29687v1","primary_category":"quant-ph","published":"2026-06-29","tags":["formalization","theorem-proving-ml","quantum"],"lean_note":"Proves the FGG conjecture on QAOA approximation ratio using Claude Fable 5 with end-to-end Lean 4 verification.","repo":"https://github.com/urikol/QuantumOptimization","thumb":"thumbs/2606.29687.jpg"},{"id":"2606.29493","title":"Faults in Our Formal Benchmarking: Dataset Defects and Evaluation Failures in Lean Theorem Proving","authors":["Pawan Sasanka Ammanamanchi","Siddharth Bhat","Stella Biderman"],"abstract":"Benchmarks for LLM-assisted theorem proving in Lean are often treated as intrinsically reliable because every solved instance comes with a machine-checked proof. However, the kernel only checks that a proof establishes a \\emph{formal} statement; it does not verify that the statement faithfully encodes the intended informal problem, nor that evaluation harnesses are robust to trivial or adversarial solutions. We audit five widely used Lean theorem-proving benchmarks and their forks, using corpus-scale static checkers to surface 4,833 findings, including 398 mechanically certified issues such as counterexamples, vacuous theorems, and unsound axioms. We also document semantic defects such as missing hypotheses, problem simplification, incomplete or incorrect translations, and Lean-specific specification hazards. Beyond dataset construction, we survey evaluation-time failure modes and show, on corrected subsets, that defects can both inflate and deflate reported prover scores. We propose a fault taxonomy, a suite of automated checkers and recall-oriented semantic audit prompts, and release standards to guide the creation of formal math datasets and to make evaluation more reproducible and trustworthy. Our checkers, audit prompts, and corrected dataset snapshots are available at https://github.com/Shashi456/atp-checkers.","abs_url":"https://arxiv.org/abs/2606.29493","pdf_url":"https://arxiv.org/pdf/2606.29493v1","primary_category":"cs.AI","published":"2026-06-28","tags":["benchmark","tooling"],"lean_note":"Audits five Lean theorem-proving benchmarks with corpus-scale static checkers implemented as Lean 4 metaprograms.","repo":"https://github.com/Shashi456/atp-checkers","thumb":"thumbs/2606.29493.jpg"},{"id":"2606.29361","title":"A sharp 5/8 bound for an Erdős-Sós pairwise-sums problem","authors":["Ricky Cipollini"],"abstract":"Let $f_3(N)$ be the least integer such that every set $A\\subseteq\\{1,\\ldots,N\\}$ of size at least $f_3(N)$ contains distinct elements $a,b,c\\in A$ such that $a+b\\in A$, $a+c\\in A$, and $b+c\\in A$. We prove that $f_3(N)\\le 5N/8+O(1)$. Together with the standard construction $[N/8,N/4]\\cup[N/2,N]$, this gives $f_3(N)=5N/8+O(1)$, resolving Erdős Problem 865. The proof is self-contained. An earlier conditional version of the reduction has also been formalized in Lean 4/Mathlib with no sorries and no added axioms.","abs_url":"https://arxiv.org/abs/2606.29361","pdf_url":"https://arxiv.org/pdf/2606.29361v1","primary_category":"math.CO","published":"2026-06-28","tags":["formalization","mathlib","combinatorics","number-theory"],"lean_note":"Formalizes a key reduction for the Erdős-Sós pairwise-sums problem in Lean 4 with Mathlib, no sorries or added axioms.","repo":"https://github.com/mrricky22/erdos-865-lean","thumb":"thumbs/2606.29361.jpg"},{"id":"2606.28990","title":"The Fundamental Theorem of Asset Pricing, Formalized in Lean 4","authors":["Raphael Coelho"],"abstract":"The Fundamental Theorem of Asset Pricing states that a market is free of arbitrage exactly when it admits an equivalent martingale measure. We formalize it in Lean 4 over Mathlib in three settings: a finite-state market over a finite horizon (Harrison-Pliska), a one-period market on an arbitrary probability space with a single scalar return (Follmer-Schied), and a one-period market with finitely many assets. The finite case is the geometry of a separating hyperplane; the scalar one-period case is an elementary change of measure. In the $d$-asset case the equivalent martingale measure is constructed explicitly, as the minimiser of the smooth convex potential $\\mathbb{E}[\\log(1+e^{\\langleθ,Y\\rangle})]$: absence of arbitrage is precisely coercivity of the potential, its first-order condition is the martingale property, and the minimiser's logistic weight is the density of the measure. The construction uses no Hahn-Banach theorem, no $L^0$-closedness argument, no measurable selection, and no non-redundancy hypothesis. To our knowledge this is the first machine-checked Fundamental Theorem of Asset Pricing in any proof assistant. The boundary is explicit: the general multi-period Dalang-Morton-Willinger theorem lies outside the development. Every theorem is sorry-free, each headline result's axioms are pinned to Mathlib's classical defaults by a build-enforced gate, and the whole is reproducible from a pinned toolchain.","abs_url":"https://arxiv.org/abs/2606.28990","pdf_url":"https://arxiv.org/pdf/2606.28990v1","primary_category":"q-fin.MF","published":"2026-06-27","tags":["formalization","mathlib","economics"],"lean_note":"Formalizes the Fundamental Theorem of Asset Pricing in Lean 4 over Mathlib in three market settings.","repo":"https://github.com/raphaelrrcoelho/formal-mathfin","thumb":"thumbs/2606.28990.jpg"},{"id":"2606.28841","title":"LAMP: Lean-based Agentic framework with MCP and Proof Repair","authors":["Santhana Srinivasan R","Maithilee Patawar"],"abstract":"Large language models are increasingly capable of mathematical reasoning, but the proofs they generate are often unreliable and hard to verify. Interactive theorem provers such as Lean 4 address this by accepting only kernel-checked proofs; however, their reach is bounded by the formalized knowledge available. While Mathlib, a repository of formalized Lean 4 theorems that covers diverse mathematical areas, certain specialized areas remain underrepresented; notably, the domain of Combinatorics on Words (CoW). CoW studies sequences, exploring their properties such as periodicity, borders, conjugacy, and morphisms. As a result, specialized provers, trained on Mathlib-centered data, lack the lemmas to operate in CoW. We present two contributions. First, we introduce a Lean 4 formalization of CoW containing eight modules and \\textbf{93} declarations of core definitions and foundational lemmas. Second, we present LAMP, a multi-agent framework that synthesizes kernel-verified Lean 4 proofs by providing explicit, structured domain knowledge at inference time through an ontology, rather than by fine-tuning a prover. LAMP coordinates a Planner, Builder, and Verifier with Model Context Protocol based access to a domain-specific CoW ontology. In a suite of 90 CoW theorems that span all eight modules and three difficulty levels, LAMP synthesizes verified proofs for 96.7% of theorems, substantially exceeding both an unscaffolded baseline and existing specialized provers. An ablation shows that removing LAMP's tool-grounded architecture or its Planner/Builder separation each cost roughly 12 percentage points, even with the backbone model held fixed.","abs_url":"https://arxiv.org/abs/2606.28841","pdf_url":"https://arxiv.org/pdf/2606.28841v1","primary_category":"cs.LO","published":"2026-06-27","tags":["theorem-proving-ml","ai-agents","tooling"],"lean_note":"Presents a multi-agent framework that synthesizes verified Lean 4 proofs using domain-specific ontology access via MCP.","repo":"https://github.com/oOo0oOo/lean-lsp-mcp","thumb":"thumbs/2606.28841.jpg"},{"id":"2606.28710","title":"The Two Genie Game: Adoption and Welfare in Audit-Grounded AI Governance","authors":["Darrell Lewis-Sandy"],"abstract":"We ask under what conditions an agent with a harm-minimizing policy can displace an approval-seeking (RLHF) agent in a competitive market, and when that policy is sufficient to prevent community harm. We use evolutionary game theory (finite-population Moran-Fermi pairwise comparison) to formalize this subject to assumptions of wisher hindsight, peer testimony, a monotone harm ledger, sufficient information density of community feedback, and a finite, depleting resource pool, in a negative-sum environment. We show that adoption is favored when the prior distributions on how readily wishers attune to community sentiment are monotone, exhibit endpoint inversion, and have a centro-symmetric pairing property, and demonstrate this with several long-tailed priors (Hill, Pareto, Lomax, Frechet). Where it is favored, a critical adoption level separates communities that drift back to the approval-seeking agent from those for which the audited agent fixes; above that level fixation is the overwhelmingly likely outcome. We derive when fixation is attainable as a bound on the effective (informational) size N_c of the community, which must be small enough to allow fixation before depletion. We present these as Theorems 5.4 and 5.5; the algebraic and finite-grid backbone is machine-checked in Lean 4, with the barrier-crossing asymptotics retained as explicit hypotheses. We show that a self-audited agent with a community ledger is not, in general, sufficient to prevent community harm. Sufficiency depends both upon the alignment of the agent's audit with community values and the timeframe over which harm is evaluated. Regardless of alignment, once adoption reaches dominance, the state is absorbing. The same policy that reduced harm under alignment becomes a trap, welfare-negative under misalignment and, even under alignment, one that locks in harm deferred past the adoption horizon.","abs_url":"https://arxiv.org/abs/2606.28710","pdf_url":"https://arxiv.org/pdf/2606.28710v1","primary_category":"cs.AI","published":"2026-06-27","tags":["verification","formalization","economics"],"lean_note":"Machine-checks the algebraic and finite-grid backbone of evolutionary game theory adoption theorems in Lean 4.","repo":"https://github.com/dlewissandy/two-genie-scripts","thumb":"thumbs/2606.28710.jpg"},{"id":"2606.28572","title":"Geometric Measurements of the Axiom of Choice in Neural Proof Embeddings","authors":["Rodrigo Mendoza-Smith"],"abstract":"The axiom of choice has divided the foundations of mathematics for over a century, but the distinction between classical and constructive proofs has remained a philosophical and methodological one. We use Lean 4's kernel-level tracking of axiom dependence to show that the axiom of choice has a measurable geometric correlate in proof space that obeys a one-parameter mixture law and has operational consequences for neural theorem provers. To do this, we partition $471{,}260$ declarations of Mathlib by transitive dependence on the axiom of choice and represent a filtered population of $42{,}355$ traced theorems by their sequences of tactic invocations. We use the constructive proofs in this dataset to train a self-supervised proof encoder and show that when using it to measure classical proofs, three complementary measurements (anomaly score, reconstruction loss, and density-superlevel containment) exhibit a common decline with the proof's distance from the axiom in the dependency graph, from sharp separation at the shallow boundary (AUC $0.847$ at distance $2$) to indistinguishability at distance~$9{+}$. Robustness controls show that the signature survives length, file, author, and topic controls, and replicates under full-source encoders trained on normalised proof source. Operationally, we show that on an evaluation sample of $251$ Mathlib theorems, Lean's \\texttt{aesop} tactic solves constructive theorems at $13\\times$ the rate of classical ones, and a neural-guided hybrid using the ReProver tactic generator compresses the gap to $5\\times$. The geometric anomaly score predicts \\texttt{aesop} failure beyond proof length, providing an operational link between the geometric signature and prover performance.","abs_url":"https://arxiv.org/abs/2606.28572","pdf_url":"https://arxiv.org/pdf/2606.28572v1","primary_category":"cs.LG","published":"2026-06-26","tags":["theorem-proving-ml","mathlib","logic"],"lean_note":"Uses Lean 4's kernel axiom tracking to measure the geometric signature of Classical.choice in 42,355 Mathlib proof embeddings.","repo":"https://github.com/rodrgo/geometric-axiom-of-choice","thumb":"thumbs/2606.28572.jpg"},{"id":"2606.28013","title":"The Signal-Coverage Matrix: Stratifying Type and Semantic Errors in Statement Autoformalization","authors":["Chengxiao Dai","Zhaokun Yan","Zhanhui Lin"],"abstract":"Headline type-correctness (TC\\%) of LLM autoformalization has climbed from $\\sim$53\\% to $\\sim$76\\% in two years, yet this scalar conceals which errors each method resolves. We propose a signal-coverage matrix that crosses the Lean elaborator (pass/fail) with a semantic-equivalence judgment (equivalent/not), sorting every output into one of four cells: true success (TS), type-only (TO), semantic-only (SO), or both fail (BF). On ProofNet\\# and MiniF2F-test with DeepSeek V4-Pro across Vanilla, Lean-Retry, Sample-Filter, and Stratified Autoformalization (SAF): (1) the +34 to +36 TS gain across the three elab-feedback methods is $\\sim$64\\% type-stratum recovery, with SO flat on net (87.5\\% of original semantic errors rescued, 8 newly created). (2) The TO-to-TS rate is 23/61 for each method (Wilson 95\\% CI [26.6\\%, 50.3\\%]), and this stratum-level recovery rate predicts $Δ$TS on held-out methods to within 2/186 and renders $Δ$TC linear in the Vanilla elab-fail rate across six (model, dataset) cells ($R^2=0.96$). (3) The two judges disagree by 26 to 37 pp on elab-feedback outputs (vs. 7 pp on Vanilla), with 30 to 56\\% of symbolic-judge false negatives traceable to elaborator-forced rewrites. The persistent residual reduces to two gold-formalization errors. TC\\% gains should be credited by which cell moved, not by the scalar alone.","abs_url":"https://arxiv.org/abs/2606.28013","pdf_url":"https://arxiv.org/pdf/2606.28013v1","primary_category":"cs.CL","published":"2026-06-26","tags":["autoformalization","benchmark","theorem-proving-ml"],"lean_note":"Uses the Lean elaborator as ground truth for type-correctness in a diagnostic matrix evaluating autoformalization methods.","thumb":"thumbs/2606.28013.jpg"},{"id":"2606.27041","title":"Formalizing a Many-Sorted Hybrid Polyadic Modal Logic in Lean","authors":["Andrei-Alexandru Oltean","Bogdan Macovei","Ioana Leuştean"],"abstract":"We present a Lean formalization of a general hybrid modal logic with many-sorted signatures and polyadic modal operators. The system borrows ideas from both algebraic specification and dynamic logics, and is designed to serve as a uniform axiomatic foundation for specifying and verifying programming languages and security protocols. We expose a DSL for users to define languages and protocols as many-sorted signatures, specify the relevant domain-specific axioms, and reason about program executions or protocol runs. We provide a machine-checked proof of its soundness theorem and showcase the framework's versatility through several applications: an imperative programming language for code verification, the BAN logic for security protocols, and the modal system S5. We have designed our formalization to be intrinsically sorted, that is, well-sorted formulas in the base language are well-typed terms in Lean. Thanks to intrinsic sorting, all domain specific applications can be easily embedded in our framework via the DSL, at no additional syntactic overhead required for the user to prove. All code presented in this paper is openly accessible in the following repository: https://github.com/alexoltean61/msphml-lean","abs_url":"https://arxiv.org/abs/2606.27041","pdf_url":"https://arxiv.org/pdf/2606.27041v1","primary_category":"cs.LO","published":"2026-06-25","tags":["formalization","verification","logic"],"lean_note":"Formalizes a many-sorted hybrid polyadic modal logic in Lean 4, with soundness proof and applications to code verification and security protocols.","repo":"https://github.com/alexoltean61/msphml-lean","thumb":"thumbs/2606.27041.jpg"},{"id":"2606.26525","title":"Theory-Scale Auto-Formalization of Logics for Computer Science","authors":["Yuming Feng","Frederick Pu","One An","Osbert Bastani","Li Zhang","Jiani Huang","Xujie Si","Ziyang Li"],"abstract":"Auto-formalization is critical for scalable formal verification, but existing progress largely focuses on isolated statements, while theory-scale auto-formalization, which coherently translates hundreds of interdependent definitions, lemmas, and theorems, remains open due to challenges in consistency, faithfulness, scalability, and correctness. In this paper, we introduce LCS-Bench, a stand-alone, theory-scale benchmark based on Logics for Computer Science. LCS-Bench is built through a novel semi-automated agentic pipeline that leverages concept graphs, formal signature planning, issue tracking, sorry-filling with counter-example search, complemented by faithfulness review from human experts. The resulting artifact covers 327 textbook items, over 4,076 Lean declarations, and more than 85K lines of Lean code. The dataset supports broad evaluation through a data engine that automatically derives five tracks of evaluation benchmarks, measuring different aspects of auto-formalization and theorem-proving capabilities. We also introduce a novel evaluation protocol featuring definitional equivalence checkers, enabling more fine-grained and faithful assessment. Through extensive evaluation on 14 models, we demonstrate that (1) LCS-Bench is of high quality, consistent, and faithful; (2) the benchmark is challenging, with state-of-the-art models achieving only 20.1% on auto-formalization tasks; and (3) our analysis reveals key findings regarding theory-scale auto-formalization and suggests promising directions for future work.","abs_url":"https://arxiv.org/abs/2606.26525","pdf_url":"https://arxiv.org/pdf/2606.26525v1","primary_category":"cs.LG","published":"2026-06-25","tags":["autoformalization","benchmark","dataset","logic"],"lean_note":"Introduces LCS-Bench, a theory-scale Lean 4 benchmark with 4,076 declarations and 85K lines of code for evaluating auto-formalization.","thumb":"thumbs/2606.26525.jpg"},{"id":"2606.26442","title":"AXLE: A Cloud Infrastructure for Lean 4 Theorem Proving Utilities","authors":["Jimmy Xin","Alex Schneidman","Chris Cummins","Karun Ram","Srihari Ganesh","Jannis Limperg"],"abstract":"We present AXLE (Axiom Lean Engine), a cloud service for Lean 4 proof manipulation, extraction, and verification. Recent progress in AI for mathematics -- reinforcement learning pipelines, agentic proving workflows, dataset curation -- demands Lean 4 tooling that scales to millions of requests while remaining correct and robust; existing infrastructure offers parallel compilation but not scalable proof verification, higher-level proof manipulation, multi-version support, or per-request isolation at the throughput modern AI workflows require. AXLE provides 14 Lean 4 metaprogramming tools spanning strict proof verification, declaration metadata extraction, semantic source manipulation, deterministic proof repair and simplification, and lemma extraction. The service runs as a multi-tenant cloud deployment with per-request isolation and concurrent support for multiple Lean 4 and Mathlib versions, accessible via a Python SDK, command-line interface, web UI, MCP server, and raw HTTP API. AXLE is publicly available and free to use at https://axle.axiommath.ai and via the axiom-axle PyPI package, with no local Lean 4 installation required. It has served over 500 million requests to date and is the underlying infrastructure for Axiom Math's proving efforts, including its 12/12 score on the 2025 Putnam competition.","abs_url":"https://arxiv.org/abs/2606.26442","pdf_url":"https://arxiv.org/pdf/2606.26442v1","primary_category":"cs.LO","published":"2026-06-24","tags":["tooling","verification","theorem-proving-ml"],"lean_note":"Presents AXLE, a scalable cloud service providing 14 Lean 4 metaprogramming tools for proof verification, manipulation, and extraction.","repo":"https://github.com/oOo0oOo/lean-lsp-mcp","thumb":"thumbs/2606.26442.jpg"},{"id":"2606.26406","title":"Beyond Feedforward Networks: Reentry Neural Systems as the Fundamental Basis of Subjecthood and Intrinsic Safety of Next-Generation AGI","authors":["A. S. Ushakov","Yu. N. Berdinsk"],"abstract":"We propose a complete architectural blueprint for safe artificial general intelligence based on a closed reentry loop (D <-> I cycle). In contrast to feedforward networks, which are directed acyclic graphs (C=0, S=0) incapable of self-reference, the proposed architecture contains a structural cycle (C >= 1) with self-sustaining amplification (rho > 1), mathematically guaranteeing the emergence of a self-model, instrumental self-preservation, and unprogrammed goal-directed behaviour. The agent's goals are encoded as a non-textual D-vector in the architecture itself, making them immune to reinterpretation and prompt injection. We present the S-measure -- a polynomial-time [O(N^3)] computable alternative to Tononi's NP-hard Phi -- with machine-verified Lean 4 proof that S>0 implies positive integrated information. The work provides full Python/NumPy implementations (Tarjan-based cycle complexity, Delta-S barrier), industrial horizontal scaling via Apache Kafka and Docker Compose, a taxonomy of six epochs of AI evolution, a zoo of future reentry architectures (RAS, diffusion attractors, fractal loops), gauge-invariant networks for safe swarms, fault-tolerance and recovery protocols, and eight falsifiable predictions. All formal proofs are machine-verified in Lean 4. This architecture is deployable today and represents a topologically protected, safe-by-design approach to AGI.","abs_url":"https://arxiv.org/abs/2606.26406","pdf_url":"https://arxiv.org/pdf/2606.26406v1","primary_category":"cs.LG","published":"2026-06-24","tags":["verification","ml-theory"],"lean_note":"Machine-verifies in Lean 4 that S>0 implies positive integrated information for a proposed reentry-based AGI safety measure.","repo":"https://doi.org/10.5281/zenodo.20767214","thumb":"thumbs/2606.26406.jpg"},{"id":"2606.26052","title":"On the existence problem of regular Gabor frames","authors":["Jaume de Dios Pont","Lukas Liehr","Mitchell A. Taylor"],"abstract":"For every dimension $d > 1$, we establish explicit criteria on lattices $Λ\\subset \\mathbb{R}^{2d}$ with density $D(Λ) > 1$ such that no function with a continuous Zak transform generates a Gabor frame along $Λ$. In particular, this gives a negative answer to the existence problem of Gabor frames with window functions in the Schwartz space, the Feichtinger algebra, and the Fourier-invariant Wiener space. Our result is based on a characterization of when a collection of quasiperiodic functions admits a common zero, which may be of independent interest. We also include a formalization of our main result in Lean 4.","abs_url":"https://arxiv.org/abs/2606.26052","pdf_url":"https://arxiv.org/pdf/2606.26052v1","primary_category":"math.FA","published":"2026-06-24","tags":["formalization","analysis"],"lean_note":"Formalizes in Lean 4 a criterion on lattices for which no Schwartz-class function generates a Gabor frame.","repo":"https://github.com/jaumededios/Schwartz_Gabor_Frames","thumb":"thumbs/2606.26052.jpg"},{"id":"2606.26035","title":"Every Nonnegative Integer Is a Sum of a Triangular, a Pentagonal, and a Heptagonal Number","authors":["Yichuan Cao","Dakai Guo","Ruichen Qiu","Ruyong Feng","Xiao-Shan Gao"],"abstract":"In this paper, it is proved that any nonnegative integer can be written in the following form $$ x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2, \\qquad x,y,z \\in \\mathbb{N}. $$ This settles the conjecture recorded as OEIS A287616. All parts of the proof have been formalized in Lean 4, with the exception of two results: one externally cited theorem and one statement verified by symbolic computation. Both the natural-language proof and the Lean formalization were generated by the MechMath Agent Team developed by the authors.","abs_url":"https://arxiv.org/abs/2606.26035","pdf_url":"https://arxiv.org/pdf/2606.26035v1","primary_category":"math.NT","published":"2026-06-24","tags":["formalization","number-theory","ai-agents"],"lean_note":"Formalizes in Lean 4 the proof that every nonnegative integer is a sum of a triangular, pentagonal, and heptagonal number.","repo":"https://github.com/MechMath/integer-sum","thumb":"thumbs/2606.26035.jpg"},{"id":"2606.25412","title":"Formalization of Line Search Methods by Lean","authors":["Yiyang Zhang","Kenneth W. Shum"],"abstract":"This paper presents a formalization of line search methods in the Lean 4 theorem prover. Our goal is to advance machine verification of nonlinear optimization theory by translating standard textbook definitions and convergence arguments into rigorous Lean code. We formalize fundamental notions related to gradient descent and descent directions, adaptive step-size selection via backtracking line search, and several classical line search criteria, including the Armijo, Goldstein, and Wolfe conditions, as well as nonmonotone variants. We further formalize a key convergence result, namely the Zoutendijk theorem, which plays a central role in the global convergence analysis of gradient-based iterative methods. By providing machine-checkable definitions and proofs for line search theory, this work complements existing formalizations of first-order optimization methods and establishes a foundation for the verified development of more advanced algorithms in nonlinear programming.","abs_url":"https://arxiv.org/abs/2606.25412","pdf_url":"https://arxiv.org/pdf/2606.25412v1","primary_category":"math.OC","published":"2026-06-24","tags":["formalization","analysis"],"lean_note":"Formalizes gradient descent, backtracking line search, Armijo/Goldstein/Wolfe conditions, and the Zoutendijk theorem in Lean 4.","repo":"https://github.com/AquilaCheung/LEAN-formalization-lineSearchMethods","thumb":"thumbs/2606.25412.jpg"},{"id":"2606.25409","title":"CV-Rules: Serializability Verification of Concurrency Control Protocols via Explicit Transaction Ordering","authors":["Takashi Hoshino","Shigeo Mitsunari","Takashi Kambayashi","Ryoji Kurosawa","Sho Nakazono"],"abstract":"We present CV-rules, an alternative characterization of serializability in which a transaction order constructed by a protocol satisfies two per-read conditions, C-rule (Causality) and V-rule (View Consistency), that constrain the reads-from relation and competing writers. While classical Multi-Version Serialization Graph (MVSG) reasoning characterizes serializability via its acyclicity, our approach requires explicit order construction, enabling direct proofs that build on the protocol's own mechanisms. We prove that CV-rules, serializability, and MVSG acyclicity are all equivalent. Moreover, the C/V separation reveals that serializability is polynomial-time decidable for any fixed bound on the width of the order forced by C-rule. We verify five protocols: Two-Phase Locking, Multi-Version Timestamp Ordering, Serial Safety Net (SSN), Aria, and SnapChain. For SSN and Aria, whose original papers defined only certification conditions, we identify explicit transaction orders arising from their mechanisms; we also prove that Aria's unique-write constraint is unnecessary for serializability. SnapChain, in contrast, is designed directly from CV-rules, enforcing V-rule by construction. All results except the complexity bounds are mechanized in Lean with no additional axioms and no admitted goals.","abs_url":"https://arxiv.org/abs/2606.25409","pdf_url":"https://arxiv.org/pdf/2606.25409v1","primary_category":"cs.LO","published":"2026-06-24","tags":["verification","logic"],"lean_note":"Mechanizes in Lean proofs of serializability equivalences and correctness of five concurrency control protocols.","repo":"https://doi.org/10.5281/zenodo.20755695","thumb":"thumbs/2606.25409.jpg"},{"id":"2606.25363","title":"TheoremGraph: Bridging Formal and Informal Mathematics","authors":["Simon Kurgan","Evan Wang","Eric Leonen","Sophie Szeto","Luke Alexander","Artemii Remizov","Jarod Alper","Giovanni Inchiostro","Vasily Ilin"],"abstract":"Mathematical knowledge is organized around statements and their dependencies, but this structure is exposed unevenly: informal papers cite mostly at the document level, while formal libraries record fine-grained dependencies over a much smaller body of mathematics. We introduce TheoremGraph, a unified statement-level dependency graph spanning both informal and formal mathematics. On the informal side, we parse 11.7M theorem-like environments from mathematics arXiv and recover 18.3M candidate directed dependencies, each labeled by the extractor that proposed it so downstream users can trade coverage for precision. On the formal side, we release LeanGraph, a Lean 4 elaborator-level extractor producing 388,105 declaration nodes and 11.3M typed edges across 25 Lean projects. We bridge the two graphs by embedding generated natural-language slogans into a shared semantic space, linking related statements across papers and across the informal/formal divide; an LLM judge affirms 47,952 such matches above a 0.8 cosine floor, with the judge-acceptance rate rising from 48% across the floor to 87% in the >=0.9 tier. On formal concept retrieval, our name-and-signature representation with graph expansion comes within 0.5pp of LeanSearch v2's reranked Recall@10 (0.775 vs. 0.780) without an LM reranker. We release the dataset, extractors, HTTP API, and MCP interface as infrastructure for mathematical search, attribution, and retrieval-augmented reasoning, available at theoremsearch.com and huggingface.co/datasets/uw-math-ai/theorem-matching.","abs_url":"https://arxiv.org/abs/2606.25363","pdf_url":"https://arxiv.org/pdf/2606.25363v1","primary_category":"cs.IR","published":"2026-06-24","tags":["tooling","dataset","mathlib"],"lean_note":"Introduces LeanGraph, an elaborator-level dependency extractor producing 388K nodes and 11.3M edges across 25 Lean projects.","thumb":"thumbs/2606.25363.jpg"},{"id":"2606.25172","title":"Exact Local Annotations for Regular Languages","authors":["Faruk Alpay","Baris Basaran"],"abstract":"A regular language is recognized by a finite monoid, but a locally checkable explanation of that recognition can have a nontrivial update geometry. We study exact bounded-arity annotations for regular word languages under one-symbol substitutions. The cost of an edit is the number of annotation cells that a canonical locally accepted representation must change, together with the corresponding bit movement and the number of local constraints that must be revalidated. For every morphism recognizing a regular language, the balanced product annotation gives constant locality, linear size, O(log n) edit stability, O(log n) revalidation, and constant access to the membership value. The matching lower bound proved here is restricted to product decompositions that expose an edit-active nontrivial group quotient as ordered product labels; in that setting one substitution changes every quotient label on an ancestor path. We also show that annotation-free bounded-window recognition is exactly strict locality, prove closure properties for a two-sided total decision variant, and formulate the remaining constant-stability boundary as a finite obstruction problem. The ancillary files include Lean, CP-SAT, and CUDA certificates, including a context-free interval-chart experiment.","abs_url":"https://arxiv.org/abs/2606.25172","pdf_url":"https://arxiv.org/pdf/2606.25172v1","primary_category":"cs.FL","published":"2026-06-23","tags":["formalization","logic"],"lean_note":"Includes Lean certificates for results on bounded-arity annotations recognizing regular languages.","thumb":"thumbs/2606.25172.jpg"},{"id":"2606.24213","title":"Kops: Safely Extending the eBPF Compilation Pipeline with Native Operations","authors":["Yusheng Zheng","Zhengjie Ji","Weichen Tao","Hao Sun","Wei Zhang","Dan Williams","Andi Quinn"],"abstract":"eBPF safely extends OS kernels in domains such as networking, observability, and security. The safety comes from an in-kernel compilation pipeline where a verifier checks every program, and a kernel just-in-time compiler (JIT) translates the verified bytecode to native code. The kernel keeps the JIT simple to stay trustworthy, translating one bytecode instruction at a time in a single pass. This single-pass design misses optimization opportunities, so eBPF runs up to twice as slow as natively compiled code in our characterization. Adding optimizations to the kernel JIT directly requires upstream acceptance and a long release cycle, enlarges the trusted computing base (TCB), and grows the per-architecture kernel code. To address this, we present Kops, an extension interface that lets userspace compilers and kernel modules introduce new operations without modifying the kernel core, while keeping a minimal trusted computing base (TCB). Each operation has two forms, a proof sequence of vanilla eBPF instructions that the existing verifier checks and a native emit of machine instructions that the JIT compiles. Because the verifier checks the proof sequence, the native emit is the only per-operation addition to the TCB. Hardware idioms are the lowest-hanging fruit for this interface. With Kops, we build EInsn, seven operations such as rotate and conditional select that CPUs execute as single instructions. Lean 4 proofs show that each native emit computes the same result as its proof sequence. On x86-64 and ARM64, EInsn speeds up eBPF microbenchmarks by up to 24% and production applications by up to 12%. The same interface also supports whole-program native replacement, reaching 2.358x at the cost of a larger TCB.","abs_url":"https://arxiv.org/abs/2606.24213","pdf_url":"https://arxiv.org/pdf/2606.24213v1","primary_category":"cs.OS","published":"2026-06-23","tags":["verification"],"lean_note":"Proves in Lean 4 that each eBPF native-emit operation computes the same result as its vanilla-bytecode proof sequence.","repo":"https://github.com/aquasecurity/tracee","thumb":"thumbs/2606.24213.jpg"},{"id":"2606.23959","title":"Does My Embedding Reflect That $A = B$? Evaluating Mathematical Equivalence in Embedding Models","authors":["Jiaying Ye","Samarth Rao","Leo Carlin","Kedar Chintalapati","Saharsh Bhargava","Rachit Jaiswal","Michael Zhou","Jared Darlington","Jiahe Lu","Jarod Alper","Vasily Ilin","Henry Kvinge"],"abstract":"Because mathematics is highly abstract, a single statement can take very different forms depending on what subfield it is framed in. There are many examples where breakthroughs occurred after researchers discovered that a question had already been answered in a different field. At the same time, the growth of new resources related to formalization has increased the need for tools that enable efficient and reliable navigation between mathematical 'languages' (e.g., from Lean to natural language). In this paper, we investigate whether current embedding models capture mathematical equivalence. To do this, we introduce the Mathematically Equivalent but Lexically Different Pairs (MELD) Dataset, a collection of mathematically equivalent statements that are expressed in very different language. We show that current state-of-the-art embedding models tend to group statements by the terminology used to make them instead of the underlying math. Motivated by this, we propose a contrastive approach to learning embeddings of mathematical text that focuses on aligning informal statements with different formalizations. Our experiments demonstrate that this leads to improvements not only on informal-formal retrieval tasks but also on MELD, which only contains natural language statements.","abs_url":"https://arxiv.org/abs/2606.23959","pdf_url":"https://arxiv.org/pdf/2606.23959v2","primary_category":"cs.CL","published":"2026-06-22","tags":["autoformalization","benchmark"],"lean_note":"Evaluates embedding models on mathematical equivalence using Lean formalizations for informal-formal retrieval alignment.","repo":"https://github.com/uw-math-ai/math2vec","thumb":"thumbs/2606.23959.jpg"},{"id":"2606.23555","title":"Cubic Jordan algebras are not a series","authors":["Bruce Westbury"],"abstract":"The idea of the exceptional series is that the exceptional simple Lie algebras should form a series. Since all four simple Lie algebras in the fourth row of the Freudenthal magic square are exceptional it is natural to ask if the remaining rows form a series. A stronger version of this question is that, for the first two rows (corresponding to the real and complex numbers), there is a category defined by a presentation which is a reasonable candidate for the series. Our main results show that neither of these candidates is a series but each consists of a finite set of points. In each case the series is defined by a parameter and we show that the relations imply that this parameter satisfies a polynomial. These two results were obtained by a computer calculation. Our calculation is supported by a website for inspection, and the calculations are certified by Lean 4.","abs_url":"https://arxiv.org/abs/2606.23555","pdf_url":"https://arxiv.org/pdf/2606.23555v1","primary_category":"math.RT","published":"2026-06-22","tags":["formalization","algebra"],"lean_note":"Certifies in Lean 4 the polynomial relations from a computer calculation showing the candidate series are finite point sets.","thumb":"thumbs/2606.23555.jpg"},{"id":"2606.23277","title":"GIF: Locally Sound Geometric Information Flow Control for LLMs","authors":["Adam Storek","Nikolaus Holzer","Zhuo Zhang","Suman Jana"],"abstract":"Large language models increasingly mediate interactions between sensitive data, untrusted inputs, and privileged actions in agentic systems, creating security and privacy risks. These range from prompt injections that manipulate downstream tool use to leakage of confidential information through model outputs. Recent Information Flow Control (IFC)-based defenses show promise but lack a principled semantic foundation for reasoning about information flow through the model itself. Since any input token may influence any output token in an autoregressive LLM, existing approaches suffer from severe taint explosion. We present Geometric Information Flow (GIF), a semantic framework for tracking information flow from input tokens to outputs. GIF uses the LLM Jacobian and local output geometry to upper-bound the Shannon mutual information between perturbed input spans and model outputs, yielding a scalable measure computable on large models via automatic differentiation and low-rank approximation. Unlike attention-based or correlational attribution heuristics, GIF satisfies local geometric soundness, and we provide a fully mechanized Lean 4 proof that it upper-bounds the true information flow induced by a given prompt under local regularity assumptions. We evaluate GIF on integrity and confidentiality tasks across multiple prompt-injection and privacy-leakage benchmarks. GIF achieves near-perfect recall even without a downstream declassifier, outperforming attention-based baselines. Combined with lightweight LLM-based declassifiers, it matches or exceeds the F1 of direct LLM-as-judge baselines such as GPT-5.5 xhigh reasoning while using up to 81x lower token cost. GIF flows detected with small surrogate models transfer to larger state-of-the-art models and other model families, even when the surrogate is up to 200x smaller, suggesting black-box deployment without gradient access.","abs_url":"https://arxiv.org/abs/2606.23277","pdf_url":"https://arxiv.org/pdf/2606.23277v1","primary_category":"cs.AI","published":"2026-06-22","tags":["verification","ml-theory"],"lean_note":"Provides a fully mechanized Lean 4 proof that the geometric information-flow measure upper-bounds true information flow under local regularity.","thumb":"thumbs/2606.23277.jpg"},{"id":"2606.22997","title":"A Greatest Common Divisor Criterion of Certain Binomial Coefficients","authors":["Dakai Guo","Ruichen Qiu","Yichuan Cao","Ruyong Feng","Xiao-Shan Gao"],"abstract":"The binomial greatest common divisor (gcd) criterion recorded as OEIS A080170 is proven. The criterion also appears as conjecture (17) in Ralf Stephan's list of OEIS conjectures. For $k\\geq 2$, put \\[ D(k)=\\gcd_{2\\leq q\\leq k+1}\\binom{qk}{k}, \\qquad n=k+1. \\] If $P$ is the largest prime-power component $p^a$ exactly dividing $n$, then the criterion asserts \\[ D(k)=1 \\quad\\Longleftrightarrow\\quad \\frac{n}{P}>P. \\] The proof is formalized in Lean and the Lean artifact is accepted as part of the Formal Conjectures project. Both the natural-language proof and the Lean formalization are generated by the MechMath Agent Team, an AI agent developed by the authors.","abs_url":"https://arxiv.org/abs/2606.22997","pdf_url":"https://arxiv.org/pdf/2606.22997v1","primary_category":"math.NT","published":"2026-06-22","tags":["formalization","number-theory","ai-agents"],"lean_note":"Formalizes in Lean the proof of the OEIS A080170 binomial gcd criterion, accepted into the Formal Conjectures project.","repo":"https://github.com/guodk/formal-conjectures","thumb":"thumbs/2606.22997.jpg"},{"id":"2606.26141","title":"Classifying the Groups of Order $p^3$ in Lean","authors":["Li Xiang"],"abstract":"This note discusses our formalisation in Lean 4 of the classification of groups of order $p^3$ for a prime number $p$, using mathlib4. We present the five isomorphism classes and give a detailed account of the formalisation, with particular emphasis on the non-abelian case, which requiring the most substantial formal development. For odd~$p$, the non-abelian groups are the Heisenberg group $\\Heis(\\Z/p\\Z)$ and the semidirect product $\\Z/p^2\\Z\\rtimes\\Z/p\\Z$; for $p=2$, they are $D_4$ and $Q_8$. We describe the construction of these concrete groups, the structural lemmas about centers, commutators, and exponents, and the explicit isomorphism constructions that classify an arbitrary non-abelian $p^3$-group.","abs_url":"https://arxiv.org/abs/2606.26141","pdf_url":"https://arxiv.org/pdf/2606.26141v1","primary_category":"math.HO","published":"2026-06-20","tags":["formalization","mathlib","algebra"],"lean_note":"Formalizes in Lean 4 with Mathlib the classification of groups of order p^3 into five isomorphism classes.","repo":"https://github.com/lixiang90/p3group","thumb":"thumbs/2606.26141.jpg"},{"id":"2606.21867","title":"ForEx: A Formal Verification Framework for Explainable Reasoning in Logical Fallacy Detection and Annotation","authors":["Pei-Cing Huang","Chienyu Liu","Chan Hsu","Ci-Siang Chen","Pei-Ju Lee","Yihuang Kang"],"abstract":"Current evaluations of Large Language Models (LLMs) on logical fallacy detection focus on predicted labels, but do not establish whether those labels are supported by the reasoning the models provide. We propose ForEx (Formal Verification for Explainable Reasoning), a framework that translates LLM-generated explanations into Lean4 and verifies whether the translated rationale is derivable under encoded premises, not the logical validity of the original natural language argument. To distinguish prediction outcomes from the formal status of the supporting reasoning, we introduce the LLM Argument Verification Matrix, which separates label consistency from formal verification status. Experiments on LOGIC-Climate show that over 90% of LLM outputs can be translated into formal reasoning chains that pass verification, while agreement with human annotations remains around 20%. These results expose a systematic gap between formal derivability and label agreement, a distinction invisible to prediction-based metrics. ForEx moves LLM evaluation beyond label correctness toward machine-checkable analysis of formalized reasoning chains.","abs_url":"https://arxiv.org/abs/2606.21867","pdf_url":"https://arxiv.org/pdf/2606.21867v1","primary_category":"cs.AI","published":"2026-06-20","tags":["autoformalization","verification","logic"],"lean_note":"Translates LLM-generated fallacy-detection explanations into Lean4 and checks whether the rationale is derivable from encoded premises.","thumb":"thumbs/2606.21867.jpg"},{"id":"2606.21583","title":"Short Second Proof of the Odd-Modulus Directed Torus Hamilton Decomposition Theorem","authors":["SangHyun Park"],"abstract":"Let $D_d(m)=\\operatorname{Cay}((\\mathbb Z/m\\mathbb Z)^d,\\{e_1,\\ldots,e_d\\})$, with all generators oriented positively. We give a second proof that $D_d(m)$ decomposes into $d$ directed Hamilton cycles for every $d\\ge 2$ and every odd $m\\ge 3$. The combinatorial core is a fixed-row-sum selection theorem for replicated supports: when each indexed support $A$ is repeated in $m$ identical rows, one can select $\\lfloor |A|/2\\rfloor$ entries from each row so that every column total is a unit modulo $m$. Applied to the Hamilton factors using a chosen coordinate direction, these selections prescribe the voltages in a cyclic lift that splits the direction into two. In fibre coordinates, the lifted successor is $\\widehat h_j(x,z)=(h_j(x),z+\\mathbf 1_{\\{j\\in M(x)\\}})$. After one traversal of the base Hamilton cycle, the fibre return is translation by the total carry. Since this carry is a unit modulo $m$, the return is a single $m$-cycle and the lifted factor is Hamilton. The new fibres also preserve the direction-constant block structure required for the next split. Iterating from a directed $m$-cycle with $d$ parallel copies of each arc yields the desired decomposition. The proof strategy was proposed with the assistance of OpenAI GPT-5.5 Pro and formally verified in Lean 4.","abs_url":"https://arxiv.org/abs/2606.21583","pdf_url":"https://arxiv.org/pdf/2606.21583v1","primary_category":"math.CO","published":"2026-06-19","tags":["formalization","combinatorics"],"lean_note":"Formally verifies in Lean 4 a second proof of the odd-modulus directed torus Hamilton decomposition theorem.","repo":"https://github.com/aria1th/Torus-Hamilton-Decomposition-Program","thumb":"thumbs/2606.21583.jpg"},{"id":"2606.20943","title":"Quantum Dust from the Curse of Dimensionality","authors":["Kenan Oggad"],"abstract":"Why do unrelated approaches to quantum gravity nearly all find spacetime two-dimensional at the shortest scales? Each theory answers only within its own dynamics; we highlight a single kinematic route to the same value, one assuming no field equation and living in the geometry of the space of states alone. That route is concentration of measure on the Fubini-Study geometry of pure states, which forces the pairwise distances of a random sample to equalize as the dimension grows, so any finite sample collapses to an equidistant dust whose thresholded metric graph is the complete graph. Handed this dust, a diffusion probe reads it as two-dimensional in the large-sample limit, the value the running spectral dimension takes at the dust's single relaxation scale, a property of the measurement rather than the structure; this convergence on two is not, by itself, evidence that spacetime is two-dimensional. Whether a given two is such an artifact is governed by the Laplacian spectrum near zero, and whether that reading carries across an emergence map is the condition we call spectral faithfulness; a single relaxation scale encodes no spectral dimension that tells one structure from another. The collapse, the probe value, and the eigenvalue-density criterion are machine-checked in Lean 4 against Mathlib, resting on the standard Beta law of overlaps; a power-law tail of small eigenvalues reads a genuine dimension, a single scale above a gap reads two at its own clock, and a gapped two-scale band reads off the universal line. These classes are run on graph-Laplacian proxies, and whether a link-graph reading carries to the physical nonlocal operator is left open. The spectral test reads the eigenvalue density near zero and separates, on a given structure, a measurement artifact from a dimension the structure genuinely expresses.","abs_url":"https://arxiv.org/abs/2606.20943","pdf_url":"https://arxiv.org/pdf/2606.20943v1","primary_category":"gr-qc","published":"2026-06-18","tags":["formalization","mathlib","physics"],"lean_note":"Machine-checks in Lean 4 against Mathlib the measure-concentration collapse, the diffusion-probe value, and the eigenvalue-density criterion.","thumb":"thumbs/2606.20943.jpg"},{"id":"2606.20439","title":"Four-digit Kaprekar dynamics in odd bases","authors":["Evan Chen","Ken Ono","Richard E. Schwartz","Dinesh S. Thakur"],"abstract":"Start with four digits, arrange them in both descending and ascending order, subtract, and repeat. This simple process is known as the Kaprekar routine, famous in base ten for sending every nonconstant four-digit string to $6174$. We show that in every odd base $B>3$, the four-digit Kaprekar map has an unexpectedly rigid structure. After at most three iterations, every nonconstant orbit enters an explicit triangular region $\\mathcal{T}_B$, and on this region the map is conjugate to projective doubling: \\[ \\{[r],[s]\\}\\longmapsto \\{[2r],[2s]\\}. \\] This gives a complete finite description of all nonconstant terminal cycles, including an explicit formula for their lengths and counts. In particular, the longest terminal cycle has length at most $(B-1)/2$, and equality can occur only when $B$ is prime. For primes $p>5$, equality occurs precisely when the least positive $m$ with $2^m\\equiv\\pm1\\pmod p$ is $m=(p-1)/2$. The results proved here were first formulated by Schwartz and Thakur. As a test case for AI-assisted formal mathematics, AxiomProver produced Lean/mathlib formalizations of these results.","abs_url":"https://arxiv.org/abs/2606.20439","pdf_url":"https://arxiv.org/pdf/2606.20439v1","primary_category":"math.NT","published":"2026-06-18","tags":["formalization","theorem-proving-ml","number-theory","ai-agents"],"lean_note":"Formalizes its odd-base four-digit Kaprekar dynamics theorems in Lean 4 with Mathlib, generated autonomously by the AxiomProver system from statements alone.","repo":"https://github.com/AxiomMath/kaprekar4","thumb":"thumbs/2606.20439.jpg"},{"id":"2606.20358","title":"Formalizing Extended Complex Numbers, Mobius Transformations, and Cross Ratio in Lean 4","authors":["Fubin Yan","Kenneth W. Shum"],"abstract":"The extended complex plane is a fundamental object in complex analysis, hyperbolic geometry, and mathematical physics. Its geometry is governed by Möbius transformations, with the cross ratio serving as a central invariant. We present a formalization of these concepts in the Lean4 theorem prover. The extended complex plane is represented using Mathlib's Option type over $\\mathbb{C}$, where the additional element represents the point at infinity. On this foundation, we define Möbius transformations, their action on the extended complex plane, and the cross ratio. We formalize several basic properties of Möbius transformations, including their group structure, and identify them with a projective general linear group. We also prove the uniqueness of a Möbius transformation mapping any three distinct points to any other three distinct points, and the invariance of the cross ratio. All proofs are machine-checked in Lean 4. The complete development comprises approximately 6,000 lines of Lean code, including about 40 definitions and 150 lemmas and theorems. This work provides a verified foundation for future formalizations of conformal geometry, hyperbolic models, modular forms, and applications in mathematical physics.","abs_url":"https://arxiv.org/abs/2606.20358","pdf_url":"https://arxiv.org/pdf/2606.20358v2","primary_category":"math.CV","published":"2026-06-18","tags":["formalization","analysis","geometry"],"lean_note":"Formalizes the extended complex plane, Mobius transformations, and the cross ratio in Lean 4 using Mathlib.","repo":"https://github.com/fubinyan/ComplexVariables","thumb":"thumbs/2606.20358.jpg"},{"id":"2606.20121","title":"BARReL: a modern backend for Atelier B in Lean","authors":["Ghilain Bergeron","Vincent Trélat"],"abstract":"BARReL is a Lean 4 library bridging Atelier B, an industrial tool for the B method, and the Lean proof assistant by enabling users to conduct their formal B developments -- up to machine refinement and implementation -- interactively inside Lean, while retaining standard B syntax. B partial operators are carefully encoded by generating explicit well-definedness conditions, leveraging Lean's dependent types to enforce a well-definedness discipline by construction. That is, proof obligations and proof steps cannot silently rely on ill-typed or ill-defined instantiations. BARReL also features basic automation to try to discharge such well-definedness conditions automatically. The implementation is written entirely using Lean meta-programming and is designed to be modular: extending the supported B fragment typically requires only adding new syntax and encoding clauses. We illustrate the approach on a small but representative case study, and argue that BARReL can act as a stepping stone towards a strongly reliable Atelier B toolchain grounded in the Lean proof assistant.","abs_url":"https://arxiv.org/abs/2606.20121","pdf_url":"https://arxiv.org/pdf/2606.20121v1","primary_category":"cs.LO","published":"2026-06-18","tags":["tooling","verification"],"lean_note":"Implements BARReL, a Lean 4 library embedding Atelier B's B method with well-definedness conditions enforced through dependent types.","thumb":"thumbs/2606.20121.jpg"},{"id":"2606.20068","title":"Process-Verified Reinforcement Learning for Theorem Proving via Lean","authors":["Minsu Kim","Se-Young Yun"],"abstract":"While reinforcement learning from verifiable rewards (RLVR) typically has relied on a single binary verification signal, symbolic proof assistants in formal reasoning offer rich, fine-grained structured feedback. This gap between structured processes and unstructured rewards highlights the importance of feedback that is both dense and sound. In this work, we demonstrate that the Lean proof assistant itself can serve as a symbolic process oracle, supplying both outcome-level and fine-grained tactic-level verified feedback during training. Proof attempts are parsed into tactic sequences, and Lean's elaboration marks both locally sound steps and the earliest failing step, yielding dense, verifier-grounded credit signals rooted in type theory. We incorporate these structured rewards into a GRPO-style reinforcement learning objective with first-error propagation and first-token credit methods that balances outcome- and process-level advantages. Experiments with STP-Lean and DeepSeek-Prover-V1.5 show that tactic-level supervision outperforms outcome-only baselines in most settings, delivering improvements on benchmarks such as MiniF2F and ProofNet. Beyond empirical gains, our study highlights a broader perspective: symbolic proof assistants are not only verifiers at evaluation time, but can also act as process-level reward oracles during training. This opens a path toward reinforcement learning frameworks that combine the scalability of language models with the reliability of symbolic verification for formal reasoning.","abs_url":"https://arxiv.org/abs/2606.20068","pdf_url":"https://arxiv.org/pdf/2606.20068v1","primary_category":"cs.AI","published":"2026-06-18","tags":["theorem-proving-ml"],"lean_note":"Uses Lean's elaboration as a process oracle, parsing proofs into tactics to supply dense tactic-level RL rewards on MiniF2F and ProofNet.","thumb":"thumbs/2606.20068.jpg"},{"id":"2606.19936","title":"Prismriver: Formalization of Music Theory and Algorithmic Composition in Lean 4","authors":["Leni Aniva","Claire Wang"],"abstract":"Music theory obeys a rich set of mathematical rules and symmetries. These symmetries follow mathematical structure which can be verified and expressioned in the precise language of a proof assistant. In this paper, we present Prismriver, a formalization of music theory in Lean 4. By formalizing music theory in Lean 4, we open the door to verifiable algorithmic composition and accompaniment generation. We also enable the analysis of monadic analysis of structures in music.","abs_url":"https://arxiv.org/abs/2606.19936","pdf_url":"https://arxiv.org/pdf/2606.19936v1","primary_category":"cs.LO","published":"2026-06-18","tags":["formalization","tooling","mathlib"],"lean_note":"Presents Prismriver, a Lean 4 library formalizing music theory with Mathlib group actions and supporting verifiable algorithmic composition.","repo":"https://codeberg.org/aniva/Prismriver","thumb":"thumbs/2606.19936.jpg"},{"id":"2606.19761","title":"Finishing Oltean's Completeness Proof in Lean 4 for Hybrid Logic $L(\\forall)$","authors":["Lars Warren Ericson"],"abstract":"We present a machine-checked completeness theorem, in Lean 4, for the hybrid logic $L(\\forall)$: propositional modal logic with nominals, the satisfaction-style binder $\\forall$, and the box modality. (Machine-checked completeness for basic hybrid logic, without binders, was pioneered by Asta Halkjær From in Isabelle/HOL.) We build on Alex Oltean's 2023 Lean 4 formalization, which mechanized the syntax, semantics, Hilbert-style proof system, and soundness following Blackburn's Hybrid Completeness (1998), but left completeness unfinished. Finishing it requires manufacturing fresh names at two structurally different points, and our central finding is that they call for two different tools. (1) The root witnessed maximal consistent set, built by an extended Lindenbaum construction, needs at each step a nominal fresh for the whole set; the right tool is structural freshness: extend the language so an infinite supply of nominals is reserved by construction. We survey the design space (Oltean's odd/even encoding inside $\\mathbb{N}$, the disjoint-sum $N \\oplus \\mathbb{N}$ parameterization suggested by Bud Mishra, and From's synthetic-completeness frameworks) and explain the encoding we adopt. (2) The witnessed $\\Diamond$-successor of a maximal consistent set cannot be obtained this way: its canonical box-reduct provably mentions every nominal, so no reserved name is fresh. Here the right tool is one Oltean chose but left incomplete: an existence-lemma Henkin construction drawing each witness from the predecessor's witnessedness through a fresh state variable; we complete it with a data-carrying witness accumulator and a compactness argument. The theorem $Γ\\models \\varphi \\to Γ\\vdash \\varphi$ is fully formalized: the development is sorry-free, and #print axioms reports only propext, Classical.choice, and Quot.sound. We port the development to Lean v4.30.0 / mathlib v4.30.0.","abs_url":"https://arxiv.org/abs/2606.19761","pdf_url":"https://arxiv.org/pdf/2606.19761v1","primary_category":"cs.LO","published":"2026-06-18","tags":["formalization","logic"],"lean_note":"Completes a machine-checked completeness theorem for hybrid logic L(forall) in Lean 4, building on Oltean's formalization.","repo":"https://github.com/catskillsresearch/hybrid_logic_lean_revisited","thumb":"thumbs/2606.19761.jpg"},{"id":"2606.18557","title":"DeFAb: A Verifiable Benchmark for Defeasible Abduction in Foundation Models","authors":["Patrick Cooper","Alvaro Velasquez"],"abstract":"A rule-based logic solver resolves every instance in our benchmark in under 50 microseconds with 100% accuracy; the best frontier language model reaches 65% at best and drops to 23.5% under rendering-robust evaluation (worst case over four surface renderings). We introduce DeFAb (Defeasible Abduction Benchmark), a dataset and generation pipeline that converts four decades of publicly funded knowledge bases into formally grounded instances for defeasible abduction: constructing hypotheses that explain anomalies by overriding defaults while preserving unrelated expectations. Because every hypothesis must pass polynomial-time checks for valid derivation, conservativity, and minimality, DeFAb makes logical rigor the instrument for measuring creativity and theoretical reasoning, scoring the disciplined construction of theory revisions rather than fluent but theory-destroying prose. The pipeline pairs taxonomic hierarchies (OpenCyc, YAGO, Wikidata) with behavioral property graphs (ConceptNet, UMLS) to produce 372,648+ instances across 33.75M materialized rules from 18 sources, in three levels with polynomial-time verifiable gold standards. Four frontier models do not reliably internalize defeasible reasoning: rendering-robust Level 2 accuracy is 7.8-23.5%; chain-of-thought variance (~36 pp) exceeds any inter-model gap; and a matched contamination control isolates a +19.4 pp Level 3 gap. We further release DeFAb-Hard (a 235-instance Level 3 difficulty variant; best model 53.3% vs 100% symbolic) and CONJURE (a kernel-verified transformative-creativity variant of 560 Lean 4/Mathlib instances whose gold answers are definitions the proof kernel did not previously contain, judge-free verifier; a pilot finds zero novel concepts). The same verifier doubles as an exact reward for preference optimization (DPO, RLVR/GRPO). Released under MIT at https://huggingface.co/datasets/PatrickAllenCooper/DeFAb.","abs_url":"https://arxiv.org/abs/2606.18557","pdf_url":"https://arxiv.org/pdf/2606.18557v1","primary_category":"cs.AI","published":"2026-06-17","tags":["benchmark","dataset","logic"],"lean_note":"The CONJURE track uses Lean 4/Mathlib as a kernel verifier for 560 transformative-creativity instances whose gold answers are new definitions.","repo":"https://github.com/PatrickAllenCooper/blanc","thumb":"thumbs/2606.18557.jpg"},{"id":"2606.18462","title":"Exact 6-cut rigidity and small-order superconnectivity for the 6-regular case of Dirac's k=4 problem","authors":["Alper Ferudun"],"abstract":"Dirac asked in 1970 whether for every k >= 4 there is a k-vertex-critical graph without critical edges; Jensen settled all k >= 5, and only k=4 remains open. Following Skottova and Steiner, call a graph G a (4,1)-graph if chi(G)=4, chi(G-v)=3 for every vertex v, and chi(G-e)=4 for every edge e; they proved delta(G) >= 6 and lambda(G) >= 6 for every (4,1)-graph and asked whether a 6-regular (4,1)-graph exists. We prove three results about this 6-regular case. Theorem A (computational): there is no 6-regular 4-vertex-critical graph on n <= 15 vertices, except for a unique graph (up to isomorphism) on n=13, whose 13 critical edges form a Hamilton cycle; hence any 6-regular (4,1)-graph has at least 16 vertices. Theorem B: in a 6-regular (4,1)-graph every 6-edge-cut is either the edge star of a vertex or has both shores of size at least 15; consequently every 6-regular (4,1)-graph on at most 29 vertices is super-6-edge-connected. Theorem C (all sizes): no shore of a nontrivial 6-edge-cut in a 6-regular (4,1)-graph induces a bipartite graph; more generally, a shore whose deficiency is concentrated on two vertices forces them to receive equal colours in every proper 3-colouring. The proof of Theorem B rests on an exact classification of the 3x3 cut matrices of 6-edge-cuts in (4,1)-graphs (exactly 21 matrices, five types up to row/column permutations) together with a boundary-shortfall lemma; the unique near-miss is K_{3,3,3} minus a rainbow 3-matching. Several supporting lemmas are machine-checked in Lean 4/Mathlib.","abs_url":"https://arxiv.org/abs/2606.18462","pdf_url":"https://arxiv.org/pdf/2606.18462v1","primary_category":"math.CO","published":"2026-06-16","tags":["formalization","combinatorics"],"lean_note":"Machine-checks several supporting local-colouring lemmas in Lean 4/Mathlib for the 6-regular case of Dirac's k=4 problem.","thumb":"thumbs/2606.18462.jpg"},{"id":"2606.17581","title":"Visored: A Controlled-Natural-Language Prover for LLM-Generated Mathematics","authors":["Xiyu Zhai","Xinyi Chen","Yiping Wang","Runlong Zhou","Liao Zhang","Simon S. Du"],"abstract":"We present a dependent-type-based prover designed around the way LLMs (and humans) tend to write mathematics, complementing existing systems such as Lean and Rocq. Its core design choices are a surface that imitates mathematical natural language and a rule-driven automation layer that closes the routine steps a textbook would omit, so that an accepted proof can be re-emitted as a checked Lean file. Early experiments suggest that, even without any prover-specific training data, LLMs can learn to use it effectively on the miniF2F benchmark. Lean output excerpts: https://github.com/xiyuzhai-husky-lang/visored/","abs_url":"https://arxiv.org/abs/2606.17581","pdf_url":"https://arxiv.org/pdf/2606.17581v1","primary_category":"cs.PL","published":"2026-06-16","tags":["autoformalization","theorem-proving-ml","tooling"],"lean_note":"A controlled-natural-language prover whose accepted proofs are re-emitted as checked Lean files, driven by an LLM on miniF2F.","repo":"https://github.com/xiyuzhai-husky-lang/visored","thumb":"thumbs/2606.17581.jpg"},{"id":"2606.18292","title":"A Formalization of Austrian Economics. Praxeological Foundations: The Base System and Its Derived Theorems","authors":["Rafał Komendarczyk","Walter Block","John Levendis","Frank Tipler"],"abstract":"This paper presents an axiomatization of Ludwig von Mises' praxeology in many-sorted first-order logic, isolating the foundational layer. We introduce a formal language with five sorts ({\\sf Actors}, {\\sf Actions}, {\\sf Ends}, {\\sf Things}, {\\sf Times}) and six primitive relations ({\\em Acts}, {\\em Avail}, {\\em EndOf}, {\\em Use}, a preference order, and a time order), together with a base axiom system organised into three layers: the structure of action itself, the actor's preference order together with its revelation in choice, and material scarcity. The base system captures purposeful action in its bare praxeological form. Working entirely within the base system we derive the core classical Misesian propositions as Hilbert-style theorems: the asymmetry of revealed preference, the existence of opportunity cost, the structural scarcity of time, the subjectivity of opportunity cost, the law of diminishing marginal utility, and the increasing marginal disutility of labour. Where a theorem requires structure beyond the praxeological core -- as with diminishing marginal utility -- the additional premises are made explicit; identifying these hidden premises is one of the methodological payoffs of the approach. A self-contained {\\em Lean} companion encodes the language as {\\em Lean} type classes and constructs concrete models -- a three-period Robinson Crusoe economy and its infinite-time extension -- whose acceptance by the type-checker is a constructive consistency proof of the full base theory.","abs_url":"https://arxiv.org/abs/2606.18292","pdf_url":"https://arxiv.org/pdf/2606.18292v1","primary_category":"econ.TH","published":"2026-06-15","tags":["formalization","logic","economics"],"lean_note":"A Lean companion encodes the praxeology axioms as type classes and constructs concrete models whose type-checking is a constructive consistency proof.","repo":"https://github.com/rafkom72/praxeology-lean","thumb":"thumbs/2606.18292.jpg"},{"id":"2606.16893","title":"Symbolic Informalization: Fluent, Productive, Multilingual","authors":["Aarne Ranta"],"abstract":"Symbolic informalization enables a reliable conversion of formal mathematics to natural language. It has the potential to make machine-checked content human-readable without loss of precision. In a traditional proof system usage, symbolic informalization generalizes the limited mechanisms of syntactic sugar into the ordinary language of mathematics. In a setting where proofs are constructed by artificial intelligence and autoformalization, symbolic informalization can explain what precisely has been constructed. This paper outlines the project Informath, which aims to show how symbolic informalization can produce fluent text with a reasonable development effort and address multiple formal and natural languages. Informath is based on an interlingual architecture, where Dedukti works as a hub between different proof systems (Agda, Lean, Rocq) and Grammatical Framework (GF) takes care of linguistic correctness and variation in different natural languages.","abs_url":"https://arxiv.org/abs/2606.16893","pdf_url":"https://arxiv.org/pdf/2606.16893v1","primary_category":"cs.AI","published":"2026-06-15","tags":["tooling","autoformalization"],"lean_note":"Treats Lean as one source proof system in an interlingual formal-to-natural-language informalization pipeline built on Dedukti.","repo":"https://github.com/GrammaticalFramework/informath","thumb":"thumbs/2606.16893.jpg"},{"id":"2606.16688","title":"A Lean-Certified Proof of $K_8(4, 2) = 23$","authors":["Andreas Florath"],"abstract":"We prove the exact octonary covering-code value $K_8(4, 2) = 23$ in Lean 4. The upper bound is given by an explicit 23-word radius-two code in $(Fin\\:8)^4$ , checked over all $8^4$ ambient words. The lower bound excludes covers with at most 22 words. A fiber-counting and missing-pair argument first rules out covers with at most 21 words. In the remaining 22-word case, the proof reduces a hypothetical cover to six missing-pair graphs coming from the coordinate-pair projections. Fiber-counting arguments constrain these graphs, and two Lean-checked Linear RAT (LRAT) refutations of stored conjunctive-normal-form (CNF) instances force a common 3 + 3 + 2 block structure. This structure is incompatible with a 22-word cover: the two three-symbol components already force 18 codewords, while the remaining two-symbol component would require a binary strength-two array of length four with at most four rows, which is impossible. The result is packaged as a proof-carrying Lean artifact: the explicit upper bound, structural lower bound, CNF instances, and LRAT refutations are checked inside Lean, with no external SAT solver used during proof replay.","abs_url":"https://arxiv.org/abs/2606.16688","pdf_url":"https://arxiv.org/pdf/2606.16688v1","primary_category":"cs.IT","published":"2026-06-15","tags":["formalization","combinatorics"],"lean_note":"Proves the octonary covering-code value K_8(4,2)=23 in Lean 4 with Lean-checked LRAT refutations.","repo":"https://github.com/florath/covering-codes-lean","thumb":"thumbs/2606.16688.jpg"},{"id":"2606.16679","title":"Formalizing chip-firing and Riemann--Roch for graphs in Lean 4","authors":["Dhyey Dharmendrakumar Mavani","Nathan Pflueger"],"abstract":"The Riemann--Roch theorem for graphs, due to Baker and Norine, is a foundational result establishing a powerful analogy between finite graphs and algebraic curves. We describe a complete formal proof of this theorem implemented in the Lean 4 theorem prover. Our formalization includes the existence and uniqueness of q-reduced divisors, a modified form of Dhar's burning algorithm, the bijection between acyclic orientations with unique source and maximal superstable configurations, and Clifford's theorem. We also include several challenges for future formalization.","abs_url":"https://arxiv.org/abs/2606.16679","pdf_url":"https://arxiv.org/pdf/2606.16679v1","primary_category":"math.CO","published":"2026-06-15","tags":["formalization","combinatorics"],"lean_note":"Formalizes the Baker-Norine Riemann-Roch theorem for graphs in Lean 4.","repo":"https://github.com/DhyeyMavani2003/chip-firing-with-lean","thumb":"thumbs/2606.16679.jpg"},{"id":"2606.16541","title":"The Faithfulness Gap: Certifying Semantic Equivalence Between Natural-Language and Formal Mathematical Statements","authors":["Noor Islam S. Mohammad","Tamim Sheikh"],"abstract":"Autoformalization, translating natural-language mathematics into formal proof assistants, is bottlenecked not by translation fluency but by \\emph{faithfulness}: a formal statement can typecheck and be provable, yet still encode a different theorem than the source intended. We introduce \\emph{Bidirectional Provability Fingerprinting} (\\bpf{}), a framework that certifies faithfulness by characterizing each candidate through its forward and backward consequence neighborhoods in the ambient theory and matching these against probes derived from the natural-language statement. We further introduce four novel components: (i) \\emph{Counterfactual Probe Generation} (\\cpg{}), a contrastive procedure that synthesizes probes targeting specific drift directions; (ii) the \\emph{Equivalence Spectrum}, a continuous faithfulness score that replaces brittle binary verdicts; (iii) \\emph{Adaptive Probe Budget Allocation} (\\apba{}), an information-theoretic budget router; and (iv) \\emph{Faithfulness-Guided Decoding} (\\fgd{}), which uses \\bpf{} signals as a reward during autoformalization. We prove a \\emph{drift detection theorem} and a \\emph{PAC-faithfulness} result establishing that the equivalence class of a natural language statement is learnable from $\\mathcal{O}(\\log(1/δ)/\\varepsilon)$ probes under mild assumptions. We release \\driftbench{}, a benchmark of $2{,}183$ NL/Lean~4 pairs with controlled drift labels across six subfields of mathlib4. \\bpf{}\\,+\\,\\cpg{} detects $89.6\\%$ of drifted formalizations at a $3.0\\%$ false-positive rate-against $41.2\\%$ for typecheck and $63.3\\%$ for LLM-judge baselines, and \\fgd{} reduces the rate at which a state-of-the-art autoformalizer emits drifted statements by $47\\%$. https://pmlrbd.github.io/BPF/","abs_url":"https://arxiv.org/abs/2606.16541","pdf_url":"https://arxiv.org/pdf/2606.16541v1","primary_category":"cs.AI","published":"2026-06-15","tags":["autoformalization","benchmark"],"lean_note":"Releases DriftBench, 2,183 NL/Lean 4 pairs over six mathlib4 subfields, to certify autoformalization faithfulness.","thumb":"thumbs/2606.16541.jpg"},{"id":"2606.16289","title":"Moment-Free Kunchenko Stochastic Polynomials via Empirical Characteristic Function","authors":["Serhii Zabolotnii"],"abstract":"We give a characteristic-function formulation of Kunchenko's stochastic-polynomial construction for settings in which raw moments may fail to exist. In the finite-variance trigonometric case, the coefficients of the Kunchenko normal system are expressed through the characteristic function and its derivative. In the moment-free case, empirical characteristic functions on a fixed finite frequency grid define a bounded discrepancy geometry that remains meaningful for Cauchy, symmetric stable, and other heavy-tailed laws. We prove well-definedness and finite-grid almost sure consistency of this empirical characteristic-function geometry. We introduce the associated minimum-CF-distance estimator and establish its identifiability, strong consistency, and asymptotic normality on a fixed grid, with a covariance built from bounded trigonometric moments that stays finite even for Cauchy and stable laws; refining the grid increases the optimal-weight information monotonically to the Fisher information, so the estimator is asymptotically efficient in the dense-grid limit. We also relate bounded sine scores to weak stochastic-polynomial estimating equations. A small Lean 4 / Mathlib supplement checks selected deterministic identities underlying the bounded-score construction; convergence arguments and statistical interpretation remain outside the formalization.","abs_url":"https://arxiv.org/abs/2606.16289","pdf_url":"https://arxiv.org/pdf/2606.16289v1","primary_category":"stat.ME","published":"2026-06-15","tags":["formalization","probability"],"lean_note":"A Lean 4/Mathlib supplement machine-checks selected deterministic identities underlying the bounded-score estimator.","repo":"https://github.com/SZabolotnii/Ku_CF-code-supplement","thumb":"thumbs/2606.16289.jpg"},{"id":"2606.16239","title":"Thakur's hypotheses on power sums of $\\mathbb{F}_q[t]$","authors":["Evan Chen","Ken Ono"],"abstract":"In his 2009 paper, Thakur posed three conjectural hypotheses for the degrees of the power sums \\[ S_d(k)=\\sum_{\\substack{a\\in \\mathbb F_q[t] \\text{ monic}\\\\ °a=d}} a^{-k} \\qquad\\text{and}\\qquad s_d(k)=-°_t S_d(k). \\] For prime fields $q=p$, we prove Hypotheses H1 and H2, giving a unique greedy description of the extremal term in Carlitz's formula and establishing the recursion \\[ s_d(k)=s_{d-1}(s_1(k))+s_1(k). \\] As consequences, the prime-field recursion gives the strict Newton-polygon convexity used in the prime-field Carlitz-Goss Riemann-hypothesis theorem, and it recovers Thakur's nonvanishing theorem for positive multizeta values over $\\mathbb F_p[t]$. We also prove Hypothesis H3 for all finite fields $q=p^f$, establishing the monotonicity \\[ s_d(k)<s_d(k+1)\\qquad (p\\nmid k). \\] We provide Lean formalizations of the arguments in this paper, generated by AxiomProver.","abs_url":"https://arxiv.org/abs/2606.16239","pdf_url":"https://arxiv.org/pdf/2606.16239v1","primary_category":"math.NT","published":"2026-06-15","tags":["formalization","number-theory"],"lean_note":"Provides Lean formalizations of the paper's arguments, generated by AxiomProver.","repo":"https://github.com/AxiomMath/zeta-h123","thumb":"thumbs/2606.16239.jpg"},{"id":"2606.16144","title":"EconCSLib: A Lean Library for Computational Economics and AI-Assisted Research","authors":["Xiaohui Bei","Jiajun Ma","Zhan Jing","Hongfei Fu","Zhihao Gavin Tang"],"abstract":"Mathematical formalization uses interactive theorem provers to turn informal mathematical statements into machine-checkable artifacts. The success of mathlib, a large collaborative library for Lean, illustrates the potential of this approach. Recent progress in AI-assisted programming and theorem proving is also making large-scale formalization more practical. This paper presents EconCSLib, an early Lean 4 library for computational economics, as both infrastructure and a case study for AI-assisted formalization. The library aims to provide reusable definitions and theorems for game theory, mechanism design, social choice, and related areas. Beyond verified proofs of existing results, the library also aims to host machine-checked open problems and formalization of modern research papers. We discuss the design principles behind the library, the lessons learned from its development, and future directions for AI-assisted formalization in computational economics.","abs_url":"https://arxiv.org/abs/2606.16144","pdf_url":"https://arxiv.org/pdf/2606.16144v1","primary_category":"cs.GT","published":"2026-06-15","tags":["tooling","formalization","economics"],"lean_note":"Presents EconCSLib, an early Lean 4 library of definitions and theorems for computational economics.","repo":"https://github.com/gametheoryinlean/EconCSLib","thumb":"thumbs/2606.16144.jpg"},{"id":"2606.16134","title":"Revisiting average case complexity of multilevel syllogistic: From the 1995 Courant Technical Report to Lean 4 Formalization","authors":["Lars Warren Ericson"],"abstract":"We describe a Lean~4 formalization revisiting NYU Courant Technical Report TR1995-711 on the average-case complexity of Multilevel Syllogistic (MLS). The development encodes Reischuk--Schindelhauer average-case classes, an axiomatic MLS/EMLS semantics layer, a partial Ferro--Omodeo--Schwartz decision procedure with proved soundness and partial completeness on a membership-free fragment, serialization and step budgets, and conditional NP-average completeness and non-AvP hardness corollaries modulo explicitly documented structural axioms. Full Lean sources are inlined in the appendix modules.","abs_url":"https://arxiv.org/abs/2606.16134","pdf_url":"https://arxiv.org/pdf/2606.16134v1","primary_category":"cs.LO","published":"2026-06-15","tags":["formalization","logic"],"lean_note":"Formalizes a 1995 average-case complexity result for Multilevel Syllogistic in Lean 4.","repo":"https://github.com/catskillsresearch/avg_case_mls","thumb":"thumbs/2606.16134.jpg"},{"id":"2606.15972","title":"Formalize Once, Edit the Rest: Efficient Lean-Based Answer Selection for Math Reasoning","authors":["Ji Feng","Zhouxing Shi"],"abstract":"With large language models (LLMs) increasingly applied to mathematical reasoning, formal proof assistants such as Lean can be leveraged to verify reasoning outputs with machine-checkable rigor, enabling use cases such as answer selection in test-time scaling with K sampled candidate answers. However, employing Lean requires that LLM outputs, originally in natural language, first be formalized. Existing Lean-based answer-selection work uses an autoformalization model to generate a formal statement in Lean for each candidate answer independently, incurring a significant computational cost. We propose BASE, a base-and-edit pipeline that formalizes a single base candidate per problem and derives the remaining K-1 statements by editing the answer expression in place. To facilitate this, we train a rewriter model LEANSCRIBE to localize the answer in the base formalization and generate a reusable edit function for the other K-1 candidates. BASE simultaneously improves selection accuracy and reduces formalization cost - a Pareto improvement that holds on all 12 (dataset, solver) configurations across four benchmarks and three solvers, cutting autoformalizer calls by about 5x at K=8, with the reduction expected to become larger as K grows. Code is available at https://github.com/ucr-rai/base-and-edit.","abs_url":"https://arxiv.org/abs/2606.15972","pdf_url":"https://arxiv.org/pdf/2606.15972v1","primary_category":"cs.CL","published":"2026-06-14","tags":["autoformalization","theorem-proving-ml"],"lean_note":"Uses Lean to verify formalized math-reasoning answers, editing one base formalization across candidates instead of formalizing each separately.","repo":"https://github.com/ucr-rai/base-and-edit","thumb":"thumbs/2606.15972.jpg"},{"id":"2606.15520","title":"A Lean 4 Formalization of Euclidean Domain Algorithms from a 1986 Icon Experimentation Package","authors":["Lars Warren Ericson"],"abstract":"We describe a Lean 4 formalization of the algorithms and domain types from NYU Computer Science Technical Report \\#232, \\emph{An ICON Package for Experimenting with Euclidean Domains} (Ericson, 1986). The original system implemented Lipson's catalog of procedures over integers, rationals, modular rings, polynomial rings, and truncated power series via a custom runtime dispatch mechanism in Icon. The present work separates three concerns: mathematical definitions grounded in Mathlib's \\texttt{EuclideanDomain} hierarchy, computable mirrors suitable for evaluation and regression testing, and report-formatting infrastructure that reproduces the 1986 benchmark output line-for-line. All fourteen application algorithms from Section 3 of the report are defined and typecheck without \\texttt{sorry}; those grounded in Mathlib -- chiefly integer gcd and extended Euclid -- additionally carry machine-checked proofs. We classify each procedure by its epistemic status relative to Mathlib, enumerate the coherence obligations between the proof and computable layers, and state precisely what is theorem-backed versus regression-trusted. The formalization makes explicit the verification boundary that the 1986 package crossed only informally.","abs_url":"https://arxiv.org/abs/2606.15520","pdf_url":"https://arxiv.org/pdf/2606.15520v1","primary_category":"cs.LO","published":"2026-06-14","tags":["formalization","mathlib","algebra"],"lean_note":"Formalizes all fourteen Euclidean-domain algorithms from a 1986 Icon package in Lean 4 atop Mathlib's EuclideanDomain hierarchy.","repo":"https://zenodo.org/records/20561267","thumb":"thumbs/2606.15520.jpg"},{"id":"2606.15159","title":"Every natural number is a sum of distinct semiprime unit fractions","authors":["Shisheng Li"],"abstract":"We prove that every natural number is a finite sum of distinct unit fractions whose denominators are semiprimes (products of two distinct primes). This is the $ω=2$ integer case of a problem of Erdős and Graham, stated only as a conjecture by Butler, Erdős and Graham (Integers 15 (2015), A51), who proved the $ω=3$ analogue. Counterintuitively the problem hardens as $ω$ decreases -- the induction's feed thins -- so $ω=2$ is the hard case; our proof adapts the Butler-Erdős-Graham induction to this thin-feed regime, where the entire content of the induction step reduces to an explicit onset inequality $Y_0(N)\\le\\min\\{β(N),β'(N)\\}$, proved for all $N\\ge10$ by Olson's addition theorem and elementary Chebyshev bounds above a finite, machine-checked base range. The same engine extends to the rationals: for every squarefree $b$, every $a/b$ above an explicit threshold $\\min\\{B_{N_b}/6,\\,1/5\\}$ is $ω=2$ representable, unconditionally. As an application we give the first complete proof of the rational $ω=3$ statement -- every $a/b$ with squarefree $b$ is a sum of distinct sphenic unit fractions -- that Butler, Erdős and Graham conjectured but left unpublished; a descent settles every $ω\\ge3$. What remains open is the $ω=2$ regime below this threshold, which we reduce to a single explicit conjecture -- that the gap-free floor of a semiprime subset-sum set tends to zero. This work is a human-AI collaboration: AI tools (notably Anthropic's Claude, used through Claude Code) contributed substantially to the Lean formalisation, the experiments, and the writing; correspondingly, every result is machine-checked in Lean 4 / Mathlib (no sorry; two cited classical axioms, plus the native_decide compiler-trust base for the finite computations), so its correctness is independent of the tools used.","abs_url":"https://arxiv.org/abs/2606.15159","pdf_url":"https://arxiv.org/pdf/2606.15159v2","primary_category":"math.NT","published":"2026-06-13","tags":["formalization","number-theory"],"lean_note":"Every result is machine-checked in Lean 4/Mathlib with no sorry, including the finite base-range computations.","thumb":"thumbs/2606.15159.jpg"},{"id":"2606.15089","title":"A Machine-Checked Itô Calculus for Brownian Motion","authors":["Raphael Coelho"],"abstract":"We present a machine-checked development of the $L^2$ Itô calculus of Brownian motion on a bounded time interval $[0,T]$, formalized in Lean 4 on top of Mathlib and the BrownianMotion package. The development contains: the construction of the Itô integral as an isometry of Hilbert spaces, from a predictable-rectangle $π$-system through the density of simple adapted processes; the Itô integral as a process, proved to be an $L^2$-continuous martingale through a single structural identity (the integral at time $t$ is the conditional-expectation projection of its terminal value onto $\\mathcal{F}t$), from which adaptedness, the martingale property, the contraction bound, and both the terminal and the time-indexed Itô isometries follow as corollaries; and Itô's formula for $C^3$ functions with bounded derivatives, including its time-dependent form $df = f_x,dB + (f_t + \\tfrac12 f{xx}),dt$, obtained by a discrete-to-continuous argument through weighted quadratic variation and explicit $L^2$ remainder bounds. To our knowledge this includes the first machine-checked proof of Itô's formula, and the first machine-checked construction of the Itô integral as a martingale-valued process, in any proof assistant. We are deliberate about the boundary: the theory is the $L^2$ theory on $[0,T]$ with bounded-derivative integrand classes; localization to the unrestricted $C^2$ formula, integrators beyond Brownian motion, and pathwise statements are out of scope, and we say precisely why and where. The development is roughly 7,200 lines of Lean across 22 modules; every theorem is sorry-free, the axioms of each headline result are pinned to Mathlib's classical defaults by a build-enforced gate, and the whole is reproducible from a pinned toolchain.","abs_url":"https://arxiv.org/abs/2606.15089","pdf_url":"https://arxiv.org/pdf/2606.15089v1","primary_category":"q-fin.MF","published":"2026-06-13","tags":["formalization","mathlib","probability","analysis"],"lean_note":"Formalizes the L^2 Itô calculus of Brownian motion in Lean 4 on Mathlib and the BrownianMotion package, including Itô's formula.","repo":"https://github.com/RemyDegenne/brownian-motion","thumb":"thumbs/2606.15089.jpg"},{"id":"2606.14867","title":"Evaluating the Robustness of Proof Autoformalization in Lean 4","authors":["Zhengtao Gui","Sheng Yang","Zhouxing Shi"],"abstract":"Proof autoformalization aims to translate a mathematical informal proof written in natural language into a formal proof in a formal language such as Lean~4. Several works have developed LLM-based models for proof autoformalization. However, existing evaluations have typically focused on translating well-formed informal proofs from curated datasets. We argue that a robust proof autoformalizer must remain faithful even for informal proofs that diverge from these idealized ones, and we present the first study on the robustness of proof autoformalization models. We formulate two categories of perturbations and evaluate robustness under each: a global perturbation paraphrases the informal proof in a different style, under which the formalization should remain consistent; a local perturbation alters a value, symbol, or proof step, possibly in a counterfactual way, and a robust formalization should faithfully reflect the perturbation rather than reverting to the original one or inferring a different one on its own. We build a benchmark with both perturbations on miniF2F and MATH-500, and automatically measure how stable a proof autoformalization's correctness is under global perturbations and how faithfully its output reflects local perturbations. We evaluate seven recent models, all of which are sensitive to global perturbations and mostly fail to remain faithful under local perturbations. Code and data are available via https://github.com/ucr-rai/robust-proof-autoformalization.","abs_url":"https://arxiv.org/abs/2606.14867","pdf_url":"https://arxiv.org/pdf/2606.14867v1","primary_category":"cs.CL","published":"2026-06-12","tags":["autoformalization","benchmark","theorem-proving-ml"],"lean_note":"Builds a benchmark measuring the robustness of LLM proof autoformalization into Lean 4 under global and local perturbations of informal proofs.","repo":"https://github.com/ucr-rai/robust-proof-autoformalization","thumb":"thumbs/2606.14867.jpg"},{"id":"2606.14000","title":"Formalizing Numerical Analysis: An Agent Pipeline and Quality Audit Beyond Kernel Acceptance","authors":["Theodore Meek","Siyuan Ge","Di Qiu Xiang","Simon Chess","Vasily Ilin"],"abstract":"Recent work has demonstrated that coding agents can formalize entire advanced mathematics textbooks in Lean 4, yet existing efforts concentrate on branches of mathematics already well-represented in mathlib and measure success solely through kernel acceptance. We address both limitations by applying a coding agent to formalize Numerical Methods for Ordinary Differential Equations, a textbook in numerical analysis that is largely absent from mathlib, stressing the agent's capacity to develop new theory from scratch. We further introduce a systematic, reproducible three-dimensional framework for evaluating the quality of agent-produced formalizations beyond compilation: semantic correctness, Mathlib reuse, and cross-file reuse via LLM-as-judge methods. Applying this framework to our own formalization and to the released outputs of RepoProver and M2F, we uncover recurring unfaithful formalization patterns, including incomplete multi-part statements, added weakening hypotheses, and parameter restrictions, that kernel acceptance entirely obscures. Our results suggest that compilation-based metrics substantially overstate formalization quality, and we provide a reproducible audit methodology to support more rigorous evaluation of future autoformalization systems.","abs_url":"https://arxiv.org/abs/2606.14000","pdf_url":"https://arxiv.org/pdf/2606.14000v1","primary_category":"cs.AI","published":"2026-06-12","tags":["autoformalization","ai-agents","meta-study","analysis"],"lean_note":"Uses a Claude-driven agent pipeline to formalize a numerical-analysis ODE textbook in Lean 4 and audits formalization faithfulness beyond compilation.","thumb":"thumbs/2606.14000.jpg"},{"id":"2606.13782","title":"MA-ProofBench: A Two-Tiered Evaluation of LLMs for Theorem Proving in Mathematical Analysis","authors":["Lushi Pu","Weiming Zhang","Xinheng Xie","Zixuan Fu","Bingxiang He","Hongya Lyu","Xin Li","Jie Zhou","Yudong Wang"],"abstract":"Large Language Models (LLMs) have made notable progress in automated theorem proving, yet existing formal benchmarks remain limited in both mathematical coverage and difficulty. Most are concentrated in areas that are easier to formalize, such as algebra and elementary number theory, and provide limited coverage of subfields that require deeper reasoning, including mathematical analysis. To address this gap, we introduce MA-ProofBench, to the best of our knowledge, the first formal theorem-proving benchmark dedicated to Mathematical Analysis. The benchmark contains 200 formalized theorems covering 6 core topics and 27 subcategories, including measure and integration theory, complex analysis, and functional analysis. The problems are divided into two difficulty levels, an undergraduate level (Level I, 100 problems) and a Ph.D. qualifying level (Level II, 100 problems), to evaluate how well LLMs perform formal reasoning at different mathematical depths. Each problem is constructed through a human-led, LLM-assisted formalization pipeline followed by independent expert review, ensuring that the formal statements remain faithful to the original mathematics. We evaluate a range of recent general-purpose reasoning models and formal theorem provers on MA-ProofBench. However, most models perform poorly: even the best-performing model, GPT-5.5, achieves only 16% Pass@8 on Level I and 5% on Level II, while most models stay close to 0% on Level II. Further analysis identifies Mathlib hallucinations and incomplete proofs as the two dominant failure modes, while an evaluation on the natural-language version of the benchmark exposes a clear gap between informal and formal reasoning. MA-ProofBench is intended to serve as a reliable reference for tracking progress in formal mathematical reasoning in advanced domains.","abs_url":"https://arxiv.org/abs/2606.13782","pdf_url":"https://arxiv.org/pdf/2606.13782v2","primary_category":"cs.AI","published":"2026-06-11","tags":["benchmark","theorem-proving-ml","analysis"],"lean_note":"Introduces MA-ProofBench, 200 Lean 4 formal theorems in mathematical analysis, and evaluates LLMs and theorem provers on them.","repo":"https://github.com/zai-org/GLM-5","thumb":"thumbs/2606.13782.jpg"},{"id":"2606.13306","title":"EconCSLib: AI-Assisted Lean Formalization for Economics & Computation research","authors":["Nikhil Garg"],"abstract":"This paper presents EconCSLib, a Lean 4 library and workflow for formalizing research papers in Economics and Computation with language-model assistance. The central design principle is a human-AI-Lean workflow: an LLM writes Lean code, Lean checks formal statements and proofs, and humans (assisted by an LLM) verify the translation boundary from paper claims to formal statements. EconCSLib is organized around research papers, preserving their formal statements and following their proof structure to the extent possible; reusable mathematical statements are elevated into shared EconCS infrastructure. The workflow is designed to be author-facing: researchers can formalize their own papers, inspect the Lean code's translations of paper-facing statements, and contribute reusable components back to the library; this is supported by post-formalization validation reports, paper result dependency graphs, and a review dashboard. The current public repository contains 11 formalized papers and 3 partially formalized papers, along with initial libraries for probability, auctions, matching markets, and graph tools. The library and workflow are available at https://github.com/nikhgarg/EconCSLib, with corresponding project webpage at https://gargnikhil.com/EconCSLib/. To our knowledge, we are also among the first applied math researchers to systematically pursue Lean formalization of one's own publications in the process of building such a community library. We welcome users and contributors to the project.","abs_url":"https://arxiv.org/abs/2606.13306","pdf_url":"https://arxiv.org/pdf/2606.13306v1","primary_category":"cs.GT","published":"2026-06-11","tags":["formalization","ai-agents","economics"],"lean_note":"Presents EconCSLib, a Lean 4 library and human-AI workflow for formalizing economics and computation research papers.","repo":"https://github.com/nikhgarg/EconCSLib","thumb":"thumbs/2606.13306.jpg"},{"id":"2606.12884","title":"Volterra--Wiener--Kunchenko Orthogonalization: From Wiener--Hermite to Distribution-Matched Volterra Bases","authors":["Serhii Zabolotnii"],"abstract":"The monomial parameterization of finite-memory Volterra identification is ill-conditioned under non-Gaussian input, and the Wiener--Hermite expansion removes this ill-conditioning only for Gaussian white-noise input. We construct the distribution-matched Volterra--Wiener--Kunchenko (VWK) basis by oriented Gram--Schmidt orthogonalization of monomials in $L^2(P)$ and use it as an arbitrary-polynomial-chaos coordinate system for finite-memory Volterra identification from data, following the generalized polynomial chaos of Xiu and Karniadakis (2002) and the data-driven arbitrary polynomial chaos of Oladyshkin and Nowak (2012). The basis itself is classical; the contribution is the Volterra-estimation reading. First, an order-2 misspecification-penalty theorem shows that a self-normalized diagonal estimator in the variance-matched Gaussian basis incurs an excess $L^2(P)$ risk governed by the skew coefficient $δ=μ_3/σ^2$, vanishing exactly for symmetric inputs. Second, conditioning experiments separate the constructional fact that the population matched Gram is the identity from the finite-sample design Gram: at $n=2000$, the centered-exponential empirical VWK Gram remains far better conditioned than the power Gram, although it degrades with degree. Third, a machine-checked Lean 4 proof establishes the Binomial$(N,p)$ Krawtchouk row for arbitrary $N$. Full least squares over a fixed span is basis-invariant, so VWK stabilizes diagonal cross-correlation and regularized coordinate fits rather than claiming universal prediction superiority. The analysis is moment-based, finite-memory, and restricted to product input laws.","abs_url":"https://arxiv.org/abs/2606.12884","pdf_url":"https://arxiv.org/pdf/2606.12884v1","primary_category":"stat.ME","published":"2026-06-11","tags":["formalization","probability"],"lean_note":"Includes a machine-checked Lean 4 proof establishing the Binomial(N,p) Krawtchouk row for arbitrary N.","repo":"https://github.com/SZabolotnii/VWK-Orthogonalization-code-supplement","thumb":"thumbs/2606.12884.jpg"},{"id":"2606.12594","title":"Pythagoras-Prover: Advancing Efficient Formal Proving via Augmented Lean Formalisation","authors":["Joshua Ong Jun Leang","Zheng Zhao","Mihaela Cătălina Stoian","Qiyuan Xu","Haonan Li","Wenda Li","Shay B. Cohen","Eleonora Giunchiglia"],"abstract":"Modern Lean theorem provers achieve strong performance only with substantial training and inference compute, driven in part by scarce verified proof data and the long reasoning traces of formal proof search, making both supervised fine-tuning (SFT) and sampling expensive. We introduce Pythagoras-Prover, a compute-efficient open-source family of Lean theorem provers built for practical compute budgets. The family spans two generation paradigms: autoregressive models at 4B and 32B parameters, and a first proof-of-concept diffusion-based prover (4B) that iteratively refines Lean proofs at inference time. For training efficiency, we build a Lean-verified corpus stratified into easy, medium, and hard problems for curriculum SFT, so models acquire proof skills progressively from shorter, simpler proofs to longer, harder ones. During SFT, a dynamic proof-reasoning filtering scheme preserves informative proof traces while keeping each instance within an 8k-token context budget. We also introduce Augmented Lean Formalisation (ALF), which expands scarce verified corpora into variants of formal statements, populated via self-distillation for extra training signal without formally verifying every mutated instance. By perturbing known problems while preserving their formal character, ALF reduces reliance on any statement's surface form. Empirically, Pythagoras-Prover-4B surpasses DeepSeek-Prover-V2-671B at pass@32 on MiniF2F-Test (86.1% vs 82.4%) with ~167x fewer parameters, while Pythagoras-Prover-32B sets the open-source state of the art at 93.0% on MiniF2F-Test and solves 93 of 672 PutnamBench problems. We release MiniF2F-ALF, an ALF-mutated contamination-sensitive benchmark on which every evaluated model loses accuracy; here our 32B remains strongest and our 4B matches the prior state of the art, Goedel-Prover-V2-32B.","abs_url":"https://arxiv.org/abs/2606.12594","pdf_url":"https://arxiv.org/pdf/2606.12594v1","primary_category":"cs.AI","published":"2026-06-10","tags":["theorem-proving-ml","benchmark"],"lean_note":"Trains compute-efficient Lean 4 theorem provers and a Lean corpus augmentation method, evaluated on MiniF2F and PutnamBench.","repo":"https://github.com/project-numina/aimo-progress-prize","thumb":"thumbs/2606.12594.jpg"},{"id":"2606.12471","title":"Identifiability Without Gaussianity: Symbolic World Models and Near-Infinite Temporal Consistency","authors":["Seth Dobrin","Łukasz Chmiel"],"abstract":"Klindt, LeCun, and Balestriero (arXiv:2605.26379) proved that Joint-Embedding Predictive Architectures (JEPAs) achieve linear identifiability, the linear recovery of the world's true latent variables, if and only if the world's latent dynamics follow a Gaussian, stationary process. This Gaussian boundary implies a fundamental limit on temporal consistency: for any non-Gaussian physical system, the representation error of a statistical World Model grows monotonically with time. We prove that this limit is an artifact of the statistical alignment mechanism, not a property of World Models in general. We introduce the Physics-Grounded Symbolic Architecture (PGSA) and prove three results: (1) a PGSA achieves exact linear identifiability for all physical regimes, regardless of the latent distribution; (2) the per-step error of a PGSA is bounded by numerical precision alone; and (3) as a direct consequence, a PGSA maintains temporal consistency for an unbounded number of transitions, a property we term near-infinite temporal consistency. We further prove that statistical World Models cannot achieve this property for any non-Gaussian system, regardless of model capacity or the volume of training data. The algebraic cores of four of the theorems are formalized in Lean 4 with Mathlib4 v4.31.0 (zero sorry placeholders); the Klindt et al. converse is taken as an external premise. The contrast establishes that symbolic grounding in the causal generator of the world's dynamics is the sufficient condition and, in non-Gaussian regimes, the only condition for near-infinite temporal consistency.","abs_url":"https://arxiv.org/abs/2606.12471","pdf_url":"https://arxiv.org/pdf/2606.12471v1","primary_category":"stat.ML","published":"2026-06-09","tags":["formalization","ml-theory","mathlib"],"lean_note":"Formalizes the algebraic cores of four theorems on symbolic world-model identifiability in Lean 4 with Mathlib4 (zero sorry).","repo":"https://github.com/ARYA-Labs-PBC/pgsa-world-model-proofs","thumb":"thumbs/2606.12471.jpg"},{"id":"2606.11430","title":"Towards a Bridge Layer Between Bibliographic and Formalized Mathematical Knowledge","authors":["A. Mayeux"],"abstract":"Mathematical knowledge is split between bibliographic databases (e.g., MathSciNet, zbMATH Open) and formal proof libraries (e.g., Lean mathlib), preventing unified access between published results and their formalizations. We propose a relational bridge-database that aligns publication metadata with formal artifacts, providing an interoperability layer between mathematical literature and machine-verifiable proofs. We introduce a paper-level formalization score that measures how much of a publication is covered in formal systems. As a feasibility study, we show how such scores can be estimated via cross-document alignment between informal texts and Lean formalizations, enabling large-scale analysis of formalization coverage. This framework is a first step toward integrating bibliographic and formal mathematical ecosystems into scalable, machine-actionable knowledge graphs linking publications to formal proof objects.","abs_url":"https://arxiv.org/abs/2606.11430","pdf_url":"https://arxiv.org/pdf/2606.11430v1","primary_category":"cs.DL","published":"2026-06-09","tags":["tooling","mathlib","dataset"],"lean_note":"Proposes a bridge-database aligning math publications with Lean mathlib formalizations and a paper-level formalization-coverage score.","thumb":"thumbs/2606.11430.jpg"},{"id":"2606.11421","title":"Second-Order Least Squares as a Special Case of the Polynomial Maximization Method","authors":["Serhii Zabolotnii"],"abstract":"We prove that optimally weighted second-order least squares (SLS) and the degree-two generalized polynomial maximization method (PMM) are the same population estimating equation for linear regression with conditionally homoskedastic non-Gaussian errors: they choose the same optimal linear combination of the first two centered residual moments, solve one population normal system, share one influence function, and attain the common asymptotic variance $c_2g_2/N$ -- the ordinary-least-squares slope-variance factor $c_2$ scaled by the PMM variance-reduction coefficient $g_2=1-γ_3^2/(2+γ_4)$ (with $γ_3,γ_4$ the error skewness and excess kurtosis). Feasible plug-in implementations are therefore first-order equivalent, with only higher-order finite-sample differences. The identity is sharp: under heteroskedasticity the unconditional PMM body and the conditional SLS weighting separate, costing efficiency for symmetric errors and consistency for asymmetric errors. Beyond degree two, PMM holds an efficiency reserve that SLS cannot reach within its second-moment span. For symmetric platykurtic errors SLS collapses to ordinary least squares for the slope, while degree-three PMM exploits kurtosis information outside the SLS moment span through a closed-form coefficient $g_3$; for canonical asymmetric laws this reserve is $30$--$50\\%$ within the degree-three polynomial moment class. The Lean 4 development machine-checks the degree-specific algebraic core -- the closed forms for $g_2$ and $g_3$, the $g_2\\le1$ result, the design cancellations, and the symmetric collapse -- while the general monotonicity $g_{S+1}\\le g_S\\le1$ is proved analytically by nesting. A Monte Carlo study illustrates the equivalence, the reserve, and the heteroskedastic boundary at finite samples.","abs_url":"https://arxiv.org/abs/2606.11421","pdf_url":"https://arxiv.org/pdf/2606.11421v1","primary_category":"stat.ME","published":"2026-06-09","tags":["formalization","verification","ml-theory"],"lean_note":"A Lean 4 development with Mathlib machine-checks the algebraic core: closed forms for g2 and g3, the g2<=1 bound, and the symmetric collapse.","repo":"https://github.com/SZabolotnii/PMM-SLS-BRIDGE-code-supplement","thumb":"thumbs/2606.11421.jpg"},{"id":"2606.11417","title":"Signed Compression Progress on a Sealed Audit is Goodhart-Resistant","authors":["Ayush Mittal","Dhruv Gupta"],"abstract":"Compression progress is a long-standing proposal for intrinsic motivation: reward an agent when its world model becomes better at predicting or compressing experience. The folk claim is that this reward is \"credible\" because it is paid only for learning. We make this precise and prove it. If intrinsic reward is the signed decrease of a fixed sealed-audit loss, r_t = E(theta_{t-1}) - E(theta_t), then cumulative reward telescopes exactly to endpoint audit improvement, so no policy can push reward up indefinitely while true audit performance stagnates or degrades. For finite audit panels the same result holds with a sharp false-positive budget: cumulative empirical reward is at most true audit improvement plus 2 Delta_n(F, delta), the uniform audit deviation of the model class. This is horizon-free: adaptivity over time costs nothing once the sealed panel uniformly controls the class. The theorem also identifies the failure modes: the guarantee disappears if progress is clipped, scored on the agent's own stream, exposed to a high-capacity model on a reusable panel, or applied to a neural class that makes Delta_n vacuous. We give a Lean 4 mechanization of the structural core (telescoping, the finite-audit bound, finite Gibbs, and the entropy floor) and an experiment suite on ARC-TGI grid-transformation generators with adaptive holdout attacks. Experiments confirm the theory: finite-audit deviation scales as n^{-0.527}; signed progress resists clip-farming, stream leakage, and noisy-TV curiosity; naive reusable audits are exploitable by black-box scalar feedback, while standard release defenses keep the attack below the 2 Delta_n threshold. Signed compression progress on a sealed audit is an accounting signal of genuine improvement.","abs_url":"https://arxiv.org/abs/2606.11417","pdf_url":"https://arxiv.org/pdf/2606.11417v1","primary_category":"cs.LG","published":"2026-06-09","tags":["formalization","verification","ml-theory"],"lean_note":"A self-contained Lean 4 / Mathlib development mechanizes the structural core: telescoping, the finite-audit bound, finite Gibbs, and the entropy floor.","thumb":"thumbs/2606.11417.jpg"},{"id":"2606.09674","title":"(Auto)formalization is supposed to be easy: Trellis process semantics for spelling out rigorous proofs","authors":["Wesley Pegden"],"abstract":"We present Trellis: an autoformalization system that leverages LLM agents in a deterministically constrained workflow to enforce incremental progress in Lean autoformalization tasks through iterative refinement of natural language proofs. Our approach is motivated by the common mathematician's notion of what it means to have a rigorous proof in the first place: namely, that it would be routine to elaborate any part of the proof in further detail. The result is a system which aims to achieve reliable autoformalization on a modest budget and with generalist agents, with specialization to autoformalization coming not from any task-specific agent training but instead from a meaning-of-rigor inspired workflow enforced by process semantics. We link to an end-to-end Lean formalization of a recent Ramsey theory breakthrough produced by the process.","abs_url":"https://arxiv.org/abs/2606.09674","pdf_url":"https://arxiv.org/pdf/2606.09674v1","primary_category":"cs.AI","published":"2026-06-08","tags":["autoformalization","ai-agents","theorem-proving-ml","combinatorics"],"lean_note":"Trellis is an LLM-agent autoformalization system that produces end-to-end Lean proofs via a process-semantics-constrained workflow.","thumb":"thumbs/2606.09674.jpg"},{"id":"2606.09600","title":"Formal Foundations and Proof-Carrying Certificates for q-ary Covering Codes in Lean 4","authors":["Andreas Florath"],"abstract":"Covering codes in finite Hamming spaces ask for small sets of words whose Hamming balls cover the whole space. This paper presents a Lean 4 formalization of the elementary theory of q-ary covering codes, centered on certificate predicates for upper bounds, lower bounds, and exact covering numbers $K_q(n,r)$. The formalization proves the q-ary Hamming-ball volume formula, the sphere-covering lower bound, elementary exact cases, product and relation rules, and selected small exact certificates. It also demonstrates an end-to-end workflow for checking explicit upper bounds transcribed from van Laarhoven et al. (1989). The accompanying database is proof-carrying: stored bounds have traces that replay to Lean proofs of the corresponding upper- or lower-bound predicates. The contribution is not new record bounds or a reproduction of known tables, but a reusable, auditable foundation for machine-checked covering-code certificates.","abs_url":"https://arxiv.org/abs/2606.09600","pdf_url":"https://arxiv.org/pdf/2606.09600v1","primary_category":"cs.IT","published":"2026-06-08","tags":["formalization","verification","combinatorics"],"lean_note":"Formalizes elementary theory of q-ary covering codes in Lean 4 with proof-carrying certificates for covering-number bounds.","repo":"https://github.com/florath/covering-codes-lean","thumb":"thumbs/2606.09600.jpg"},{"id":"2606.09450","title":"TheoremBench: Evaluating LLMs on Theorem Proving in Formal Mathematics","authors":["QuocViet Pham","Elvir Karimov","Andrey Galichin","Ivan Oseledets"],"abstract":"LLMs have recently achieved strong results on formal proving benchmarks. However, existing evaluations remain heavily concentrated on competition-style problems and often fail to capture how models behave on longer, more dependency-rich mathematical developments. We introduce TheoremBench, a Lean4 benchmark designed to evaluate theorem provers beyond contest settings. The benchmark is built from nearly one hundred classical theorems and is released in two complementary forms: a plain main version containing one target theorem per instance, and a premised version that expands each theorem into a structured family of related proving tasks consisting of the main theorem together with automatically extracted supporting subtheorems. This design enables evaluation of not only whether the final theorem was proved from scratch, but also of partial progress through the internal proof structure of a theorem. Our experiments show that explicit premises substantially improve performance for Lean4-capable prover models. To provide a comprehensive evaluation, we introduce theorem-level coverage and token-efficiency metrics that expose qualitative differences in proof behavior. The results show that current provers remain strongly biased toward easy subtheorems and often solve theorems through long and inefficient tactic traces rather than compact proof plans. TheoremBench therefore provides a more fine-grained view of formal reasoning ability and highlights the importance of structural benchmark design for evaluating Lean4 theorem provers.","abs_url":"https://arxiv.org/abs/2606.09450","pdf_url":"https://arxiv.org/pdf/2606.09450v1","primary_category":"cs.AI","published":"2026-06-08","tags":["benchmark","theorem-proving-ml"],"lean_note":"Introduces TheoremBench, a Lean4 benchmark of ~100 classical theorems with extracted premises for evaluating LLM provers.","thumb":"thumbs/2606.09450.jpg"},{"id":"2606.09045","title":"Embedding Finite Functions into Low-Degree Polynomial Functions over Commutative Rings","authors":["Roman Bacik"],"abstract":"A function $f \\colon X^k \\to X$ on a finite set embeds into a polynomial of total degree $d$ over a commutative ring $R$ if there is an injection $j \\colon X \\to R$ and a polynomial $g$ of total degree at most $d$ with $j \\circ f = g \\circ j^k$, where $j^k$ applies $j$ in each coordinate. These are the transition functions of $k$-neighbour cellular automata, and the injection $j$ is an enlargement of the alphabet that preserves the transitions. We prove three results, all verified in Lean~4 with Mathlib~\\cite{bacik2026finbin}. Every unary function $f \\colon X \\to X$ embeds into a polynomial of total degree $1$. Every binary Kronecker delta embeds into a polynomial of total degree $4$. For every $d$ there is a binary function that does not embed into any polynomial of total degree $d$.","abs_url":"https://arxiv.org/abs/2606.09045","pdf_url":"https://arxiv.org/pdf/2606.09045v1","primary_category":"math.CO","published":"2026-06-08","tags":["formalization","combinatorics","algebra"],"lean_note":"Proves three embedding results for finite functions, all verified in Lean 4 with Mathlib.","repo":"https://github.com/roman3017/FinBin","thumb":"thumbs/2606.09045.jpg"},{"id":"2606.08902","title":"A Kernel-Clean Lean Mechanization of Classical Lottery in Action and the Wakker--Debreu--Koopmans Representation Layer","authors":["Jingyuan Li","Ilia Tsetlin","Fan Wang"],"abstract":"We present a Lean 4/Mathlib formalization of the additive representation theory behind Classical Lottery in Action and the Wakker-Debreu-Koopmans (WDK) layer it relies on. Our central result is a machine-checked proof that the cross-pair Thomsen / double-cancellation (hexagon) condition is irreducible from the ordinal axioms of additive conjoint measurement (weak order, restricted solvability, Archimedean condition, and tradeoff consistency). We exhibit an explicit verified counter-model (additiveRealBoolPref) satisfying all ordinal axioms yet failing the cross-pair condition, with every strict standard sequence being an arithmetic progression and hence non-dense. Around this boundary we mechanize the full derivable construction: continuous Debreu/Eilenberg utility from separability, standard-sequence grids, bisection methods from connectedness, and global additive gluing. All public theorems are sorry-free conditional wrappers over this single irreducible structural input. The development is kernel-clean, depending only on standard Lean foundations (propext, Classical.choice, Quot.sound). The companion file ClassicalLotteryInAction.lean formalizes local classical-lottery constructions, average-utility results, matching-frequency lemmas, and ambiguity-attitude statements used by the Management Science paper. This draws a precise, machine-certified line between what additive conjoint measurement can prove and what it must assume.","abs_url":"https://arxiv.org/abs/2606.08902","pdf_url":"https://arxiv.org/pdf/2606.08902v1","primary_category":"cs.LO","published":"2026-06-08","tags":["formalization","economics"],"lean_note":"Formalizes additive conjoint measurement and the Wakker-Debreu-Koopmans representation layer in Lean 4 with Mathlib.","repo":"https://github.com/jingyuanli-hk/classical-lottery-in-action-lean-artifact","thumb":"thumbs/2606.08902.jpg"},{"id":"2606.06807","title":"Halving the original Kalton--Roberts upper bound for nearly additive set functions","authors":["Boon Suan Ho","Tomasz Kania"],"abstract":"Let $K_\\mathrm{KR}$ denote the optimal Kalton--Roberts constant for approximately additive real-valued set functions on algebras of sets. Kalton and Roberts proved $K_\\mathrm{KR}\\le89/2$, and Bondarenko, Prymak, and Radchenko improved the upper bound to $38.8$. We prove that $$K_\\mathrm{KR}\\le\\frac{694,198,146,664,396,294,486,127,753}{34,994,834,677,886,019,996,000,000}\\,\\approx 19.837.$$ Thus the original Kalton--Roberts upper bound is more than halved. The proof changes the source collections fed into the expander-recombination step however still uses expander graphs as the other proofs do. The four expander families used in the final recombination are certified by exact rational interval arithmetic, and the proof has been formalised in Lean.","abs_url":"https://arxiv.org/abs/2606.06807","pdf_url":"https://arxiv.org/pdf/2606.06807v1","primary_category":"math.CO","published":"2026-06-05","tags":["formalization","mathlib","combinatorics","analysis"],"lean_note":"The improved Kalton-Roberts upper bound, including exact rational interval-arithmetic certificates, is formalised in about 6,200 lines of Lean and Mathlib.","repo":"https://github.com/boonsuan/KaltonRoberts","thumb":"thumbs/2606.06807.jpg"},{"id":"2606.06468","title":"Goedel-Architect: Streamlining Formal Theorem Proving with Blueprint Generation and Refinement","authors":["Jui-Hui Chung","Ziyang Cai","Zihao Li","Qishuo Yin","Rohit Agarwal","Simon Park","Rodrigo Porto","Narutatsu Ri","Ziran Yang","Shange Tang","Xingyu Dang","Hongzhou Lin","Mengdi Wang","Danqi Chen","Chi Jin","Liam H Fowl","Sanjeev Arora"],"abstract":"We introduce Goedel-Architect, an agentic framework for formal theorem proving in Lean 4 centered on blueprint generation and refinement. A blueprint is a dependency graph of definitions and lemmas that builds up to the main theorem. First, Goedel-Architect generates a blueprint of formally stated definitions and lemmas, along with declared dependencies. This blueprint is optionally guided by a natural language proof. Then, a tool-equipped Lean prover component closes each open lemma node in parallel using relevant dependencies. Failed lemmas in turn drive refinement of the global blueprint. This strategy contrasts with other mainstream approaches which use recursive lemma decomposition, and can inefficiently loop on dead-end strategies. Using the open-weight DeepSeek-V4-Flash (284B-A13B) as the backbone, Goedel-Architect attains 99.2% pass@1 on MiniF2F-test and 75.6% pass@1 on PutnamBench. With an optional natural-language proof seeding the initial blueprint on the harder problems, we additionally close the remaining two MiniF2F-test problems (reaching 100%), lift PutnamBench to 88.8% (597/672), and solve 4/6 on IMO 2025, 11/12 on Putnam 2025, and 3/6 on USAMO 2026. This represents state-of-the-art performance for an open-source pipeline at a price point up to 500x less than comparable open-source pipelines.","abs_url":"https://arxiv.org/abs/2606.06468","pdf_url":"https://arxiv.org/pdf/2606.06468v1","primary_category":"cs.AI","published":"2026-06-04","tags":["theorem-proving-ml","ai-agents","benchmark"],"lean_note":"Agentic blueprint-generation framework that proves theorems in Lean 4, evaluated on MiniF2F, PutnamBench, and IMO 2025.","thumb":"thumbs/2606.06468.jpg"},{"id":"2606.05796","title":"GCD: Garbled, Corrected, Demonstrandum -- Fixing and Proving Go's Extended GCD Implementation","authors":["Linard Arquint"],"abstract":"We verify the 'extendedGCD' implementation in Go's standard library ('crypto/internal/fips140/bigmod'), which plays a crucial role in the generation of RSA key pairs. Even though the Go implementation is supposedly a direct port from BoringSSL's implementation, we uncovered two deviations that each break the algorithm's invariants: (1) the Go implementation deviates in the way coefficients are updated, and (2) it permits a larger input domain. We address both deviations; the first by fixing the Go implementation, which results in an on average 24% speedup, and the second deviation by porting an existing proof for BoringSSL and extending it to cover the larger input domain. We prove correctness and termination of the fixed Go implementation using Gobra, a deductive program verifier for Go. Where necessary, we used Lean to prove key lemmata on non-linear arithmetic, which we import into Gobra. Our verification effort reveals three key insights: subtle bugs can slip into even well-reviewed code with surprising ease; formal verification is a powerful tool for uncovering them; and AI agents can facilitate the verification process by iteratively refining invariants and lemmata based on Gobra's error messages.","abs_url":"https://arxiv.org/abs/2606.05796","pdf_url":"https://arxiv.org/pdf/2606.05796v2","primary_category":"cs.CR","published":"2026-06-04","tags":["verification","cryptography"],"lean_note":"Lean proves key non-linear arithmetic lemmas that are imported into the Gobra verifier for Go's extended-GCD code.","repo":"https://github.com/arquintl/go-gcd","thumb":"thumbs/2606.05796.jpg"},{"id":"2606.05632","title":"Evaluation of LLMs for Mathematical Formalization in Lean","authors":["Tyson Klingner","Drew Bladek","Escher Crawford","Bohao Chen","Ariel Fu","Kaira Nair","Jarod Alper","Giovanni Inchiostro","Vasily Ilin"],"abstract":"Within the past few years, the ability of Large Language Models (LLMs) to generate formal mathematical proofs has improved drastically. We provide a comparison of various LLMs' effectiveness in producing formal proofs in Lean 4 with the goal of assisting those seeking to use LLMs to support their own projects. We utilize both pass@$k$ and refine@$k$ metrics as the benchmark for our comparison and evaluate on subsets of both miniF2F and miniCTX datasets. Our testing shows that overall, Gemini 3.1 Pro and Claude Opus 4.7 perform best. Gemini 3.1 Pro achieved a 92\\% success rate on miniF2F via refine@32 whereas Opus 4.7 achieved a 86\\% success rate on miniCTX via refine@32. When taking cost into account, NVIDIA Nemotron 3 Super and GPT-OSS 120B were the most efficient, with competitive accuracies and average costs of $<\\$0.01$ per correct proof.","abs_url":"https://arxiv.org/abs/2606.05632","pdf_url":"https://arxiv.org/pdf/2606.05632v1","primary_category":"cs.AI","published":"2026-06-04","tags":["theorem-proving-ml","benchmark"],"lean_note":"Compares LLMs at generating Lean 4 proofs, evaluated on miniF2F and miniCTX with pass@k and refine@k.","repo":"https://github.com/uw-math-ai/LLMsLean","thumb":"thumbs/2606.05632.jpg"},{"id":"2606.05629","title":"An automated proof that R(B_8,B_10)=37","authors":["Jeremy Kalfus","Bernard Lidický"],"abstract":"We present a short proof that the book Ramsey number $R(B_8,B_{10})$ equals 37. The lower bound $R(B_8,B_{10}) \\ge 37$ is already available in the literature, so it is enough to rule out a 37-vertex graph containing neither a copy of $B_8$ nor a copy of $B_{10}$ in its complement. The problem as well as the proof were found with AutoMath, an AI-assisted mathematical discovery workflow developed by the first author. A Lean formalization of the upper-bound argument is available in the accompanying repository.","abs_url":"https://arxiv.org/abs/2606.05629","pdf_url":"https://arxiv.org/pdf/2606.05629v1","primary_category":"math.CO","published":"2026-06-04","tags":["formalization","combinatorics"],"lean_note":"Formalizes the upper-bound argument that every 37-vertex graph contains B_8 or B_10 in its complement, settling R(B_8,B_10)=37.","repo":"https://github.com/BLL4/RamseyB8B10","thumb":"thumbs/2606.05629.jpg"},{"id":"2606.05400","title":"LeanMarathon: Toward Reliable AI Co-Mathematicians through Long-Horizon Lean Autoformalization","authors":["Yuanhe Zhang","Yuekai Sun","Taiji Suzuki","Jason D. Lee","Fanghui Liu"],"abstract":"Long-horizon autoformalization of research mathematics fails not only at hard lemmas, but at scale: statements drift, dependencies tangle, context decays, and local repairs corrupt distant work. We present LeanMarathon, a multi-agent harness for reliable research-level Lean autoformalization. Its core abstraction is an evolving blueprint: a Lean file that serves simultaneously as formal proof skeleton, natural-language proof graph, and shared system of record. Four contract-scoped agents construct, audit, prove, and repair this blueprint. These agents are coordinated by a two-stage orchestrator that first stabilizes target fidelity through adversarial review and then discharges the proof directed acyclic graph (DAG) from its dynamic leaves upward in parallel CI-gated rounds. LeanMarathon turns one brittle multi-hour run into many local, recoverable, parallel transactions. We evaluate LeanMarathon on two recent research papers spanning four Erdős problems (#1051, #1196, #164, #1217). Across three autonomous runs, it formalizes all seven target theorems with no sorry, proving 258 lemmas and theorems. These results show that reliable AI co-mathematics requires not only stronger provers, but durable harnesses that preserve target fidelity across long mathematical developments. The code can be found at https://github.com/YuanheZ/LeanMarathon.","abs_url":"https://arxiv.org/abs/2606.05400","pdf_url":"https://arxiv.org/pdf/2606.05400v1","primary_category":"cs.AI","published":"2026-06-03","tags":["autoformalization","ai-agents","theorem-proving-ml","number-theory"],"lean_note":"Multi-agent harness that autoformalizes research mathematics into Lean via an evolving blueprint, with CI-gated agents writing and proving lemmas.","repo":"https://github.com/YuanheZ/LeanMarathon","thumb":"thumbs/2606.05400.jpg"},{"id":"2606.05117","title":"A problem of Andrews and Dhar on partitions","authors":["Simon Mahns","Ken Ono","Jujian Zhang"],"abstract":"This paper is motivated by a broad question about AI-assisted mathematics: can an AI system help discover and certify an explicit bijection between two infinite sequences of complicated combinatorial sets already known to be equinumerous? The challenge is to find a reversible structure explaining that equality uniformly across the sequence. We give an affirmative test case in the setting of a partition problem. Andrews and Dhar introduced two partition families $\\mathcal{C}_3(n)$ and $\\mathcal{D}_3(n)$, and for \"nonexceptional'' $n$, they asked for a bijective proof of their equality \\[ |\\mathcal{C}_3(n)|=\\frac{|\\mathcal{D}_3(n)|}{3}. \\] We prove a residue-class equidistribution theorehm for $\\mathcal{D}_3(n)$ that identifies a \"canonical third'' subset $\\mathcal{D}_3^{(0)}(n)\\subseteq \\mathcal{D}_3(n)$. Answering their question, we construct a bijection \\[ ι_n:\\mathcal{C}_3(n)\\longrightarrow \\mathcal{D}_3^{(0)}(n) \\] as a highly structured composition of four maps. AxiomProver autonomously produced and Lean-verified the equidistribution theorem. The bijection was found through human--AxiomProver collaboration, and the theorem was autoformalized and verified by the system.","abs_url":"https://arxiv.org/abs/2606.05117","pdf_url":"https://arxiv.org/pdf/2606.05117v1","primary_category":"math.CO","published":"2026-06-03","tags":["formalization","theorem-proving-ml","combinatorics","ai-agents"],"lean_note":"AxiomProver autonomously generated and Lean-verified a residue-class equidistribution theorem and autoformalized the partition bijection proof.","repo":"https://github.com/AxiomMath/andrews_dhar_problem","thumb":"thumbs/2606.05117.jpg"},{"id":"2606.05030","title":"Imbuing Large Language Models with Bidirectional Logic for Robust Chain Repair","authors":["Zehua Cheng","Wei Dai","Jiahao Sun","Thomas Lukasiewicz"],"abstract":"Autoregressive chain-of-thought (CoT) reasoning in large language models (LLMs) is fundamentally forward-directed: each step conditions only on prior tokens. This unidirectional inductive bias renders even capable models susceptible to error snowballing, wherein a single logical or arithmetic mistake in an early step irreversibly corrupts the entire reasoning chain. We introduce Teleological Reasoning Infilling (\\TRI{}), a training framework that endows decoder-only transformers with a native \\emph{goal-conditioned bridging} capability. The key insight is to reframe erroneous reasoning segments as fill-in-the-middle (FIM) tasks: given a verified prefix premise $P$, a verified downstream milestone $S$, and the original query $Q$, the model must synthesise the logical bridge $M$ that connects $P$ to $S$ rigorously and completely. To achieve this with standard causal architectures, we introduce a Prefix-Suffix-Middle (PSM) sequence rearrangement with three non-overlapping sentinel tokens, enabling $M$ to attend to both $P$ and $S$ without any structural modification to the self-attention mechanism. Training proceeds in two stages: (i) Supervised Fine-Tuning (SFT) on symbolically verified $(P, S, M)$ triples extracted from formal mathematics corpora, and (ii) Direct Preference Optimisation (DPO) with a deterministic symbolic verifier (Lean 4 / Python) as the sole reward oracle, eliminating LLM-judge sycophancy. At inference, TRI operates as a surgical repair module within a dual-system loop: a causal draft model generates an initial trace, the verifier pinpoints failures, and TRI infills only the damaged segment, leaving verified sections intact. Comprehensive experiments on three benchmarks demonstrate that TRI achieves state-of-the-art performance across all tasks, while reducing per-problem token expenditure by 31.2%.","abs_url":"https://arxiv.org/abs/2606.05030","pdf_url":"https://arxiv.org/pdf/2606.05030v1","primary_category":"cs.CL","published":"2026-06-03","tags":["theorem-proving-ml","verification"],"lean_note":"Uses a Lean 4 symbolic verifier as the reward oracle for DPO training on formal math reasoning triples.","thumb":"thumbs/2606.05030.jpg"},{"id":"2606.04883","title":"Optimizing the Cost-Quality Tradeoff of Agentic Theorem Provers in Lean","authors":["Kári Rögnvaldsson","Chenhao Sun","Jasper Dekoninck","Martin Vechev"],"abstract":"Large language models (LLMs) are increasingly used in workflows for generating formal proofs in Lean. These workflows often decompose problems into smaller lemmas, sample many proof attempts, and use compiler feedback to guide search. However, they can be prohibitively expensive, often spending substantial compute on attempts that ultimately fail. In this work, we address this problem with an action routing agent that consists of a data plane and a control plane. The data plane generates natural-language lemma decompositions, formalizes them in Lean, and samples proof attempts for the resulting theorem and lemma targets. The control plane observes previous failed Lean attempts, estimates both the likelihood of success and cost of another attempt, and decides whether to continue proving the current target or restart from a new breakdown. On a subset of PutnamBench, our agent decreases the cost by $25.8\\%$ over a fixed-step baseline on average, preserving performance while using substantially less compute. These results suggest that failed Lean trajectories provide actionable signals for cost-aware resource allocation in agentic theorem proving.","abs_url":"https://arxiv.org/abs/2606.04883","pdf_url":"https://arxiv.org/pdf/2606.04883v1","primary_category":"cs.CL","published":"2026-06-03","tags":["theorem-proving-ml","ai-agents","benchmark"],"lean_note":"An action-routing agent decides whether to continue or restart Lean proof attempts, allocating compute across lemma decompositions on PutnamBench.","repo":"https://github.com/eth-sri/optimizing-lean-agents","thumb":"thumbs/2606.04883.jpg"},{"id":"2606.04311","title":"Formal verification of the S-two AIR","authors":["Jeremy Avigad","Anat Ganor","Lior Goldberg","David Levit","Ohad Nir","Yoav Seginer","Alon Titelman"],"abstract":"StarkWare's S-two prover provides an efficient means for establishing, on blockchain, that a program written in the Cairo virtual machine language runs to completion. The latter claim is encoded by an algebraic intermediate representation (AIR) that captures the semantics of the Cairo language. The AIR asserts the existence of tables of values from a finite field satisfying certain algebraic constraints. A cryptographic interactive proof system, circle STARK, provides an efficiently-checked certificate that the AIR is satisfied. We describe our verification, using the Lean 4 proof assistant, that the AIR encoding is sound, which is to say, the satisfiability of the AIR implies the computational claim.","abs_url":"https://arxiv.org/abs/2606.04311","pdf_url":"https://arxiv.org/pdf/2606.04311v1","primary_category":"cs.CR","published":"2026-06-03","tags":["verification","cryptography"],"lean_note":"Verifies in Lean 4 that StarkWare's S-two AIR encoding of the Cairo VM is sound.","repo":"https://github.com/starkware-libs/formal-proofs","thumb":"thumbs/2606.04311.jpg"},{"id":"2606.06523","title":"Lean4Agent: Formal Modeling and Verification for Agent Workflow and Trajectory","authors":["Ruida Wang","Jerry Huang","Pengcheng Wang","Xuanqing Liu","Luyang Kong","Tong Zhang"],"abstract":"Equipping Large Language Models (LLMs) to execute reliable multi-step workflows has become a central challenge in artificial intelligence. Despite recent advances in LLMs' agentic capabilities, most agent systems still lack formal methods for specifying, verifying, and debugging their workflow and execution trajectories. This challenge mirrors a long-standing problem in mathematics, where the ambiguity of natural languages (NLs) motivates the development of formal languages (FLs). Inspired by this paradigm, we propose **Lean4Agent**, to the best of our knowledge, the first framework that uses Lean4, a dependent-type FL to model and verify agent behavior. **Lean4Agent** launches **FormalAgentLib**, an extensible Lean4 library for formally modeling and verifying agent workflows' semantic consistency under explicit assumptions, and enabling localization of execution-time failures revealed by trajectories. Building on **FormalAgentLib**, we further develop **LeanEvolve**, which applies results in **FormalAgentLib** to revise workflows to enhance its capability. Extensive experiments on a hard problem subset of SWE-Bench-Verified and a subset of ELAIP-Bench across 5 leading LLMs indicate that the verification-passing workflows outperform the failing ones by an average of **11.94%**, and **LeanEvolve** further improves SWE performance by **7.47%** on average. Furthermore, **Lean4Agent** establishes a foundation for a new field of using expressive dependent-type FL to formally model and verify agent behavior.","abs_url":"https://arxiv.org/abs/2606.06523","pdf_url":"https://arxiv.org/pdf/2606.06523v2","primary_category":"cs.AI","published":"2026-06-02","tags":["verification","ai-agents"],"lean_note":"Uses Lean4 and the FormalAgentLib library to formally model and verify LLM-agent workflows and execution trajectories.","repo":"https://github.com/RickySkywalker/Lean4Agent","thumb":"thumbs/2606.06523.jpg"},{"id":"2606.03743","title":"Proof-Refactor: Refactoring Generated Formal Proofs into Modular Artifacts","authors":["Yiming Fu","Peixuan Liu","Zichen Wang","Kun yuan"],"abstract":"While Large Language Models (LLMs) have shown strong performance in generating formal proofs, their outputs often remain less readable, modular, maintainable, and reusable than proofs in mature formal mathematics libraries. We argue that this gap stems in part from the compile-first objective implicit in most proof-generation pipelines, which encourages monolithic or ad hoc proof scripts rather than library-quality artifacts. Existing approaches to proof-quality improvement often rely on explicit, computable optimization objectives. In practice, however, the most tractable and experimentally validated objectives are largely length-based, while higher-level qualities such as readability, modularity, maintainability, and reusability are difficult to reduce to reliable automatic metrics. Instead of optimizing proof improvement against a single proxy metric, we take a process-guided approach inspired by human proof-refactoring workflows. We propose an agentic framework $\\textbf{Proof-Refactor}$ that decomposes proof refactoring into four phases: extracting candidate proof fragments, designing helper declarations, formally proving the extracted and designed components, and repairing the original proof using the verified components. On generated Lean proofs from PutnamBench and Putnam2025, Proof-Refactor improves rubric-based refactoring scores over a strong Claude Code refactoring baseline, with the largest gains in signature quality and human readability. These results suggest that process-guided refactoring can improve proof structure without treating proof length as the primary objective.","abs_url":"https://arxiv.org/abs/2606.03743","pdf_url":"https://arxiv.org/pdf/2606.03743v1","primary_category":"cs.AI","published":"2026-06-02","tags":["tactic-automation","ai-agents","theorem-proving-ml","mathlib"],"lean_note":"Refactors LLM-generated Lean proofs from PutnamBench into modular artifacts using a four-phase agent on Claude Code with lean-lsp-mcp.","repo":"https://github.com/pelicanhere/proof-refactor","thumb":"thumbs/2606.03743.jpg"},{"id":"2606.03303","title":"LEAP: Supercharging LLMs for Formal Mathematics with Agentic Frameworks","authors":["Po-Nien Kung","Linfeng Song","Dawsen Hwang","Jinsung Yoon","Chun-Liang Li","Simone Severini","Mirek Olšák","Edward Lockhart","Quoc V Le","Burak Gokturk","Thang Luong","Tomas Pfister","Nanyun Peng"],"abstract":"Large Language Models (LLMs) exhibit strong informal mathematical reasoning but struggle to generate mechanically verifiable proofs in formal languages like Lean. We present LEAP, an agentic framework that enables general-purpose foundation models to achieve state-of-the-art performance on automated formal theorem proving. LEAP leverages foundation model capabilities, such as informal reasoning, instruction following, and iterative self-refinement. By decomposing complex problems into smaller units, the system bridges formal proof construction with informal blueprints through continuous interaction with the Lean compiler. To provide a rigorous evaluation beyond increasingly saturated benchmarks, we introduce Lean-IMO-Bench, a benchmark of IMO-style problems formalized in Lean, with short statements yet highly non-routine and multi-step proofs across a wide range of difficulty levels. Empirically, on the latest 2025 Putnam Competition, an annual mathematics competition for undergraduate students in North America, LEAP solves all 12 problems, matching recent breakthroughs by frontier formal mathematical models. On Lean-IMO-Bench, LEAP boosts the one-shot formal solve rate of general-purpose LLMs from below 10% to 70%, notably surpassing the 48% benchmark set by a specialized, gold-medal-caliber IMO system. Furthermore, we demonstrate LEAP's research-level utility by autonomously formalizing complex proofs for open combinatorial challenges, including a verified proof for a key subproblem in Knuth's Hamiltonian decomposition of even-order Cayley graphs.","abs_url":"https://arxiv.org/abs/2606.03303","pdf_url":"https://arxiv.org/pdf/2606.03303v2","primary_category":"cs.AI","published":"2026-06-02","tags":["ai-agents","theorem-proving-ml","autoformalization","benchmark"],"lean_note":"Presents an agentic framework that drives a general-purpose LLM to construct Lean proofs via compiler feedback, lemma decomposition, and LeanSearch retrieval.","repo":"https://github.com/AxiomMath/putnam2025","thumb":"thumbs/2606.03303.jpg"},{"id":"2606.02392","title":"Topology as Logic: Structural Role Geometry Across Formal, Software, Biological, and Prebiotic Systems","authors":["Vladi Ivanov"],"abstract":"We ask whether dependency topology correlates with functional load-bearing organization as recoverable geometry -- not as a metaphor, but as a measurable structural property detectable by multilayer network analysis. Across seven independent substrates, we show that hub persistence and rank divergence under the Functional Proximity Law recover operational organization that domain experts describe as logic: axiomatic load-bearing structure in formal mathematics, control and contract structure in legacy software, conserved hub grammar across approx. 600 million years of neural evolution, catalytic role organization in a published prebiotic autocatalytic network, carry-path dominance in a 4-bit digital circuit, betweenness persistence in the ISCAS85 c432 standard benchmark (n=196), and a directional formal-systems replication in the Coq Corelib (n=17). A key methodological finding: degree-based hub persistence is weak between physical wiring and simulation state-correlation layers (r=0.21 in c432), while betweenness-based persistence is stronger (r=0.77 in the 4-bit ALU post-hoc; r=0.34 in c432). The ISCAS85 pre-registered primary hypothesis was CONFIRMED (degree r=0.426, p=0.002, Spearman r=0.551). The formal-systems claim is supported by two proof-assistant corpora: Lean 4 mathlib4 (CONFIRMED, r=0.777, p=0.004) and Coq Corelib (PARTIAL, direction confirmed, r=0.288, p=0.287, n=17, underpowered). All seven experiments were pre-registered before analysis.","abs_url":"https://arxiv.org/abs/2606.02392","pdf_url":"https://arxiv.org/pdf/2606.02392v1","primary_category":"cs.SI","published":"2026-06-01","tags":["meta-study","mathlib"],"lean_note":"Uses the Lean 4 mathlib4 dependency graph as a proof-assistant corpus for network hub-persistence analysis.","repo":"https://github.com/vladi160/preregistrations","thumb":"thumbs/2606.02392.jpg"},{"id":"2606.01794","title":"Tridirectional Discriminating-Power Formal Verification of Smart Contract Reentrancy Defense Against Production-Deployed Solidity Source","authors":["Ray Iskander"],"abstract":"We present the first machine-checked correctness proof of the OpenZeppelin reentrancy-guard pattern against a Lean 4 state-machine model of production-deployed Solidity source. All thirteen theorems are machine-checked with zero sorry, zero user-introduced axioms, and an axiom footprint bounded by [propext] (a standard mathlib4 axiom), gated under continuous integration. Smart contract reentrancy has caused over US$500M in documented losses since 2016, with the DAO 2016 attack draining ~3.6M ETH and forcing the hard fork that split Ethereum. The OpenZeppelin ReentrancyGuard pattern is the de facto defense across production DeFi, yet no prior work has established its discriminating power: that the guard blocks attacks on vulnerable instances, preserves correct execution for non-attacking transactions, and distinguishes adjacent safe and vulnerable variants. Prior efforts formalized either guard correctness on toy contracts or attack feasibility on isolated instances - not both directions plus boundary cases against production source. We verify three production instantiations - DAO 2016, Compound v2, and Aave V3 flashLoan - plus a minimal-diff mutant of Aave V3's flashLoan (flashLoanVulnerable) isolating one security-critical difference, via mutation testing. The tridirectional structure pairs (a) attack reproduction of the DAO 2016 pattern, (b) a correctness proof for Compound v2, and (c) a boundary-case proof distinguishing Aave V3's CEI-correct flashLoan from the mutant. A capstone meta-theorem composes the three under a no-retrofit discipline, demonstrated at the first cross-protocol stress test (Compound v2 to Aave V3); broader-family portability is future work. Full Lean 4 source, CI config and reproduction commands are at https://github.com/rayiskander2406/qanary-contracts, reproducible at v1.6-phase7-closure (substrate: v1.3-layer6-closure).","abs_url":"https://arxiv.org/abs/2606.01794","pdf_url":"https://arxiv.org/pdf/2606.01794v2","primary_category":"cs.CR","published":"2026-06-01","tags":["verification","cryptography"],"lean_note":"Machine-checks OpenZeppelin reentrancy-guard correctness against a Lean 4 state-machine model of production Solidity.","repo":"https://github.com/rayiskander2406/qanary-contracts","thumb":"thumbs/2606.01794.jpg"},{"id":"2606.04016","title":"Witness-split + window-cardinality refinement for $r_3(N)$: Architecture, empirical results, and a structural hard pocket","authors":["Mehmet Ergezer"],"abstract":"We describe a reproducible computational framework for upper-bound searches on r_3(N), the maximum size of a 3-term-arithmetic-progression-free subset of [1,N]. The framework combines a verified lower-bound witness, endpoint forcing, depth-d witness-variable splitting, OEIS A003002 window-cardinality pruning, and recursive refinement of timed-out subproblems. Applied to the frontier case N = 212, K = 44, it found no feasible 44-set across millions of CP-SAT subproblems, supporting but not proving the conjectural value r_3(212) = 43. A 300-second recap leaves 45 resistant chunks; one-hour HiGHS MIP closes none of them; the full eight-hour HiGHS audit closes 25/45 and leaves 20/45 with dual bounds still pinned at 0.0. A CDCL/SAT re-attack on those LP-paradigm-resistant chunks closes 18 via conflict-driven clause learning; all eighteen carry independently verified DRAT proofs. The remaining two chunks (T1c) resist every tested paradigm under generous wall caps. We release the witness, solver scripts, result logs, tiered benchmark instances, verified DRAT/LRAT proofs, and a Lean formal-proof-search encoding of T1c, and frame the unit-gap problem r_3(212) in {43,44} as a target for stronger additive-combinatorial bounds, custom branch-and-bound, or formal proof-search systems.","abs_url":"https://arxiv.org/abs/2606.04016","pdf_url":"https://arxiv.org/pdf/2606.04016v1","primary_category":"cs.LO","published":"2026-05-31","tags":["benchmark","theorem-proving-ml","combinatorics","number-theory"],"lean_note":"Releases a Lean formal-proof-search encoding of the two resistant subproblems from an upper-bound search on r_3(212).","repo":"https://doi.org/10.5281/zenodo.20463334","thumb":"thumbs/2606.04016.jpg"},{"id":"2606.01438","title":"Formalizing multi-graded Brenner-Schröer Proj schemes and dilatations of rings in Lean4","authors":["Arnaud Mayeux","Jujian Zhang"],"abstract":"We present a detailed formalization in Lean4 of some multigraded algebraic geometry constructions, focusing on the Brenner--Schröer Proj construction and algebraic dilatations of rings.","abs_url":"https://arxiv.org/abs/2606.01438","pdf_url":"https://arxiv.org/pdf/2606.01438v1","primary_category":"cs.LO","published":"2026-05-31","tags":["formalization","algebra"],"lean_note":"Formalizes multigraded Brenner-Schroeer Proj schemes and algebraic dilatations of rings in Lean4.","thumb":"thumbs/2606.01438.jpg"},{"id":"2606.01356","title":"A Formally Verified Library of Mathematical Finance in Lean 4","authors":["Raphael Coelho"],"abstract":"We describe a library of mathematical finance built in the Lean 4 proof assistant, on top of Mathlib and the BrownianMotion package. It is broad: more than two hundred sorry-free theorems across eleven areas, from the measure-theoretic foundations of continuous-time stochastic calculus through derivative pricing to applied risk, portfolio, and fixed-income theory, and, to our knowledge, the most comprehensive machine-checked development of mathematical finance to date. Breadth is the setting, not the point. Two things make it more than a catalogue. It reaches into the continuous theory far enough to construct the L2 Itô integral as a bounded linear isometry and to derive, rather than assume, the risk-neutral pricing measure. And it audits its own faithfulness: every result is classified by how its Lean statement relates to the mathematics it claims, and a build-enforced gate pins the axioms each proof actually uses, so a reader can see precisely what has been proved and what has only been proved under added hypotheses. We close with a candid finding: a formal base over classical financial mathematics yields certified unification of known results rather than new financial theory. The contribution is therefore methodological and infrastructural, reusable verified foundations for mathematical finance, together with the faithfulness audit.","abs_url":"https://arxiv.org/abs/2606.01356","pdf_url":"https://arxiv.org/pdf/2606.01356v1","primary_category":"q-fin.MF","published":"2026-05-31","tags":["formalization","mathlib","economics"],"lean_note":"Builds a sorry-free mathematical finance library in Lean 4 on Mathlib and BrownianMotion, covering stochastic calculus and pricing.","repo":"https://github.com/raphaelrrcoelho/formal-mathfin","thumb":"thumbs/2606.01356.jpg"},{"id":"2606.01160","title":"Expected Value Alignment for Generative Reward Modeling in Formal Mathematics Verification","authors":["Shihao Ji","Haotao Tan","Zihui Song","Mingyu Li"],"abstract":"Large Language Models (LLMs) are increasingly used with formal interactive theorem provers such as Lean 4. Scaling these systems with reinforcement learning or search methods requires process reward models (PRMs) that can evaluate intermediate reasoning steps. Existing reward-model designs expose a practical trade-off. Value-head models provide continuous scores but modify the generative model interface, while generative reward models preserve textual rationales but are poorly matched to continuous floating-point regression because numeric values are split across tokens. We introduce Expected Value Alignment (EVA), a reward-modeling procedure that keeps the surface output discrete while extracting continuous scores from the model's token distribution. The model emits integer scores in a structured JSON format, and EVA computes a continuous score as the expectation over the logits of the corresponding anchor tokens. Training combines the causal language modeling objective with an auxiliary mean squared error loss on these expected values. We instantiate EVA in \\textit{Leibniz}, a reward model for Lean 4 formal verification, and evaluate it against zero-shot and reward-modeling baselines. The evaluation demonstrates that continuous logit-based scoring significantly reduces discretization artifacts while retaining the interpretability of generative critiques.","abs_url":"https://arxiv.org/abs/2606.01160","pdf_url":"https://arxiv.org/pdf/2606.01160v1","primary_category":"cs.AI","published":"2026-05-31","tags":["theorem-proving-ml"],"lean_note":"Introduces EVA reward modeling and the Leibniz reward model for LLM-based Lean 4 formal verification.","thumb":"thumbs/2606.01160.jpg"},{"id":"2606.01008","title":"FVSpec: Real-World Property-Based Tests as Lean Challenges","authors":["Quinn Dougherty","Max von Hippel","Hazel Shackleton","Mike Dodds"],"abstract":"We present a benchmark for evaluating AI models and agents on real-world formal software verification tasks. We first scrape 11,039 property-based tests (PBTs) from real-world Python repositories, then automatically translate 2,772 of them (25%) into 9,415 Lean 4 specifications with sorry placeholders (about 3 formalizations/PBT; we retain multiple attempts when none dominates on quality metrics). Translating PBTs into Lean specifications is challenging: it requires modeling Python semantics in Lean, inferring the logical property encoded in an imperative PBT, and handling the inherent difficulties of dependently-typed programming in a seldom-used language. We describe a three-agent LLM pipeline for transpiling PBTs into Lean specifications, evaluate coverage and quality metrics, and provide baselines for proof generation using several automated and model based approaches. All code (scraper and agents) and data (PBTs and Lean specifications) are open source. Our benchmark aims to drive progress on the underexplored problem of AI-assisted formal verification of real-world software, which is of increasing interest as AI produces more and more of the world's code.","abs_url":"https://arxiv.org/abs/2606.01008","pdf_url":"https://arxiv.org/pdf/2606.01008v1","primary_category":"cs.SE","published":"2026-05-31","tags":["benchmark","verification","autoformalization"],"lean_note":"Builds a benchmark translating Python property-based tests into Lean 4 specifications with sorry placeholders.","repo":"https://github.com/GaloisInc/fvspec","thumb":"thumbs/2606.01008.jpg"},{"id":"2605.30914","title":"Automating Formal Verification with Reinforcement Learning and Recursive Inference","authors":["Max Tan"],"abstract":"Automated formal verification remains challenging for large language models because data for proof assistants and verification-aware languages is scarce, and correctness depends on satisfying precise machine-checkable specifications rather than producing plausible code. This thesis studies how verifier environments can improve LLM generation of verified programs and proofs through reinforcement learning from verifiable rewards (RLVR) and verifier-guided inference-time search. First, we train open-source models in Dafny with RLVR using Group Relative Policy Optimization (GRPO) and related variants, assembling generated candidates into complete programs and scoring them with compiler and verifier outcomes. Initial experiments on an APPS-derived Dafny dataset increased verified reward from 2.2% to 58.1%, but revealed specification hacking, where models exploit weak formal specifications instead of implementing the intended solutions. After filtering underspecified and vulnerable tasks, multi-turn RLVR on the refined benchmark improves the verified pass rate from 9.7% to 31.1%. Second, we develop a verifier-guided inference scaffold in Lean that treats proof generation as structured search over decomposed subgoals, verifier feedback, diagnostics, and repair. With a fixed base model, the full scaffold with proof reviser improves pass rate on an initial VeriCoding pilot set from 46.2% under direct repair to 69.2%. On the larger VERINA dataset, whole-task decomposition plus proof reviser solves 7 of 42 previously unsolved tasks. We also introduce Dalek-Bench, a repository-scale Lean benchmark derived from the Rust $\\texttt{curve25519-dalek}$ verification project; preliminary results remain weak, indicating that stronger progress evaluation and task-specific tool-use policies are still needed.","abs_url":"https://arxiv.org/abs/2605.30914","pdf_url":"https://arxiv.org/pdf/2605.30914v1","primary_category":"cs.LG","published":"2026-05-29","tags":["theorem-proving-ml","verification","benchmark"],"lean_note":"A verifier-guided inference scaffold in Lean treats proof generation as decomposed subgoal search with feedback and repair, evaluated on VeriCoding, VERINA, and a new Dalek-Bench.","thumb":"thumbs/2605.30914.jpg"},{"id":"2605.30861","title":"Distilling LLM Feedback for Lean Theorem Proving","authors":["Gaetan Narozniak","Gérard Biau","Rémi Munos","Ahmad Rammal","Pierre Marion"],"abstract":"Post-training for reasoning models typically combines supervised fine-tuning with reinforcement learning from verifiable rewards, most commonly with GRPO. However, this algorithm suffers from sparse rewards, limited exploration, and mode collapse. Building upon recent works on self-distillation, we propose Feedback Distillation, a training method where the model is trained to match, at the token level, its own distribution conditioned on privileged feedback produced by a language model. Feedback Distillation offers token-level supervision and can inject external knowledge. Evaluating our method for Lean4 theorem-proving, we find that Feedback Distillation maintains greater diversity in generated trajectories than GRPO, yielding higher policy entropy and better pass@k scaling. The two methods are complementary: initializing GRPO from a Feedback Distillation checkpoint outperforms either method alone. All in all, our results suggest a promising avenue to improve post-training for complex reasoning.","abs_url":"https://arxiv.org/abs/2605.30861","pdf_url":"https://arxiv.org/pdf/2605.30861v1","primary_category":"cs.AI","published":"2026-05-29","tags":["theorem-proving-ml","benchmark"],"lean_note":"Proposes Feedback Distillation for post-training LLMs on Lean 4 theorem proving, evaluated on MiniF2F and LeanWorkbook using a Lean verifier.","thumb":"thumbs/2605.30861.jpg"},{"id":"2605.30333","title":"COMPOSE: Composing Future Theorems from Citations and Formal Structure","authors":["David Busbib","Michael Werman"],"abstract":"A plausible future mathematical claim must satisfy two constraints: it should follow the direction of prior work and respect the formal dependencies that constrain what can validly follow. Existing approaches typically model only one of these sources, producing claims that are either weakly grounded or insufficiently motivated. We introduce grounded future mathematical generation, where the goal is to generate a plausible future theorem-like claim for an anchor paper using two complementary sources of context: its scientific citation graph and aligned formal theorem dependency graph. To address this setting, we propose COMPOSE, a dual-graph framework that conditions a language model on both scientific citation context and formal theorem structure. To support this setting, we construct a dataset of 108K paired scientific-formal graph examples from arXiv and Mathlib, together with a benchmark of 47K future papers from 2024--2025. Experiments show that COMPOSE outperforms strong baselines on retrieval to real future papers and achieves the best overall performance under LLM-judge evaluation, producing more grounded and mathematically richer outputs. These results show that future mathematical generation benefits from combining scientific context with formal structure. Project page is available at https://david-busbib.github.io/COMPOSE-page/.","abs_url":"https://arxiv.org/abs/2605.30333","pdf_url":"https://arxiv.org/pdf/2605.30333v1","primary_category":"cs.CL","published":"2026-05-28","tags":["dataset","mathlib"],"lean_note":"Builds a dataset of paired scientific-formal graphs from arXiv and Mathlib to generate future theorem-like claims.","thumb":"thumbs/2605.30333.jpg"},{"id":"2605.30106","title":"A Rust-to-Lean Verification Pipeline with AI Provers: An Experience Report","authors":["Natalia Klaus","Palina Tolmach","Juan Conejero"],"abstract":"We describe a verification pipeline that takes production Rust cryptographic code and produces machine-checked correctness proofs in Lean 4. The pipeline combines three components: symbolic extraction tools (Charon and Aeneas, or Hax) that lift Rust into Lean 4; formal cryptographic specification libraries (ArkLib and CompPoly, from the Verified zkEVM project) that provide the mathematical targets; and AI provers (Aristotle from Harmonic AI and Aleph from Logical Intelligence) that close the resulting proof obligations. Every proof is checked by the Lean kernel, so AI output cannot compromise soundness. Within the scope of the Ethereum Foundation's zkEVM Verification Project, we applied the pipeline to cryptographic primitives in Plonky3 (FRI folding, Mersenne31 and KoalaBear field arithmetic, Horner polynomial evaluation) and RISC Zero (Merkle inclusion verification). In addition, Aleph authored proofs of two bounds-style theorems in Plonky3's compute_log_arity_for_round that previously stood as sorry. The paper describes the architecture, walks through a running example based on Aleph's two proofs, reports which classes of proof obligations AI closed and which required manual work, and discusses the engineering gaps we encountered: Lean 4 toolchain drift across tools and specific Aeneas/Hax extraction limits. We also document concrete missing lemmas, tactic gaps, and code-generation friction points discovered during proof development. We hope this contribution lowers the barrier to adoption of formal verification and facilitates more effective use of AI in this pipeline. The result is a working pipeline for formal verification of Rust, with kernel-checked proofs and reproducible artefacts.","abs_url":"https://arxiv.org/abs/2605.30106","pdf_url":"https://arxiv.org/pdf/2605.30106v1","primary_category":"cs.LO","published":"2026-05-28","tags":["verification","cryptography"],"lean_note":"Pipeline lifts Rust crypto code into Lean 4 and closes correctness proofs with AI provers, kernel-checked.","repo":"https://github.com/Verified-zkEVM/rust-lean","thumb":"thumbs/2605.30106.jpg"},{"id":"2605.29955","title":"Formalizing Mathematics at Scale","authors":["Ahmad Rammal","Niket Patel","Fabian Gloeckle","Amaury Hayat","Julia Kempe","Remi Munos","Charles Arnal","Vivien Cabannes"],"abstract":"We present AutoformBot, a multi-agent system for building an Autoformalized Textbook Library At Scale (Atlas) in Lean 4. AutoformBot orchestrates thousands of LLM agents, equipped with formal verification tools, dependency-aware task scheduling, and collaborative version control, to translate informal textbook prose into machine-checked definitions and proofs. We apply our methods to a corpus of 26 open-access textbooks spanning analysis, algebra, topology, combinatorics, and probability, producing Atlas: a verified library of over 45,000 Lean 4 declarations and 500 thousand lines of code. We release two artifacts: (i) AutoformBot, the open-source multi-agent framework; and (ii) Atlas, the resulting formal library. Our results suggest that autoformalizing the core content of graduate-level mathematics at scale is now economically and technically feasible. This opens the door to the automated verification of both human- and machine-generated mathematics at a research level.","abs_url":"https://arxiv.org/abs/2605.29955","pdf_url":"https://arxiv.org/pdf/2605.29955v1","primary_category":"cs.AI","published":"2026-05-28","tags":["autoformalization","ai-agents","dataset"],"lean_note":"Builds AutoformBot multi-agent system producing Atlas, a verified library of 45,000+ Lean 4 declarations from textbooks.","repo":"https://github.com/bors-ng/bors-ng","thumb":"thumbs/2605.29955.jpg"},{"id":"2605.29001","title":"FormInv: A Measurement Protocol for Semantic Invariance in Mathematical Reasoning Benchmarks","authors":["Nishal Thomas","Noel Thomas"],"abstract":"A paraphrase-quality audit of MathCheck (ICLR 2025) detected 4 semantically incorrect paraphrases in 129 groups (3.1%); removing them drops GPT-4o from rank 2 to rank 4 and elevates Claude Haiku and DeepSeek V3 above it; these ranking changes are invisible to any single-model evaluation. Cross-model unanimity found these errors automatically (>= 3/4 models for MathCheck; >= 6/9 for our primary evaluation) for under $10; in our own dataset the same protocol found that 47% of auto-generated connective-variation paraphrases were semantically incorrect. That flaw compounds a deeper measurement gap: Claude Haiku 4.5 achieves 86% accuracy yet SCR=50%, meaning half its theorems are answered differently under semantically equivalent restatements, while aggregate accuracy across 9 models spans only 86-96% yet Semantic Consistency Rates (SCR) span 50-82% -- a 32-point gap invisible to standard benchmarks. Formally, for any target ranking over 9 frontier models there exists a weighting over paraphrase families that realizes it (No-Free-Benchmark corollary), because no model Pareto-dominates all families -- so benchmark designers who select families are implicitly choosing which model wins. FormInv supplies the audit protocol (replicated on external benchmarks at 100% recall), SCR and per-theorem Cochran's Q as primary invariance measures evaluated on 9 models across 366-811 items (on Lean4-verified theorems), and FormInvSelector for regime-aware model selection.","abs_url":"https://arxiv.org/abs/2605.29001","pdf_url":"https://arxiv.org/pdf/2605.29001v1","primary_category":"cs.LG","published":"2026-05-27","tags":["benchmark","mathlib"],"lean_note":"Builds the FormInv invariance benchmark on 103 Lean4-verified Mathlib4 theorems to measure LLM consistency across paraphrases.","thumb":"thumbs/2605.29001.jpg"},{"id":"2605.28586","title":"Stabilizer rank bounds for magic-state orbits","authors":["Farrokh Labib","Vincent Russo"],"abstract":"Distinct Clifford orbits of magic states can exhibit different stabilizer ranks at small tensor powers. We establish this for qutrits, where the single-qutrit Clifford group has four inequivalent orbits of magic states: Strange, Norrell, Hadamard-eigenstate, and the qutrit T-state, but a nontrivial upper bound on the asymptotic exponent had been pinned down for only the qutrit T-state. For the other three orbits we give explicit stabilizer decompositions, yielding upper bounds on the per-copy asymptotic stabilizer-rank exponent: $γ_S \\le \\log_3(2)/2 \\approx 0.316$ for the Strange state, and $γ_{H_3}, γ_N \\le \\log_3(4)/3 \\approx 0.421$ for the Hadamard-eigenstate and Norrell orbits, all strictly below the prior $γ_{T_3} \\le 1/2$ baseline. We also prove the first nontrivial $Ω(m / \\log m)$ asymptotic lower bounds for the Hadamard-eigenstate and Norrell orbits, and exhibit two-qutrit Clifford circuits that convert two copies of these states into an injectable phase state with constant success probability, enabling constant-overhead injection of one non-Clifford diagonal gate per orbit. In the case of qubits, we give a closed-form decomposition of the qubit T-type orbit at four copies matching the existing $γ_T \\le \\log_2(3)/4 \\approx 0.396$ exponent via a direct algebraic identity rather than an entangled cat-state construction. An open-source library stabrank accompanies the paper, with Lean 4 proof formalizations of all the decompositions.","abs_url":"https://arxiv.org/abs/2605.28586","pdf_url":"https://arxiv.org/pdf/2605.28586v1","primary_category":"quant-ph","published":"2026-05-27","tags":["formalization","quantum"],"lean_note":"Accompanies stabrank library with Lean 4 formalizations of all stabilizer-rank decompositions.","repo":"https://github.com/unitaryfoundation/stabrank","thumb":"thumbs/2605.28586.jpg"},{"id":"2605.28512","title":"On Compositional Learning Behaviours in Formal Mathematics","authors":["Kevin Yandoka Denamganaï"],"abstract":"Self-evolving scientific agents capable of conquering the hard tail of formal mathematics require Compositional Learning Behaviours (CLBs) -- the capacity to ground and recombine novel symbolic structures in context, beyond mere recombination of prelearned atoms. We propose \\textbf{S2B-LM}, an adaptation of the Symbolic Behaviour Benchmark that removes numerical processing as a confound and adds chain-of-thought scaffolding to elicit rather than merely probe latent CLB competency. Cross-evaluating ten Lean~4 theorem provers on CLB competency (adj-ZSCT) and miniF2F whole-proof performance, exact permutation tests establish a hierarchical necessity structure: search-heavy models cover the tractable bulk without detectable CLBs, yet every model breaking into the Olympiad-level tier (miniF2F $>75\\%$) is among the five highest CLB scorers ($p=0.004$). After ruling out model scale as a confound, our results show that CLB competency is \\emph{necessary but not sufficient} for the hard tail of formal mathematical verification.","abs_url":"https://arxiv.org/abs/2605.28512","pdf_url":"https://arxiv.org/pdf/2605.28512v1","primary_category":"cs.CL","published":"2026-05-27","tags":["theorem-proving-ml","benchmark"],"lean_note":"Cross-evaluates ten Lean 4 theorem provers on compositional learning behaviours and miniF2F whole-proof performance.","thumb":"thumbs/2605.28512.jpg"},{"id":"2605.28365","title":"Risk-Controlled Lean-as-Judge for Natural-Language Mathematical Reasoning","authors":["Pauline Bourigault","Xiaotong Ji","Matthieu Zimmer","Rasul Tutunov","Haitham Bou Ammar"],"abstract":"Lean is increasingly used to judge natural-language mathematical answers, but its signal is partial: many answers never formalize, and a failed proof may reflect an ill-typed statement or a missing library fact, not a wrong answer. On MATH-500 we show this signal is (i) sharply coverage-dependent, that is the proof-winning answer is correct 96% of the time at high proved coverage but 20% at low, and (ii) sparse and often unfaithful: a 7B autoformalizer proves a class for only 28% of problems, and a manual audit finds only approximately 43% of those proofs faithful. We propose COVCAL, a selector over Lean-trace diagnostics that certifies a finite-sample selective-risk bound on accepted answers or abstains, under two regimes (a conservative Bonferroni bound and a tighter dev-then-cal rule). Feasibility depends on autoformalization coverage: with the 7B formalizer the signal is too sparse and Bonferroni abstains on all 20 bootstrap partitions, whereas a prover-specialized formalizer reaches 79% coverage and flips it to feasible on 17 of 20, accepting approximately 48% of problems at 0.98 accepted accuracy. Since self-consistency alone is already 91% accurate, our contribution is a precise account of when, and with which formalizer, a partial formal signal can be trusted under risk control.","abs_url":"https://arxiv.org/abs/2605.28365","pdf_url":"https://arxiv.org/pdf/2605.28365v1","primary_category":"cs.AI","published":"2026-05-27","tags":["theorem-proving-ml","autoformalization","meta-study"],"lean_note":"Treats Lean as a partial-observation judge of natural-language math answers, computing coverage diagnostics from Lean autoformalization and proof-search status logs.","repo":"https://github.com/paulinebourigault/covcal-lean","thumb":"thumbs/2605.28365.jpg"},{"id":"2605.27485","title":"Automating Formal Verification with Agent-Guided Tree Search","authors":["Leo Yao"],"abstract":"Formal verification offers a path to provably correct software, but writing verified code remains expensive enough that the technique is rarely used in production. Recent large language models can accelerate this work, and recent benchmarks measure their ability to translate specifications into code and machine-checked proofs of correctness. This thesis evaluates the state of such LLM-driven verified-code generation (\"vericoding\") in Lean and develops search-based methods for improving verification performance. We first reproduce a subset of the vericoding-benchmark Lean leaderboard on a current cross-vendor model pool, finding that non-reasoning performance remains roughly steady on US closed-source models while open-weight models have slightly improved. We update the iterative methodology of vericoding-benchmark with an agentic loop equipped with mathlib search, finding that model performance greatly improves and scales with agent budget. GPT-5.4 nearly saturates the benchmark at 95.0% on 423 specs with $K=50$ LLM calls. We then design two agent-directed tree-search formulations: a state-based orchestrator that branches on partial-proof states, and a context-based orchestrator that branches on full subagent contexts. Compared against the agent baseline, the context-based design solves a wider range of intermediate-difficulty specs at lower token cost, while the agent baseline retains an advantage on the hardest specs, where uninterrupted iteration matters most. We conclude that search structure has selective advantages over a strong agent baseline, and that more challenging benchmarks drawn from modern code are important to measure and drive further progress in automated formal verification. Code available upon request by contacting the author at leoy@mit.edu.","abs_url":"https://arxiv.org/abs/2605.27485","pdf_url":"https://arxiv.org/pdf/2605.27485v1","primary_category":"cs.LO","published":"2026-05-26","tags":["theorem-proving-ml","verification","mathlib"],"lean_note":"Evaluates LLM vericoding in Lean and develops agentic tree-search proof methods using mathlib search.","thumb":"thumbs/2605.27485.jpg"},{"id":"2605.26959","title":"MerLean-Prover: A Recursive Looping Harness for Lean 4 Theorem Proving","authors":["Jinzheng Li","Zeru Zhu","Yuanjie Ren"],"abstract":"MerLean-Prover is an end-to-end Lean4 theorem prover that replaces sorry declarations with kernel-checkable proofs. It is built from three agent types (Planning, Check, and Lean) composed by a recursive outer loop whose unit of revision is the proof plan itself, and uses no fine-tuning, no custom RL objective, and no theorem-specific scaffolding. On FormalQualBench, a benchmark of 23 PhD-qualifying-exam theorems, MerLean-Prover solves 10/23, surpassing the strongest published open-source baseline (OpenGauss, 8/23). On Putnam2025, the same harness closes 12/12 with substantially lower total wall-clock than the next-best system that closes the full set. The harness also transfers to smaller models: Sonnet closes all four tested FormalQualBench problems, and Haiku closes the two short ones. These results suggest that harness design is a central factor in end-to-end Lean4 theorem proving, alongside raw model capability, and that a relatively simple harness can already be effective.","abs_url":"https://arxiv.org/abs/2605.26959","pdf_url":"https://arxiv.org/pdf/2605.26959v2","primary_category":"cs.LO","published":"2026-05-26","tags":["theorem-proving-ml","ai-agents","benchmark"],"lean_note":"Presents MerLean-Prover, an end-to-end agentic harness replacing sorry with kernel-checked Lean 4 proofs, evaluated on Putnam2025.","thumb":"thumbs/2605.26959.jpg"},{"id":"2605.25556","title":"Keep the Proof State Live: Snapshotting for Efficient Tactic Search in Lean 4","authors":["Austin Shen","Yunong Shi"],"abstract":"Automated theorem proving systems built on Lean 4 increasingly rely on parallel tactic search over partially specified proofs, such as those generated by Draft-Sketch-Prove (DSP) pipelines. In current systems, each search branch reconstructs a proof state by re-running elaboration, leading to substantial per-branch overhead. In Lean 4 with Mathlib, this cost has two components: (1) import loading, which deserializes pre-compiled libraries (~60 s per branch); and (2) theorem-body elaboration, which re-checks the theorem context up to the target goal (estimated 18-735 s depending on proof complexity). Together, these account for >99% of per-branch wall time, making portfolio-based search impractical at scale. We observe that this overhead arises from a mismatch between the structure of proof search and its execution model: branching is implemented via repeated reconstruction of proof states rather than direct reuse. To address this, we introduce proof-state snapshotting, which captures the elaborated proof state once and reuses it across branches via a small extension to the Lean 4 language server. Across 48 miniF2F-v2 problems (45 prove-phase benchmarks and 3 full end-to-end runs), our approach achieves a 5.6-50x wall-time speedup over the standard fallback (average 14x, median 9.7x). Speedup increases with the number of proof branches. Our method is orthogonal to import-level caching (e.g., Kimina Lean Server), which avoids import loading but not theorem-body elaboration. The patched Lean binary and the Snapshot-DSP pipeline will be released as open source upon publication.","abs_url":"https://arxiv.org/abs/2605.25556","pdf_url":"https://arxiv.org/pdf/2605.25556v2","primary_category":"cs.LO","published":"2026-05-25","tags":["tooling","tactic-automation"],"lean_note":"Introduces proof-state snapshotting via a Lean 4 language-server extension to speed parallel tactic search.","repo":"https://github.com/A2DR1/Lean_Snapshot","thumb":"thumbs/2605.25556.jpg"},{"id":"2605.23772","title":"Agentic Proving for Program Verification","authors":["Alessandro Sosso","Akhil Arora","Bas Spitters"],"abstract":"Agentic systems have recently emerged as state-of-the-art approaches for automated theorem proving in formal mathematics. To assess how far these capabilities extend to program verification, we evaluate Claude Code in an agentic proving framework on CLEVER, a Lean 4 benchmark for verifiable code generation. Our results show that Claude generates arguably valid specifications for 98.8% of problems (with 81.3% also accepted by CLEVER's isomorphism-based scoring on the correct portion of the benchmark), certifies implementations against correct ground-truth specifications for 87.5% of problems, and reaches a 98.1% success rate on the end-to-end program generation and verification pipeline over entries with self-consistent premises. Across all stages, Claude further provides high-quality feedback on its own attempts (as confirmed under manual review), identifying underlying causes of failure and lingering bugs in the dataset. These findings highlight a growing mismatch between the difficulty of existing program verification benchmarks and the capabilities of modern agentic provers, and point to the need for more rigorous, bug-resilient evaluation methodologies, and in particular for alternatives to isomorphism-based scoring of generated specifications. More broadly, our results provide empirical evidence that tight compiler-in-the-loop agentic paradigms are currently the most effective approach for foundational program verification.","abs_url":"https://arxiv.org/abs/2605.23772","pdf_url":"https://arxiv.org/pdf/2605.23772v1","primary_category":"cs.AI","published":"2026-05-22","tags":["ai-agents","verification","benchmark"],"lean_note":"Evaluates an agentic prover on CLEVER, a Lean 4 benchmark for verifiable code generation and specification.","repo":"https://github.com/cameronfreer/lean4-skills","thumb":"thumbs/2605.23772.jpg"},{"id":"2605.23419","title":"Generalized Stochastic Approximation of the Log-Likelihood Ratio for Robust Sequential Change-Point Detection","authors":["Serhii Zabolotnii"],"abstract":"Sequential change-point detection in non-Gaussian stochastic processes is challenging because the underlying densities are rarely known in real time. Classical parametric procedures such as CUSUM lose optimality under distributional mismatch, whereas nonparametric alternatives often react slowly. We develop a unified framework that approximates the log-likelihood ratio (LLR) on a generalized stochastic basis -- polynomial, logarithmic, or fractional-power -- using only moments up to order 3s, with no analytic form of the distribution, and thereby adapts the classical CUSUM, GRSh, and SRP procedures to non-Gaussian data. The convergence functional J(s) = K^T Y is interpreted as the projection of the Kullback-Leibler divergence onto the basis span, yielding a formal criterion for selecting the approximation order. We target the regime of small relative change-points, where the signal energy changes little but the shape of the distribution -- tail structure and modality -- does. A robust threshold follows from Kunchenko's probability-error bound (KU-PE), which controls the false-alarm rate without empirical tuning. On nine public benchmarks across four domains, the method is, to our knowledge, the only one operative on extremely heavy-tailed data (excess kurtosis gamma_4 > 20), where classical methods produce 100% false alarms, while reducing the detection delay at a guaranteed false-alarm level. The core theorems are formally verified in Lean 4.","abs_url":"https://arxiv.org/abs/2605.23419","pdf_url":"https://arxiv.org/pdf/2605.23419v2","primary_category":"stat.ME","published":"2026-05-22","tags":["formalization","probability"],"lean_note":"Core theorems of a sequential change-point detection framework are formally verified in Lean 4.","repo":"https://github.com/SZabolotnii/KuYuPe-Change_Point-code-supplement","thumb":"thumbs/2605.23419.jpg"},{"id":"2605.22885","title":"ImProver 2: Iteratively Self-Improving LMs for Neurosymbolic Proof Optimization","authors":["Riyaz Ahuja","Tate Rowney","Jeremy Avigad","Sean Welleck"],"abstract":"Formal mathematics libraries are rapidly expanding, creating a growing need to refactor verified proofs for maintainability and to improve training data quality for neural provers. However, scalable proof optimization is hindered by heterogeneous and heuristically specified objectives, scarce data, and high training and inference costs. To overcome these challenges, we introduce ImProver 2, a neurosymbolic framework for automated proof optimization in Lean 4. ImProver 2 combines a data-efficient expert-iteration pipeline with a scaffold that exposes formal structure alongside lightweight informal abstractions. We further introduce a suite of metrics capturing structural proof properties. Using ImProver 2, we train a 7B-parameter model that outperforms orders-of-magnitude larger models within the same model family, and is competitive with mid-tier frontier models across metrics. We additionally demonstrate that our neurosymbolic scaffold significantly improves performance across both small and frontier models. We show that with proper scaffolding and training, small models can effectively restructure research-level proofs over complex and varied metrics, matching substantially larger systems and establishing proof optimization as a scalable, learnable task.","abs_url":"https://arxiv.org/abs/2605.22885","pdf_url":"https://arxiv.org/pdf/2605.22885v1","primary_category":"cs.AI","published":"2026-05-21","tags":["theorem-proving-ml","tooling"],"lean_note":"Introduces ImProver 2, a neurosymbolic framework for automated proof optimization in Lean 4.","repo":"https://github.com/riyazahuja/improver","thumb":"thumbs/2605.22885.jpg"},{"id":"2605.22763","title":"Advancing Mathematics Research with AI-Driven Formal Proof Search","authors":["George Tsoukalas","Anton Kovsharov","Sergey Shirobokov","Anja Surina","Moritz Firsching","Gergely Bérczi","Francisco J. R. Ruiz","Arun Suggala","Adam Zsolt Wagner","Eric Wieser","Lei Yu","Aja Huang","Miklós Z. Horváth","Andrew Ferraiuolo","Henryk Michalewski","Edward Lockhart","Codrut Grosu","Thomas Hubert","Matej Balog","Pushmeet Kohli","Swarat Chaudhuri"],"abstract":"Large language models (LLMs) increasingly excel at mathematical reasoning, but their unreliability limits their utility in mathematics research. A mitigation is using LLMs to generate formal proofs in languages like Lean. We perform the first large-scale evaluation of this method's ability to solve open problems. Our most capable agent autonomously resolved 9 of 353 open Erdős problems at the per-problem cost of a few hundred dollars, proved 44/492 OEIS conjectures, and is being deployed in combinatorics, optimization, graph theory, algebraic geometry, and quantum optics research. A basic agent alternating LLM-based generation with Lean-based verification replicated the Erdős successes but proved costlier on the hardest problems. These findings demonstrate the power of AI-aided formal proof search and shed light on the agent designs that enable it.","abs_url":"https://arxiv.org/abs/2605.22763","pdf_url":"https://arxiv.org/pdf/2605.22763v2","primary_category":"cs.AI","published":"2026-05-21","tags":["theorem-proving-ml","ai-agents","combinatorics","number-theory"],"lean_note":"LLM agents generate Lean proofs verified by its compiler to autonomously resolve open Erdős and OEIS conjectures.","thumb":"thumbs/2605.22763.jpg"},{"id":"2606.02588","title":"Lean-GAP: A Dataset of Formalized Graduate Algebra Problems","authors":["Seewoo Lee","Byung-Hak Hwang","Hyojae Lim","Jihoon Hyun","Ilkyoo Choi","Yeachan Park","Jineon Baek","Hyukpyo Hong","Keewoo Lee","Jaeseong Heo","Hyungryul Baik","Chul-hee Lee","Kyu-Hwan Lee"],"abstract":"We present Lean-GAP (Lean-Graduate Agebra Problems), 430 formalized graduate-level algebra problems from the textbook Abstract Algebra by Dummit and Foote. We develop a scalable pipeline consisting of PDF-to-LaTeX preprocessing, autoformalization into Lean 4, and verification of informal-formal correspondence. While the preprocessing and autoformalization stages can be largely automated, we find that verification remains the most subtle and labor-intensive component, requiring careful human oversight. Our contributions include (i) the construction of a structured dataset of formalized exercises, (ii) a systematic methodology for formalizing textbook mathematics, and (iii) an analysis of recurring challenges in the formalization process. We also compare the performance of different autoformalization models and highlight key bottlenecks in translating informal statements into formal language.","abs_url":"https://arxiv.org/abs/2606.02588","pdf_url":"https://arxiv.org/pdf/2606.02588v1","primary_category":"cs.LO","published":"2026-05-20","tags":["dataset","autoformalization","algebra"],"lean_note":"Builds Lean-GAP, 430 graduate algebra problems autoformalized into Lean 4 with a verification pipeline.","repo":"https://github.com/riccardobrasca/constructible","thumb":"thumbs/2606.02588.jpg"},{"id":"2605.21718","title":"Reciprocals of Partition Polynomials","authors":["Evan Chen","Ken Ono","Jujian Zhang"],"abstract":"Ballantine--Beck--Feigon--Maurischat introduced the subsum polynomial \\[ \\operatorname{sp}(λ,x):=\\prod_i (1+x^{λ_i}) \\] attached to an integer partition $λ$, and studied rational functions obtained by summing reciprocals of these polynomials over natural classes of partitions. They posed ten conjectures which naturally divide into coprimality and divisibility questions, special-value and recurrence formulas, and coefficient-shape problems. We prove all of the conjectures in the first two families: the ordinary and binary coprimality/divisibility conjectures, and the odd and ternary special-value/recurrence conjectures. AxiomProver autonomously produced Lean/mathlib formalizations and machine-checkable proofs of these six conjectures, and also discovered a counterexample to the statement as printed; the corrected form remains open.","abs_url":"https://arxiv.org/abs/2605.21718","pdf_url":"https://arxiv.org/pdf/2605.21718v2","primary_category":"math.CO","published":"2026-05-20","tags":["autoformalization","combinatorics","number-theory"],"lean_note":"AxiomProver produced Lean/mathlib formalizations of six partition-polynomial conjectures.","repo":"https://github.com/AxiomMath/PartitionPolynomial","thumb":"thumbs/2605.21718.jpg"},{"id":"2605.20527","title":"$L^2$-Stability for STFT phase retrieval","authors":["Susanna Bertolini","Jaume de Dios Pont","Ben Pineau","Mitchell A. Taylor","João P. G. Ramos"],"abstract":"We prove that the short-time Fourier transform with Gaussian window performs $L^2$-local stable phase retrieval at the constant function. The proof involved significant interplay between mathematicians and LLMs. An autoformalization in Lean 4 of an extension of our result to $L^2$-local stable phase retrieval for all Hermite windows and all elements in the finite span of the canonical basis vectors is also presented.","abs_url":"https://arxiv.org/abs/2605.20527","pdf_url":"https://arxiv.org/pdf/2605.20527v1","primary_category":"math.FA","published":"2026-05-19","tags":["autoformalization","analysis"],"lean_note":"Presents a Lean 4 autoformalization of an STFT phase-retrieval stability result for Hermite windows.","repo":"https://github.com/susannabertolini/PhaseRetrieval","thumb":"thumbs/2605.20527.jpg"},{"id":"2605.20440","title":"Group-Algebraic Tensors: Provably-optimal Equivariant Learning and Physical Symmetry Discovery","authors":["Paulina Hoyos","Shashanka Ubaru","Dongsung Huh","Vasileios Kalantzis","Kenneth L. Clarkson","Misha Kilmer","Haim Avron","Lior Horesh"],"abstract":"We introduce the $\\star_G$ tensor algebra, in which any finite group $G$ defines the multiplication rule, making equivariance an intrinsic algebraic property rather than an architectural constraint. The framework rests on three machine-verified theoretical pillars: (i)~an Eckart-Young optimality guarantee for the $\\star_G$-SVD: the first such result for symmetry-preserving tensor approximation, exact and polynomial-time; (ii)~a Kronecker factorization that composes multiple symmetries by replacing $F_G$ with $F_{G_1} \\otimes F_{G_2}$ with no architectural redesign; and (iii)~a 600-line Lean~4 formalization of the $\\star_G$ algebra. The framework provides capabilities that equivariant neural networks (ENNs) structurally cannot: a closed-form per-irreducible-representation decomposition of every prediction, and data-driven discovery of the symmetry group that best fits a dataset. As a non-trivial empirical demonstration, decomposing QM9 molecular geometry over the chiral octahedral subgroup of SO(3) recovers the Wigner--Eckart selection rules of angular momentum from data alone, with no quantum mechanical input: scalar properties are A$_1$-dominated, dipole components are T$_1$-dominated, the isotropic polarizability is uniquely insensitive to $l\\!=\\!1$ as the rank-2-trace decomposition $l\\!=\\!0 \\oplus l\\!=\\!2$ requires, and the T$_1$/A$_1$ predictive-power ratio separates vector observables from scalar observables by a factor of five. On full QM9 (130{,}831 molecules), $\\star_G$-SVD with ridge regression provides closed form predictions at $\\sim50-90\\times$ fewer parameters than parameter-matched MLPs. Algebraic equivariance thus complements architectural equivariance not as a faster-better-cheaper alternative but as a different mathematical affordance: provably-optimal symmetry-preserving compression, per-irrep interpretability, and data-driven physical discovery.","abs_url":"https://arxiv.org/abs/2605.20440","pdf_url":"https://arxiv.org/pdf/2605.20440v1","primary_category":"cs.LG","published":"2026-05-19","tags":["formalization","algebra","ml-theory"],"lean_note":"Includes a 600-line Lean 4 formalization of the star-G tensor algebra underpinning equivariant learning.","repo":"https://github.com/e3nn/e3nn","thumb":"thumbs/2605.20440.jpg"},{"id":"2605.20120","title":"Using Aristotle API for AI-Assisted Theorem Proving in Lean 4: A Formalisation Case Study of the Grasshopper Problem","authors":["Gabriel Rongyang Lau"],"abstract":"AI-assisted theorem proving can now generate substantial Lean developments for olympiad-level mathematics, but the evidential status of such developments depends on which declarations are actually verified. This paper reports a Lean 4 formalization case study of an Aristotle API proof attempt for the Grasshopper problem, originally posed as IMO 2009 Problem 6. The generated artifact states a generalized Lean version of the theorem, contains four verified helper lemmas for local components of a maximality and adjacent-swap exchange strategy, and leaves the main theorem grasshopper closed directly by one unresolved sorry. The verified components establish that the final partial sum equals the total sum, that an adjacent transposition can affect only the relevant intermediate partial sum, that the changed partial sum has the expected form, and that maximality at a position admitting an adjacent successor swap forces a corresponding forbidden-set membership fact. The Aristotle output summary identifies the intended remaining mathematical step as the global counting step needed to show that these membership facts produce at least n distinct forbidden values, contradicting the cardinality assumption |M| < n; the Lean source itself does not reduce the main theorem to a separately encoded counting lemma. This case study gives an inspectable example of a central limitation in AI-assisted formalization, namely that local proof search can succeed while the global combinatorial bookkeeping required for a theorem remains unresolved. The paper contributes a reproducible Lean artifact and a precise analysis of its verified and unverified proof content.","abs_url":"https://arxiv.org/abs/2605.20120","pdf_url":"https://arxiv.org/pdf/2605.20120v1","primary_category":"cs.AI","published":"2026-05-19","tags":["theorem-proving-ml","ai-agents","combinatorics"],"lean_note":"Case study formalizing an Aristotle API proof attempt for the IMO Grasshopper problem in Lean 4.","thumb":"thumbs/2605.20120.jpg"},{"id":"2605.19632","title":"Executable Boundary Contracts for Sound Event Traces","authors":["Faruk Alpay","Hamdi Alakkad"],"abstract":"Sound event reports often compress timed boundary behavior into frame, segment, or event scores. This paper defines executable boundary contracts for finite sound event traces. The frame fragment is a bounded Boolean fragment embeddable in STL after grid projection. The event layer adds declared interval matching, duration clauses, fragmentation clauses, and obligation restricted vector scoring. The aim is measurement, not a new general temporal logic and not a challenge leaderboard. The artifact evaluates controlled Mini LibriSpeech seeded scenes, MAESTRO Real soundscapes, frozen pretrained timing probes, and an official DCASE 2024 Task 4 baseline track. Across these tracks, standard scores and contract coordinates disagree in interpretable ways. The strongest real corpus finding is that union activity can hide typed boundary failure, while external DCASE outputs provide a class indexed challenge level reference. Code, generated tables, manifests, and Lean checks for the finite frame core are supplied as ancillary material.","abs_url":"https://arxiv.org/abs/2605.19632","pdf_url":"https://arxiv.org/pdf/2605.19632v1","primary_category":"cs.LO","published":"2026-05-19","tags":["verification","logic"],"lean_note":"A Lean 4 development machine-checks the finite frame core of the trace contract: obligation counting, neighborhood-witness monotonicity, and the Boolean implication evaluator.","repo":"https://zenodo.org/records/11034682","thumb":"thumbs/2605.19632.jpg"},{"id":"2605.20244","title":"Lean Refactor: Multi-Objective Controllable Proof Optimization via Agentic Strategy Search","authors":["Jialin Lu","Soonho Kong","Rodrigo Stehling","Kaiyu Yang","Zhangyang Wang","Weiran Sun","Wuyang Chen"],"abstract":"We present Lean Refactor, a plug-and-play retrieval-augmented agentic framework for multi-objective, controllable, and version-robust refactoring of Lean proofs. LLM-generated proofs are notoriously correct-but-verbose and brittle across library versions, yet existing refactoring works overlook three practical challenges: 1) Lean refactoring is natively multi-objective (proof length, compilation cost, and version compatibility are often in tension); 2) Lean repositories have fragile compatibility, whereas LLM releases are unaware of Lean/Mathlib versions; 3) Training-based pipelines require repeated fine-tuning with each new LLM release, scaling neither with model churn nor with Lean's release cycle. Lean Refactor steers a frozen agentic LLM with retrievals from a curated database of multi-objective refactoring strategies, each densely annotated with metadata such as supported Lean/Mathlib versions and expected compilation-cost reduction. Experiments show over $70\\%$ token-level compression on competition benchmarks, over $20\\%$ on research repositories, and up to $60\\%$ compilation-time reduction, outperforming prior work and Claude Code. Version-filtered retrieval further improves compression on the target Lean version, and refactored miniF2F proofs exhibit stronger zero-shot version transfer to future Lean releases than their unrefactored counterparts.","abs_url":"https://arxiv.org/abs/2605.20244","pdf_url":"https://arxiv.org/pdf/2605.20244v1","primary_category":"cs.LO","published":"2026-05-18","tags":["tooling","ai-agents","theorem-proving-ml"],"lean_note":"Presents Lean Refactor, a retrieval-augmented agentic framework for multi-objective refactoring of Lean/Mathlib proofs.","repo":"https://github.com/oOo0oOo/lean-lsp-mcp","thumb":"thumbs/2605.20244.jpg"},{"id":"2605.19063","title":"Mapping Uncharted Symmetries: Machine Discovery in Combinatorics","authors":["Eugenio Cainelli","Lorenzo Luccioli","Alessandro Iraci","Michele D'Adderio","Giovanni Paolini"],"abstract":"Inspired by long-standing open problems in algebraic combinatorics, we show that modern machine learning can meaningfully contribute to verifiable mathematical discoveries. In particular, we focus on the construction of simple mathematical functions under exact distributional constraints, a setting we formalize as Simple Learning Under Rigid Proportions (SLURP). We tackle this problem by introducing two methods: MapSeek-Functional, which models the desired function alternating pseudo-labeling and supervised training steps; and MapSeek-Symbolic, designed to directly produce symbolic formulas. We successfully apply both methods to a research problem in algebraic combinatorics, discovering a new combinatorial interpretation of the $q,t$-Narayana polynomials arising from representation theory. To our knowledge, this is the first such interpretation based on noncrossing partitions. Using one discovered statistic, we find a combinatorial proof of the symmetry of these polynomials in a previously unsolved case. To streamline verification and reproducibility, we release all code, including a formalization of all the mathematical discoveries of this paper in Lean 4.","abs_url":"https://arxiv.org/abs/2605.19063","pdf_url":"https://arxiv.org/pdf/2605.19063v1","primary_category":"cs.LG","published":"2026-05-18","tags":["formalization","combinatorics"],"lean_note":"Formalizes in Lean 4 the new combinatorial interpretation and symmetry proof of q,t-Narayana polynomials.","thumb":"thumbs/2605.19063.jpg"},{"id":"2605.17283","title":"OProver: A Unified Framework for Agentic Formal Theorem Proving","authors":["David Ma","Kaijing Ma","Shawn Guo","Yunfeng Shi","Enduo Zhao","Jiajun Shi","Zhaoxiang Zhang","Gavin Cheung","Jiaheng Liu","Zili Wang"],"abstract":"Recent progress in formal theorem proving has benefited from large-scale proof generation and verifier-aware training, but agentic proving is rarely integrated into prover training, appearing only at inference time. We present OProver, a unified framework for agentic formal theorem proving in Lean 4, in which failed proof attempts are iteratively revised using retrieved compiler verified proofs and Lean compiler feedback. OProver is trained through continued pretraining followed by iterative post-training: each iteration runs agentic proving, indexes newly verified proofs into OProofs and the retrieval memory, uses repair trajectories as SFT data, and uses unresolved hard cases for RL. OProofs is built from public Lean resources, large-scale proof synthesis, and agentic proving traces, containing 1.77M Lean statements, 6.86M compiler-verified proofs, and serialized trajectories with retrieved context, failed attempts, feedback, and repairs. Across five benchmarks, OProver-32B attains the best Pass@32 on MiniF2F (93.3%), ProverBench (58.2%), and PutnamBench (11.3%), and ranks second on MathOlympiad (22.8%) and ProofNet (33.2%) more top placements than any prior open-weight whole-proof prover.","abs_url":"https://arxiv.org/abs/2605.17283","pdf_url":"https://arxiv.org/pdf/2605.17283v1","primary_category":"cs.CL","published":"2026-05-17","tags":["theorem-proving-ml","ai-agents"],"lean_note":"Presents OProver, an agentic framework training and proving formal theorems in Lean 4 with compiler feedback.","thumb":"thumbs/2605.17283.jpg"},{"id":"2605.17255","title":"CAM-Bench: A Benchmark for Computational and Applied Mathematics in Lean","authors":["Wentao Long","Yunfei Zhang","Chenyi Li","Li Zhou","Chumin Sun","Zaiwen Wen"],"abstract":"Formal theorem-proving benchmarks enable mechanically verifiable evaluation of mathematical reasoning in large language models. However, existing benchmarks mainly focus on Olympiad-style problems and algebraic domains, leaving computational and applied mathematics underrepresented. We introduce CAM-Bench, a Lean 4 theorem-proving benchmark of 1,000 Lean proof targets in computational and applied mathematics, with coverage spanning optimization, numerical linear algebra, and numerical analysis. These problems are adapted from textbook exercises and often depend on locally introduced definitions, notation, algorithms, and elementary results. To construct CAM-Bench, we develop a dependency-recovery pipeline that reconstructs the local textbook context needed to state each problem faithfully. It then normalizes each problem into a standalone informal theorem and translates it into a Lean target. We validate the resulting formal problems through Lean compilation and semantic review, checking both formal correctness and semantic alignment with the original exercises. For each problem, we release the raw exercise, recovered context, normalized informal theorem, and final Lean target. CAM-Bench complements existing formal mathematics benchmarks by targeting applied mathematics problems that rely on textbook concepts and elementary theorems, many of which are not directly available as standard Mathlib4 lemmas. We evaluate widely used large language models and formalization agents on CAM-Bench, and analyze common failure modes in tracking local assumptions, applying elementary results, decomposing proofs, and maintaining long-horizon control in Lean.","abs_url":"https://arxiv.org/abs/2605.17255","pdf_url":"https://arxiv.org/pdf/2605.17255v1","primary_category":"cs.AI","published":"2026-05-17","tags":["benchmark","autoformalization"],"lean_note":"Introduces CAM-Bench, 1000 Lean 4 proof targets in computational and applied mathematics with a dependency-recovery pipeline.","repo":"https://github.com/optpku/CAM-Bench","thumb":"thumbs/2605.17255.jpg"},{"id":"2605.16846","title":"Efficient frequentist fractional polynomials for skewed dose-response and survival data: a variance-reducing alternative to OLS-FP","authors":["Serhii Zabolotnii"],"abstract":"Fractional polynomials (FP) are a standard tool for modelling nonlinear dose-response and covariate effects, implemented in the widely used mfp package. The conventional FP fit estimates its coefficients by ordinary least squares (OLS-FP), which is statistically inefficient when the regression errors are skewed or heavy-tailed, a common situation for survival times, concentrations and biomarkers. We present a drop-in replacement that keeps the identical FP model and design but estimates the coefficients with a moment-based score tuned to the residual skewness and kurtosis, giving a closed-form efficiency factor g2 = 1 - gamma3^2/(2+gamma4) relative to OLS-FP. Across skewed error laws the method reduces slope-coefficient variance by 10-20% for mildly skewed errors and up to roughly 60% for heavy-tailed log-normal errors, at realistic sample sizes, while keeping confidence-interval coverage close to nominal, and it reverts exactly to OLS-FP under symmetry, so it is never harmful when no gain is available. On the German Breast Cancer Study Group cohort it narrows the tumour-size confidence interval by 26% (bootstrap variance ratio 0.53 against the predicted 0.56), and a primary-biliary-cirrhosis cohort reproduces the gain. The estimator is closed-form, runs in milliseconds, and is released as a reproducible R package (pmm_fp in EstemPMM) with a one-command replication bundle; its core variance identity is machine-checked in Lean 4.","abs_url":"https://arxiv.org/abs/2605.16846","pdf_url":"https://arxiv.org/pdf/2605.16846v3","primary_category":"stat.ME","published":"2026-05-16","tags":["formalization","probability"],"lean_note":"The estimator's core variance identity is machine-checked in Lean 4.","repo":"https://github.com/SZabolotnii/Ku-PMM-FP-code-supplement","thumb":"thumbs/2605.16846.jpg"},{"id":"2605.16523","title":"End-to-End Formalization of Quantum Error Correction","authors":["Mattias Ehatamm","Yi Lee","Xiaodi Wu","Runzhou Tao"],"abstract":"Quantum error-correcting codes (QECCs) sit between noisy quantum hardware and reliable computation, so the code parameters used in practice must be trustworthy. The single number that summarizes a code's strength is its distance, yet certifying a distance lower bound is NP-hard in general, placing it beyond the reach of pen-and-paper proofs as well as direct proof-assistant scripting. As a result, distance values in the literature come either from non-scaling hand proofs, or from unverified solvers that leave a trust gap exactly where the code is supposed to provide a guarantee. We present Lean-QEC, the first Lean 4 formalization of stabilizer-code theory that delivers end-to-end, machine-checked distance certificates at industrial code sizes. Lean-QEC formalizes the linear algebra of qubit states, the Pauli group, stabilizer codes, the binary symplectic representation, classical coding theory, and the CSS and Bivariate Bicycle families. To break the combinatorial barrier, Lean-QEC translates the distance condition into a Boolean satisfiability formula through a verified reduction. The pipeline scales through a BitVec-flattened encoding that replaces Lean's Matrix representation, and an error-location encoding that reduces the variable count from $n$ to $k\\lceil \\log_2 n\\rceil$. With these, we obtain automatically-generated Lean-checked distance proofs for a large range of industrially viable qLDPC codes within the Bivariate Bicycle and Generalized Bicycle families, including [[90, 8, 10]] and [[70, 6, 9]] BB codes, with the formulation scaling up to 144 qubits when performed outside the Lean kernel. The resulting library is reusable and is designed to plug into broader Lean-based efforts toward end-to-end verification of fault-tolerant quantum computation.","abs_url":"https://arxiv.org/abs/2605.16523","pdf_url":"https://arxiv.org/pdf/2605.16523v1","primary_category":"quant-ph","published":"2026-05-15","tags":["formalization","verification","quantum"],"lean_note":"Presents Lean-QEC, a Lean 4 formalization of stabilizer-code theory delivering machine-checked quantum code distance certificates.","thumb":"thumbs/2605.16523.jpg"},{"id":"2605.16170","title":"BAPR: Bayesian amnesic piecewise-robust reinforcement learning for non-stationary continuous control","authors":["Yifan Zhang","Liang Zheng"],"abstract":"Real-world control systems frequently operate under \\emph{piecewise stationary} conditions, where dynamics remain stable for extended periods before undergoing abrupt regime changes. Standard robust RL methods face a fundamental dilemma: a globally conservative policy wastes performance during stable periods, while a locally adaptive policy risks catastrophic failure when the regime changes undetected. We propose \\textbf{BAPR} (Bayesian Amnesic Piecewise-Robust SAC), which unifies Bayesian Online Change Detection (BOCD) with robust ensemble RL. The BAPR operator -- a convex combination of mode-conditional Bellman operators weighted by a frozen belief distribution -- is a $γ$-contraction. A complementary counterexample, machine-verified in Lean~4, establishes a \\emph{sharp boundary}: when beliefs depend on the Q-function, the contraction factor becomes $γ+ λΔ$ (where $Δ$ is the mode reward gap), and contraction fails exactly when $γ+ λΔ\\geq 1$. We derive a \\emph{component-wise} formal error budget for the abstract operator -- every component machine-verified -- bounding post-switch recovery; the budget applies to the abstract mode-mixture operator and inherits to the implemented shared-critic algorithm only through the frozen-parameter design intuition. All results are formally verified with no \\texttt{sorry} (1,145 lines across 3 Lean~4 files, 22 machine-verified theorems). BOCD drives an adaptive conservatism mechanism: the policy becomes maximally conservative after detected change-points and smoothly relaxes as confidence grows, with detection delay $O(\\log(1/δ))$. A context-conditioning module trained via RMDM loss provides mode-aware representations from simulator-provided mode IDs at training time and requires no mode labels at deployment.","abs_url":"https://arxiv.org/abs/2605.16170","pdf_url":"https://arxiv.org/pdf/2605.16170v2","primary_category":"cs.LG","published":"2026-05-15","tags":["verification","ml-theory"],"lean_note":"Machine-verifies a contraction counterexample and error budget in Lean 4 across three files with 22 theorems.","repo":"https://github.com/erzhu419/BAPR","thumb":"thumbs/2605.16170.jpg"},{"id":"2605.16169","title":"LeanBET: Formally-verified surface area calculations in Lean","authors":["Ejike D. Ugwuanyi","Colin T. Jones","John Velkey","Tyler R. Josephson"],"abstract":"The Brunauer--Emmett--Teller (BET) method is a standard tool for estimating surface areas from adsorption isotherms, yet practical implementations involve multiple algorithmic steps whose correctness is rarely made explicit. In this work, we present a fully executable and formally verified BET analysis pipeline implemented in the Lean~4 theorem prover. Our formalization covers the complete BET Surface Identification (BETSI)-style workflow, including window enumeration, monotonicity checks, knee selection, and linear regression. We carry out computations in floating-point arithmetic and develop the corresponding correctness proofs over the real numbers, using a shared polymorphic implementation that supports both. On the proof side, we show that the regression coefficients returned by the algorithm agree with their specification-level definitions and minimize the least-squares error under the stated assumptions. We also formalize the algebraic derivation of the BET linearized expression and connect that result directly to the executable analysis pipeline. We further prove that the window enumeration is sound and complete, and that the admissibility checks and knee-based selection satisfy their formal specifications. We evaluate the implementation against the BETSI reference method on benchmark adsorption isotherms. Compared to BETSI, LeanBET agrees to machine precision for 18 of the 19 isotherms, with only a 0.03\\% deviation for the UiO-66 dataset. This demonstrates that a scientific computing workflow can be built in Lean, yielding both formal verification guarantees and numerical agreement with an established Python reference implementation.","abs_url":"https://arxiv.org/abs/2605.16169","pdf_url":"https://arxiv.org/pdf/2605.16169v1","primary_category":"cs.LO","published":"2026-05-15","tags":["verification","analysis"],"lean_note":"Implements a formally verified BET surface-area analysis pipeline in Lean 4 with correctness proofs over the reals.","repo":"https://github.com/ATOMSLab/LeanBET","thumb":"thumbs/2605.16169.jpg"},{"id":"2605.23983","title":"Saturating Scaling Laws for Equational Discovery: A Phenomenology of Growth Dynamics in Three Toy Substrates with Two Real-World Replications","authors":["Fabio Rovai"],"abstract":"We investigate growth dynamics in deterministic equational discovery substrates. Across three toy domains (arithmetic, boolean, higher-order list; n=592 trajectories), short-range substrate sizes fit a power-law N(t) proportional to t^b. Within each substrate b is architecture-sensitive (cross-validated R^2 approximately 0.82); the regression does not transfer across substrates (arith+bool to list yields R^2 approximately -0.84). A heuristic mean-field closure model predicts a saturating power-law dN/dt = K N^k exp(-mu N) of which the pure power-law is the short-range approximation. Three robustness checks: bootstrap intervals on (k, mu) are tight in 4/5 toy trajectories and degenerate in 1/5; out-of-sample forecasting on toy data (fit first 100 epochs, predict next 400) is won by pure power-law 5/5, indicating the toy trajectories do not reach saturation; on two real-world growth proxies the result splits. New Mathlib/*.lean file additions per month (mathlib4, 60 months, 9701 files) support the saturating form on OOS forecasting by approximately 7x over pure power-law; Coq mathcomp monthly commits (129 months, 3083 commits) favour pure power-law on both tests with mu collapsing to zero. The dynamics are substrate-conditional at two levels: within-substrate architecture-to-b regressions do not transfer, and the preferred functional family for N(t) itself (pure vs. saturating power-law) differs by substrate. We propose \"saturating power-law growth with substrate-conditional (k, mu), observable when the substrate has reached its saturation regime\" as a working framing.","abs_url":"https://arxiv.org/abs/2605.23983","pdf_url":"https://arxiv.org/pdf/2605.23983v1","primary_category":"cs.AI","published":"2026-05-14","tags":["meta-study","mathlib"],"lean_note":"Studies Mathlib4 monthly file-addition counts as a real-world growth dataset to test saturating power-law models.","thumb":"thumbs/2605.23983.jpg"},{"id":"2605.15445","title":"From LLM-Generated Conjectures to Lean Formalizations: Automated Polynomial Inequality Proving via Sum-of-Squares Certificates","authors":["Ruobing Zuo","Hanrui Zhao","Gaolei He","Zhengfeng Yang","Jianlin Wang"],"abstract":"Automated proving of polynomial inequalities is a fundamental challenge in automated mathematical reasoning, where rich algebraic structure and a rapidly growing certificate search space hinder scalability. Purely symbolic approaches provide strong guarantees but often scale poorly as the number of variables or the degree increases, due to expensive algebraic manipulations and rapidly growing intermediate expressions. In parallel, LLM-guided methods have made notable progress, particularly on competition-style inequalities with a small number of variables. To address the remaining scalability challenges, we propose NSPI, a neuro-symbolic framework that combines the complementary strengths of LLMs and symbolic computation for polynomial-inequality proving. Concretely, an LLM proposes a conjecture in the form of an approximate polynomial Sum-Of-Squares (SOS) decomposition; we refine it via symbolic computation to obtain an exact polynomial SOS representation, which directly proves the target inequality, and we further certify the proof in Lean, yielding an end-to-end pipeline from heuristic discovery to machine-checked proof. Experiments on challenging benchmarks involving polynomials with up to 10 variables demonstrate the effectiveness and scalability of the proposed method.","abs_url":"https://arxiv.org/abs/2605.15445","pdf_url":"https://arxiv.org/pdf/2605.15445v1","primary_category":"cs.AI","published":"2026-05-14","tags":["theorem-proving-ml","autoformalization","benchmark","algebra"],"lean_note":"LLM-proposed sum-of-squares decompositions are refined symbolically and the resulting inequality proofs are machine-checked in Lean 4.","thumb":"thumbs/2605.15445.jpg"},{"id":"2605.15163","title":"Automating Bitvector and Finite Field Equivalence Proofs in Lean","authors":["Elizaveta Pertseva","Valentin Robert","Clark Barrett","James Parker"],"abstract":"Efforts to verify Zero-Knowledge Proof circuit encodings have highlighted the challenge of proving the correctness of quantifier-free statements that make use of both bitvector and finite field operations. Existing verification workflows are either manual or rely on SMT solvers, which scale poorly on some classes of problems for reasons that include difficulties with conversion operators and challenges reasoning about inequalities. To address these limitations, we present a novel Lean tactic BitModEq that leverages range lemmas and case analysis to produce verified translations from finite fields to bitvectors. Our approach, combined with bit-blasting, outperforms state-of-the-art SMT solvers, solving 19% more ZKP arithmetization benchmarks.","abs_url":"https://arxiv.org/abs/2605.15163","pdf_url":"https://arxiv.org/pdf/2605.15163v1","primary_category":"cs.LO","published":"2026-05-14","tags":["tactic-automation","verification","cryptography"],"lean_note":"Introduces a Lean tactic BitModEq using range lemmas and case analysis to verify translations between finite fields and bitvectors for ZKP circuits.","thumb":"thumbs/2605.15163.jpg"},{"id":"2605.14549","title":"CSLibPremiseBench: Structure-Guided Premise Retrieval and Label Robustness for Lean 4 Computer-Science Theorems","authors":["Junye Ji"],"abstract":"CSLib is an emerging Lean 4 library for computer-science formalization, but its premise-retrieval behavior is not well represented by broad mathematical theorem-proving benchmarks. We introduce CSLibPremiseBench, a reproducible CSLib-specific benchmark and empirical study for source-visible premise retrieval over Lean 4 theorem and lemma declarations. The benchmark pins CSLib v4.29.0 at commit 0d37cc7fcc985cfc53b155e7eef2453f846c6da2, builds with Lean 4.29.0, and evaluates a strict import/source-order task set with 801 proxy-labelable tasks and 1875 CSLib candidate declarations. The labels are source-visible CSLib proof-reference proxies, not elaborated Lean dependency traces. We audit label robustness using stricter source-visible matching and a 300-task Lean environment expression probe, then compare BM25, symbol/name overlap, namespace/module and import-graph heuristics, PageRank-style module priors, fixed hybrids, and CSG-Rerank, a structure-guided graph-lexical reranker. CSG-Rerank gives a modest early-rank MRR gain over lexical BM25 under the strict policy, but does not reliably outperform BM25+symbol and does not improve Recall@10. A context-packet audit similarly finds stronger module/family concentration without reliable top-k proxy-gold coverage or token-utility gains. We position CSLibPremiseBench as a benchmark and audit paper: repository structure and candidate-policy design materially shape CSLib premise retrieval, proxy labels require explicit caveats, and proof-generation or proof-repair performance is not claimed.","abs_url":"https://arxiv.org/abs/2605.14549","pdf_url":"https://arxiv.org/pdf/2605.14549v1","primary_category":"cs.LO","published":"2026-05-14","tags":["benchmark","dataset","theorem-proving-ml"],"lean_note":"Introduces CSLibPremiseBench, a benchmark for premise retrieval over Lean 4 CSLib theorem and lemma declarations.","repo":"https://github.com/JJYYY-JJY/CSLibPremiseBench","thumb":"thumbs/2605.14549.jpg"},{"id":"2605.16407","title":"Proof-Carrying Certificates for LLM Pipelines: A Trust-Boundary Architecture","authors":["George Koomullil"],"abstract":"We present a framework for verifying the deterministic structured computations surrounding a large language model rather than the model itself, extending a Lean 4 trust-boundary architecture to the generic interfaces of modern LLM pipelines. Certificate validity is a Lean 4 kernel type-check plus a sorry-free transitive axiom audit against the trusted set {propext, Classical.choice, Quot.sound}; other assumptions are declared and partitioned by tier (mathematical placeholders, cryptographic assumptions, ML/human oracles). The technical contribution comprises three local certificate families and two operators. The families are conflict-aware bilattice grounding (with an emission-gate soundness lemma), embedding sensitivity and paraphrase stability, and Hoare-style agent action. The operators are a Maximal Certifiable Residue, which turns abstention into the maximum-weight certifiable residue with audit-logged dropped claims, and a Compositional Stability theorem, which yields a closed-form pipeline-wide perturbation budget from per-layer gains and margins. The three families plus a Universal Assurance Card consolidator form the per-call deliverable for high-stakes deployments: patent and legal retrieval, regulated finance, clinical decision support, and agentic systems with irreversible side effects. A compiled Lean 4 reference artifact (Lean v4.30.0-rc2, Mathlib) covers all 22 certificate types, with 17 of 46 kernel-audited declarations axiom-free, the rest depending only on the trusted set and declared assumptions, and zero uses of sorryAx or Lean.ofReduceBool. The three families are empirically tested through four registered pilots: bilattice grounding on adversarially perturbed HotpotQA, embedding sensitivity in short- and long-form settings, and Hoare-style agent action on a filesystem sandbox with adversarial prompt injection.","abs_url":"https://arxiv.org/abs/2605.16407","pdf_url":"https://arxiv.org/pdf/2605.16407v1","primary_category":"cs.LO","published":"2026-05-13","tags":["verification"],"lean_note":"Extends a Lean 4 trust-boundary architecture producing kernel-type-checked certificates for LLM pipelines.","repo":"https://github.com/gkoomullil/proof-carrying-certificates","thumb":"thumbs/2605.16407.jpg"},{"id":"2605.13171","title":"Formal Conjectures: An Open and Evolving Benchmark for Verified Discovery in Mathematics","authors":["Moritz Firsching","Paul Lezeau","Salvatore Mercuri","Miklós Z. Horváth","Yaël Dillies","Calle Sönne","Eric Wieser","Fred Zhang","Thomas Hubert","Blaise Agüera y Arcas","Pushmeet Kohli"],"abstract":"As automated reasoning systems advance rapidly, there is a growing need for research-level formal mathematical problems to accurately evaluate their capabilities. To address this, we present Formal Conjectures, an evolving benchmark of currently 2615 mathematical problem statements formalized in Lean 4. Sourced from areas of active mathematical research, the dataset features 1029 open research conjectures providing a zero-contamination benchmark for mathematical proof discovery, and 836 solved problems for proof autoformalization. Notably, the repository provides a structured interface connecting mathematicians who formalize and clarify problems with the AI systems and humans attempting to solve them. Demonstrating its immediate utility, the benchmark has already been leveraged to make new mathematical discoveries, including the resolution of open research conjectures. We describe our approach to ensuring the correctness of these formalizations in a collaborative open-source project where contributions stem from an active community. In this framework, AI-generated proofs and disproofs serve as a valuable auditing mechanism to iteratively improve the fidelity of the benchmark. Finally, we provide a standardized evaluation setup and report baseline results on frozen evaluation subsets, demonstrating a climbable signal that measures the current frontier of automated reasoning on research-level mathematics.","abs_url":"https://arxiv.org/abs/2605.13171","pdf_url":"https://arxiv.org/pdf/2605.13171v1","primary_category":"cs.AI","published":"2026-05-13","tags":["benchmark","dataset"],"lean_note":"Presents Formal Conjectures, an evolving benchmark of 2615 research-level problems formalized in Lean 4.","thumb":"thumbs/2605.13171.jpg"},{"id":"2605.13137","title":"LeanSearch v2: Global Premise Retrieval for Lean 4 Theorem Proving","authors":["Guoxiong Gao","Zeming Sun","Jiedong Jiang","Yutong Wang","Jingda Xu","Peihao Wu","Bryan Dai","Bin Dong"],"abstract":"Proving theorems in Lean 4 often requires identifying a scattered set of library lemmas whose joint use enables a concise proof -- a task we call global premise retrieval. Existing tools address adjacent problems: semantic search engines find individual declarations matching a query, while premise-selection systems predict useful lemmas one tactic step at a time. Neither recovers the full premise set an entire theorem requires. We present LeanSearch v2, a two-mode retrieval system for this task. Its standard mode applies a hierarchy-informalized Mathlib corpus with an embedding-reranker pipeline, achieving state-of-the-art single-query retrieval without domain-specific fine-tuning (nDCG@10 of 0.62 vs. 0.53 for the next-best system). Its reasoning mode builds on standard mode as its retrieval substrate, targeting global premise retrieval through iterative sketch-retrieve-reflect cycles. On a 69-query benchmark of research-level Mathlib theorems, reasoning mode recovers 46.1% of ground-truth premise groups within 10 retrieved candidates, outperforming strong reasoning retrieval systems (38.0%) and premise-selection baselines (9.3%) on the same benchmark. In a controlled downstream evaluation with a fixed prover loop, replacing alternative retrievers with LeanSearch v2 yields the highest proof success (20% vs. 16% for the next-best system and 4% without retrieval), confirming that retrieval quality propagates to proof generation. We have open-sourced all code, data, and benchmarks. Code and data: https://github.com/frenzymath/LeanSearch-v2 . The standard mode is publicly available with API access at https://leansearch.net/ .","abs_url":"https://arxiv.org/abs/2605.13137","pdf_url":"https://arxiv.org/pdf/2605.13137v2","primary_category":"cs.IR","published":"2026-05-13","tags":["tooling","mathlib","theorem-proving-ml"],"lean_note":"LeanSearch v2 is a global premise retrieval system over Mathlib for Lean 4 theorem proving.","repo":"https://github.com/frenzymath/LeanSearch-v2","thumb":"thumbs/2605.13137.jpg"},{"id":"2605.11905","title":"Rethinking Supervision Granularity: Segment-Level Learning for LLM-Based Theorem Proving","authors":["Shuo Xu","Jiakun Zhang","Junyu Lai","Chun Cao","Jingwei Xu"],"abstract":"Automated theorem proving with large language models in Lean 4 is commonly approached through either step-level tactic prediction with tree search or whole-proof generation. These two paradigms represent opposite granularities for constructing supervised training data: the former provides dense local signals but may fragment coherent proof processes, while the latter preserves global structure but requires complex end-to-end generation. In this paper, we revisit supervision granularity as a training set construction problem over proof trajectories and propose segment-level supervision, a training data construction strategy that extracts locally coherent proof segments for training policy models. We further reuse the same strategy at inference time to trigger short rollouts for existing step-level models. When trained with segment-level supervision on STP, LeanWorkbook, and NuminaMath-LEAN, the resulting policy models achieve proof success rates of 64.84%, 60.90%, and 66.31% on miniF2F, respectively, consistently outperforming both step-level and whole-proof baselines. Goal-aware rollout further improves existing step-level provers while reducing inference costs. It increases the proof success rate of BFS-Prover-V2-7B from 68.77% to 70.74% and that of InternLM2.5-StepProver from 59.59% to 60.33%, showing that appropriate supervision granularity better aligns model learning with proof structure and search. Code and models are available at https://github.com/NJUDeepEngine/SEG-ATP.","abs_url":"https://arxiv.org/abs/2605.11905","pdf_url":"https://arxiv.org/pdf/2605.11905v1","primary_category":"cs.AI","published":"2026-05-12","tags":["theorem-proving-ml","dataset"],"lean_note":"Proposes segment-level supervision for training LLM policy models on Lean 4 theorem proving, evaluated on miniF2F.","repo":"https://github.com/NJUDeepEngine/SEG-ATP","thumb":"thumbs/2605.11905.jpg"},{"id":"2605.10141","title":"FormalRewardBench: A Benchmark for Formal Theorem Proving Reward Models","authors":["Zeynel A. Uluşan","Burak S. Akbudak","Can S. Erer","Gözde Gül Şahin"],"abstract":"Recent neural theorem provers use reinforcement learning with verifiable rewards (RLVR), where proof assistants provide binary correctness signals. While verifiable rewards are cheap and scalable without reward hacking issues, they suffer from sparse credit assignment: models receive no learning signal from difficult problems where partial progress goes unrewarded. This motivates learned reward models that can evaluate proof quality beyond binary verification. However, comparing reward models is challenging since it typically requires expensive RL training ablations. To address this, we introduce \\textbf{FormalRewardBench}, the first benchmark for evaluating reward models in formal theorem proving with Lean 4. Our benchmark consists of 250 preference pairs where correct proofs are paired with incorrect variants generated through five expert curated error injection strategies: forced mistakes, minimal single-point variations, verbose incorrect proofs, natural language justification, and Python code injection. We evaluate frontier LLMs (e.g., Claude Opus 4.5), judge LLMs (e.g., CompassJudger-1-14B), general-purpose LLMs (e.g., Qwen2.5-72B-Instruct), and specialized theorem proving models (e.g., DeepSeek-Prover-V2-7B). Our results reveal that frontier LLMs achieve the highest performance (59.8\\%) while specialized theorem provers perform the worst (24.4\\%), suggesting that theorem proving ability does not transfer to proof evaluation. We provide further insights on various error injection mechanisms, highlighting the challenging nature of most injection mechanisms. We release \\textbf{FormalRewardBench} publicly to encourage more research on developing reward models in formal mathematics.","abs_url":"https://arxiv.org/abs/2605.10141","pdf_url":"https://arxiv.org/pdf/2605.10141v1","primary_category":"cs.AI","published":"2026-05-11","tags":["benchmark","theorem-proving-ml"],"lean_note":"Introduces FormalRewardBench, a benchmark of Lean 4 proof preference pairs for evaluating reward models.","repo":"https://github.com/GGLAB-KU/formal_rewardbench","thumb":"thumbs/2605.10141.jpg"},{"id":"2605.08749","title":"The Wristband Gaussian Loss: Deterministic, Composable Latents via a Sphere-Interval Decomposition","authors":["Mikhail Parakhin","André M. Carvalho","Patrick Haluptzok"],"abstract":"We present the Wristband Gaussian Loss, a deterministic batch loss for Gaussianizing point embeddings without sampling, KL terms, or iterative transport. Each $x \\in \\mathbb{R}^d$ is mapped to a direction $u = x/\\|x\\|$ and a CDF-transformed radius $t = F_{χ^2_d}(\\|x\\|^2)$ on the wristband $S^{d-1} \\times [0,1]$. We prove (and machine-verify in Lean~4) that for $d \\ge 2$ the pushforward wristband map equals $σ_{d-1} \\otimes \\mathrm{Unif}[0,1]$ iff the source is $\\mathcal{N}(0, I_d)$, and that the Neumann-reflected wristband repulsion energy is uniquely minimized at the uniform target. We compute this reflected-kernel objective in two ways: a nearest three-image pairwise truncation at $O(N^2 d)$, and a spectral Neumann path joining angular and radial Mercer modes (spherical-harmonic and cosine) at $O(N d K)$, with empirically matched gradients. A 1D Wasserstein radial term and a moment penalty serve as finite-sample accelerators with the same optimum, and Monte-Carlo null calibration turns the components into a single standardized statistic. We evaluate direct point-cloud Gaussianization with a calibrated barycentric $W_2$ score: a deterministic Gaussian reference batch is built by recursive Hungarian averaging, with each method reported as a $z$-score against same-size Gaussian batches. On the axis-uniform X benchmark, Wristband is competitive in 2D and gives the best 10D score. On a harder radial--angular-copula impostor whose Gaussian radial and angular marginals are correct but dependent, Wristband gives the best 10D and 128D scores. Coupled with learnable-key Euclidean attention and exact invertible flows, the resulting Deterministic Gaussian Autoencoder delivers a Gaussian-latent interface for counterfactual sampling with independent factors and a context/residual construction for dependent factors.","abs_url":"https://arxiv.org/abs/2605.08749","pdf_url":"https://arxiv.org/pdf/2605.08749v1","primary_category":"cs.LG","published":"2026-05-09","tags":["formalization","probability","ml-theory"],"lean_note":"Machine-verifies in Lean 4 the pushforward and energy-minimization properties of the proposed Wristband Gaussian Loss.","thumb":"thumbs/2605.08749.jpg"},{"id":"2605.08609","title":"Sheaves as a Means of Maintaining Consistency in Model-based Systems Engineering","authors":["Josh Gibson"],"abstract":"We propose that the sheaf condition on a presheaf of design spaces provides a mathematical model for multi-view consistency in the architecture of cyber-physical systems (CPS). In model-based systems engineering, multiple engineering views -- electrical, thermal, mechanical, and software -- must be kept mutually consistent, yet current practice relies on informal procedures without a precise semantic account of global consistency. We construct an architectural site: a topological space whose points are pairwise interfaces between engineering domains and whose open sets represent engineering views. A design presheaf assigns to each view its local design space and to each inclusion the corresponding restriction map. We show that the sheaf condition on this presheaf is equivalent to compatibility on pairwise overlaps, yielding a local criterion for global multi-view consistency. The equivalence and a concrete three-view worked example are machine-verified in Lean 4 using Mathlib. The formalization establishes that the design presheaf is a sheaf, that the sheaf condition is equivalent to pairwise overlap compatibility, and that compatible local design families glue to unique global designs. Global consistency of an arbitrary number of views can be certified by checking only pairwise interface compatibility; compatible local designs determine a unique global design; derived properties computed by limit-preserving functors inherit the same consistency guarantee; and the entire verification chain admits machine-checkable proofs in Lean.","abs_url":"https://arxiv.org/abs/2605.08609","pdf_url":"https://arxiv.org/pdf/2605.08609v1","primary_category":"cs.LO","published":"2026-05-09","tags":["formalization","category-theory"],"lean_note":"Machine-verifies a sheaf-consistency equivalence and worked example in Lean 4 using Mathlib.","thumb":"thumbs/2605.08609.jpg"},{"id":"2605.07782","title":"CktFormalizer: Autoformalization of Natural Language into Circuit Representations","authors":["Jing Xiong","Qi Han","Chenchen Ding","He Xiao","Zunhai Su","Chaofan Tao","Ngai Wong"],"abstract":"LLMs can generate hardware descriptions from natural language specifications, but the resulting Verilog often contains width mismatches, combinational loops, and incomplete case logic that pass syntax checks yet fail in synthesis or silicon. We present CktFormalizer, a framework that redirects LLM-driven hardware generation through a dependently-typed HDL embedded in Lean 4. Lean serves three roles: (i) type checker:dependent types encode bit-width constraints, case coverage, and acyclicity, turning hardware defects into compile-time errors that guide iterative repair; (ii) correctness firewall:compiled designs are structurally free of defects that cause silent backend failures (the baseline loses 20% of correct designs during synthesis and routing; CktFormalizer preserves all of them); (iii) proof assistant:the agent constructs machine-checked equivalence proofs over arbitrary input sequences and parameterized widths, beyond the reach of bounded SMT-based checking. On VerilogEval (156 problems), RTLLM (50 problems), and ResBench (56 problems), CktFormalizer achieves simulation pass rates competitive with direct Verilog generation while delivering substantially higher backend realizability: 95--100% of compiled designs complete the full synthesis, place-and-route, DRC, and LVS flow. A closed-loop PPA optimization stage yields up to 35% area reduction and 30% power reduction through validated architecture exploration, with automated theorem proof ensuring that each optimized variant remains functionally equivalent to its formal specification.","abs_url":"https://arxiv.org/abs/2605.07782","pdf_url":"https://arxiv.org/pdf/2605.07782v3","primary_category":"cs.CL","published":"2026-05-08","tags":["verification","autoformalization"],"lean_note":"Routes LLM-driven hardware generation through a dependently-typed HDL embedded in Lean 4 with equivalence proofs.","repo":"https://github.com/project-oak/silveroak","thumb":"thumbs/2605.07782.jpg"},{"id":"2605.07662","title":"Direction-Preserving Number Representations","authors":["Bardia Zadeh","George A. Constantinides"],"abstract":"Low-precision number formats are widely used in modern machine learning systems due to their efficiency. Accurate direction representation is key to the accuracy of vector operations. This work precisely explores the extent to which the direction of a vector can be represented by selecting its scalar elements from a common finite alphabet of a given size. This is standard practice in machine learning, where low-precision significands may be narrow-width floating-point or integer values. A geometric framework is introduced for analyzing the directional coverage of such product-structured codes. This work analytically quantifies the suboptimality gap between such product-structured codes and spherical codes for the vector as a whole, in both low and asymptotically high dimensions. Furthermore, within the product code class, it is proven that the standard formats of two's complement, fixed-point, and floating-point are suboptimal, again with quantified gap, pointing to the potential to develop new scalar number formats. Such scalar alphabets are numerically optimized across multiple block dimensions for directional coverage, including the dimension used in NVIDIA's NVFP4 format. Experimental results are presented comparing the performance of standard formats and the optimized alphabet. We find that for four bits, NVIDIA's choice of E2M1 closely approximates the optimized alphabet, providing a geometric explanation for its strong performance in low-precision machine learning workloads and an analytical understanding of the link between that superiority and block size. We provide open-source formal proofs in Lean for the theorems in this work, along with the experimental code and the optimized alphabets obtained.","abs_url":"https://arxiv.org/abs/2605.07662","pdf_url":"https://arxiv.org/pdf/2605.07662v1","primary_category":"cs.LG","published":"2026-05-08","tags":["formalization","ml-theory","geometry"],"lean_note":"Provides open-source end-to-end Lean formal proofs of the paper's theorems on directional coverage of product-structured number codes.","repo":"https://github.com/bardia01/Direction-Preserving-Number-Representations","thumb":"thumbs/2605.07662.jpg"},{"id":"2605.07147","title":"MathlibPR: Pull Request Merge-Readiness Benchmark for Formal Mathematical Libraries","authors":["Zixuan Xie","Xinyu Liu","Shangtong Zhang"],"abstract":"The ecosystem of Lean and Mathlib has become the de facto standard for large language model (LLM) assisted formal reasoning with remarkable successes in recent years. Those successes, however, only consume Mathlib as an essential dependency but do not directly contribute to it. In the meantime, the growth of Mathlib has recently been bottlenecked by the review process, which requires human reviewers to judge whether proposed pull requests (PRs) follow the Mathlib's conventions and are worth integrating as part of a shared mathematical infrastructure. This leads to our central question: can LLMs help review Mathlib PRs? To this end, we introduce MathlibPR, a benchmark built from real Mathlib4 PR histories. We further propose a staged evaluation protocol and use it to evaluate both LLM models (e.g., DeepSeek, Qwen, Goedel, and Kimina) and LLM agents (e.g., Codex and Claude Code). Surprisingly, both LLM models and LLM agents struggle to distinguish merge-ready PRs from build-passing PRs that were revised or never merged. By turning Mathlib PR histories into a supervised signal, MathlibPR provides a step toward reviewer assistants and reward models that could help evaluate PRs and steer LLMs toward producing merge-ready Mathlib contributions.","abs_url":"https://arxiv.org/abs/2605.07147","pdf_url":"https://arxiv.org/pdf/2605.07147v2","primary_category":"cs.LO","published":"2026-05-08","tags":["benchmark","mathlib","dataset"],"lean_note":"MathlibPR is a benchmark built from real Mathlib4 pull-request histories for judging PR merge-readiness.","thumb":"thumbs/2605.07147.jpg"},{"id":"2605.04734","title":"Hamilton decompositions of all directed tori at odd modulus","authors":["SangHyun Park"],"abstract":"Let $D_d(m) = \\mathrm{Cay}((\\mathbb{Z}/m\\mathbb{Z})^d, {e_0, \\ldots, e_{d-1}})$ denote the directed Cayley graph on the positive coordinate basis, equivalently the Cartesian product of $d$ directed cycles of length $m$. The equal side directed Hamilton decomposition problem asks when the arc set of $D_d(m)$ partitions into $d$ directed Hamilton cycles. We prove that such a decomposition exists for every $d \\geq 2$ and every odd $m \\geq 3$, settling the equal side directed Hamilton decomposition problem at all odd moduli. The proof combines root flat certificate theorem, a prefix count primitivity criterion, and a modular trade lifting theorem with two closure principles: the Cartesian product and the successor step $b \\mapsto 2b+1$. Together these propagate the small base dimensions $d \\in {2, 3, 5, 7}$ to all $d \\geq 2$. The boundary cases $D_7(3)$ and $D_7(5)$, where the prefix-count family saturates its zero symbol budget, are handled by explicit non prefix zero set root flat certificates whose zero set compiler. An accompanying Lean 4 formalization verifies the main theorem and the finite certificate predicates.","abs_url":"https://arxiv.org/abs/2605.04734","pdf_url":"https://arxiv.org/pdf/2605.04734v2","primary_category":"math.CO","published":"2026-05-06","tags":["formalization","combinatorics"],"lean_note":"Accompanying Lean 4 formalization verifies the main directed-tori Hamilton decomposition theorem and finite certificate predicates.","repo":"https://github.com/aria1th/Torus-Hamilton-Decomposition-Program","thumb":"thumbs/2605.04734.jpg"},{"id":"2605.04472","title":"Automated Formal Proofs of Combinatorial Identities via Wilf-Zeilberger Guidance and LLMs","authors":["Beibei Xiong","Hangyu Lv","Junqi Liu","Yisen Wang","Shaoshi Chen","Jianlin Wang","Zhengfeng Yang","Lihong Zhi"],"abstract":"Automating formal proofs of combinatorial identities is challenging for LLM-based provers, as long-horizon proof planning is required and unconstrained search quickly explodes. Symbolic methods such as the Wilf-Zeilberger (WZ) method can achieve a mechanized proof of combinatorial identities by constructing special auxiliary functions and demonstrating that they satisfy specific recurrence relations. We propose WZ-LLM, a neuro-symbolic framework that turns WZ proof plans into executable proof sketches in Lean 4 and uses an LLM-based prover to discharge the resulting machine-checkable subgoals. We also train a dedicated WZ-Prover via a Lean-kernel-verified bootstrapping loop with expert-verified iteration, followed by DAPO-based refinement. Experiments show that WZ-LLM achieves a 34% proof success rate on LCI-Test (100 classic combinatorial identities), outperforming strong baselines such as DeepSeek-V3 and Goedel-Prover-V2, and delivering consistent gains on CombiBench and PutnamBench-Comb. These results indicate that our framework provides two complementary strengths: improved direct proving for identities beyond the scope of WZ, and substantially higher end-to-end success when WZ sketches guide a specialized prover.","abs_url":"https://arxiv.org/abs/2605.04472","pdf_url":"https://arxiv.org/pdf/2605.04472v1","primary_category":"cs.LG","published":"2026-05-06","tags":["theorem-proving-ml","combinatorics"],"lean_note":"WZ-LLM turns Wilf-Zeilberger plans into Lean 4 proof sketches discharged by an LLM prover.","thumb":"thumbs/2605.04472.jpg"},{"id":"2605.03274","title":"Formalizing Singer Sidon Constructions and Sidon Set Infrastructure in Lean 4","authors":["David B. Hulak","Arthur F. Ramos","Ruy J. G. B. de Queiroz"],"abstract":"Erdős Problem 30 asks for sharp asymptotics of the Sidon extremal function $h(N)$, and Singer's construction is the classical source of lower-bound examples matching the main term. We present a Lean 4 formalization of Singer's Sidon set construction, together with reusable Sidon-set infrastructure for additive combinatorics. For every prime power $q=p^k$, we prove the existence of a Sidon set modulo $q^2+q+1$ of cardinality $q+1$; the prime-field case $q=p$ is the base presentation. The proof proceeds through a non-trivial algebraic chain: construction of the base field and its degree-three extension, analysis of the trace kernel as a 2-dimensional subspace over the base field, a geometric argument via subspace intersections establishing the multiplicative Sidon property in the quotient group, and a transfer from quotient multiplication to modular integer addition. Around this central result, we develop a reusable Sidon set library. It comprises interval and modular Sidon sets, the extremal function $h(N)$, Lindström's cross-difference inequality, a Johnson-route shift-incidence upper bound of the form $h(N)\\leq\\sqrt{N}+N^{1/4}+O(1)$, representation-function identities, and unconditional two-sided $h(N)=Θ(\\sqrt{N})$ bounds with exact floor-rounded finite statements for $N\\geq 5$. We further formalize a conditional reduction: subpolynomial prime gaps together with a full subpolynomial upper-error hypothesis for $h(N)$ imply the Erdős Problem 30 estimate $h(N)=\\sqrt{N}+O_ε(N^ε)$ for every $ε>0$. The Singer/Sidon modules and transfer lemmas comprise 7,541 lines of Lean 4 with zero active uses of sorry. We describe the mathematical lessons learned, focusing on how formalization clarifies the precise scope of classical arguments and forces explicit treatment of the passage from the field-theoretic construction to integer Sidon predicates.","abs_url":"https://arxiv.org/abs/2605.03274","pdf_url":"https://arxiv.org/pdf/2605.03274v2","primary_category":"math.CO","published":"2026-05-05","tags":["formalization","combinatorics","number-theory"],"lean_note":"Formalizes Singer's Sidon set construction and reusable Sidon-set infrastructure in 7541 lines of Lean 4.","repo":"https://github.com/d0d1/singer-theorem-lean","thumb":"thumbs/2605.03274.jpg"},{"id":"2605.12537","title":"Biprofile Deviation Logic: Report-Replacement Frames and Audit Witnesses","authors":["Faruk Alpay","Baris Basaran"],"abstract":"Biprofile deviation logic models strategic social choice states as pairs $(R,P)$, where $R$ is the true profile used for welfare comparisons and $P$ is the submitted report profile used by the rule. Coalition modalities replace only the reports of the coalition, and their relations satisfy the fixed law $E_C \\circ E_D = E_{C \\cup D}$. The paper proves soundness and completeness of $H_{\\mathrm{bp}}$ for the abstract frame class $\\mathrm{Dev}(N)$, with the reverse-composition midpoint displayed inside the canonical proof. It then separates abstract $\\mathrm{Dev}(N)$-components from genuine report-coordinate products by coordinate separation. On the social-choice side, the classical facts supply the source notions; the paper-specific contribution is the audit layer for representation changes: typed manipulation witnesses, a boundary-row theorem for off-domain extensions, and a factor-closure criterion for public deletions. The ancillary material contains the input formats, an executable certificate checker, Lean and Alloy companions for the finite relational lemmas and update patterns, recorded run logs, and checksums.","abs_url":"https://arxiv.org/abs/2605.12537","pdf_url":"https://arxiv.org/pdf/2605.12537v1","primary_category":"cs.LO","published":"2026-05-04","tags":["formalization","logic"],"lean_note":"Provides Lean and Alloy companions verifying finite relational lemmas for biprofile deviation logic.","thumb":"thumbs/2605.12537.jpg"},{"id":"2605.03034","title":"Stable Agentic Control: Tool-Mediated LLM Architecture for Autonomous Cyber Defense","authors":["Kerri Prinos","Lilianne Brush","Cameron Denton","Zhanqi Wang","Joshua Knox","Snehal Antani","Anton Foltz","Amy Villaseñor"],"abstract":"Agentic systems involved in high-stake decision-making under adversarial pressure need formal guarantees not offered by existing approaches. Motivated by the operational needs of security operations centers (SOCs) that must configure endpoint detection and response (EDR) policies under adversarial pressure, we present a tool-mediated architecture: LLM agents use deterministic tools (Stackelberg best-response, Bayesian observer updates, attack-graph primitives) and select from finite action catalogs enforced at the tool-output interface. A composite Lyapunov function machine-checked in Lean 4 with zero sorry certifies controllability, observability from asymmetric sensor data, and Input-to-State Stability (ISS) robustness under intelligent adversarial disturbance, with two corollaries extending the certificate to any controller or adversary from the catalogs. On 282 real enterprise attack graphs, the claims hold with margin. On paired offensive/defensive telemetry, a tool-mediated Claude Sonnet 4 controller reduces the attacker's expected payoff (game value) by 59% relative to a deterministic greedy baseline, with zero variance across 40 runs at four temperatures. A Claude Haiku 4.5 controller converges to suboptimal game values but stays catalog-bounded over an additional 40 runs, demonstrating that architectural stability is not dependent on the controller capability. The LLM agent's non-determinism furthers creative exploration of strategies, while the tool-mediated architecture ensures system stability.","abs_url":"https://arxiv.org/abs/2605.03034","pdf_url":"https://arxiv.org/pdf/2605.03034v1","primary_category":"cs.AI","published":"2026-05-04","tags":["verification"],"lean_note":"A composite Lyapunov function certifying stability of the agentic control architecture is machine-checked in Lean 4 with zero sorry.","repo":"https://github.com/Orange-Cyberdefense/GOAD","thumb":"thumbs/2605.03034.jpg"},{"id":"2605.02856","title":"The 1-Bit Barrier is Universal: k-Stage Pipeline Composition and Unified Leakage Bounds for Standard Modular Reductions in PQC Hardware","authors":["Ray Iskander","Khaled Kirah"],"abstract":"This is Paper 7 of a series of formally-verified analyses of masked NTT hardware for post-quantum cryptography; Paper 1 [1] established structural dependency analysis of the QANARY platform, and Paper 2 [2] quantified security margins under partial NTT masking. Arbitrary-depth $k$-stage masked NTT pipelines with fresh inter-stage masking and per-stage PF-PINI($\\leq 2$) gadgets satisfy a per-observation cardinality bound of $2 \\cdot q^{2k-2}$ on the preimage of any output value, machine-checked in Lean 4 with zero \\texttt{sorry}. Under the standard (informal) semantic translation that divides this cardinality by the total mask-tuple space size $q^{2k-1}$, the per-observation conditional probability bound is $2/q$, independent of pipeline depth $k$. The QANARY program has previously established machine-checked cardinality bounds on the per-observation leakage of masked NTT hardware: PF-PINI(2) for Barrett reduction (Paper 5 [3]), 2-stage composition with fresh inter-stage masking (Paper 6 [4]), an underlying universality theorem (Paper 3 [5]), and PF-PINI(1) for butterfly wires (Paper 4 [6]). This paper closes the program with four contributions. First, a $k$-stage composition theorem generalizing Paper 6's two-stage result to arbitrary $k \\geq 1$ gives the last-stage-determined bound $G_{k-1}.\\texttt{maxMult} \\cdot q^{2k-2}$: only the last stage's PF-PINI parameter survives, with intermediate parameters erased by fresh inter-stage masking. Second, Montgomery reduction satisfies PF-PINI(2) with tight max-multiplicity 2. Third, we assemble these into the end-to-end bound $2 \\cdot q^{2k-2}$ for any depth-$k$ PF-PINI($\\leq 2$) pipeline under fresh inter-stage masking. Fourth, a Lean-verified hypothesis-violation conditional anchors the prior empirical and structural Adams Bridge analyses ([1, 2, 7, 8]).","abs_url":"https://arxiv.org/abs/2605.02856","pdf_url":"https://arxiv.org/pdf/2605.02856v2","primary_category":"cs.CR","published":"2026-05-04","tags":["verification","cryptography"],"lean_note":"Machine-checks in Lean 4 cardinality and leakage bounds for masked NTT pipelines in post-quantum cryptography hardware.","thumb":"thumbs/2605.02856.jpg"},{"id":"2605.02331","title":"Bennett's Conjecture in Lean 4: Counter-Models for the PSR-Reducibility of Spinoza's Propositions V and XIV","authors":["Yuki Nakamura"],"abstract":"In A Study of Spinoza's Ethics (1984, §17), Jonathan Bennett argues that the demonstration of Proposition V of Spinoza's Ethica contains identifiable invalid moves and that, even granted those moves, \"cannot yield more than the conclusion that two substances could not have all their attributes in common\" -- while Spinoza concludes that they cannot share any. Bennett doubts that any valid reconstruction is available from Spinoza's stated resources without importing further commitments. Michael Della Rocca (Spinoza, 2008, ch. 2) responds that the proposition can be derived if the Principle of Sufficient Reason (PSR) is committed substantively. The debate has remained at the level of prose argument for forty years. This paper provides the first machine-checked evidence in the debate. We formalise Ethica Pars I in Lean 4, encoding Bennett's reading of Spinoza's stated axioms as a typeclass and Della Rocca's substantive PSR as an extension class. The derivation attempt yields a partial result -- substances sharing all attributes are identical -- but cannot reach the full \"sharing-any-attribute -> identity\" content of Proposition V, mechanically tracking Bennett's own all-attributes ceiling. A four-element counter-model satisfying both axiom sets while falsifying Proposition V's content establishes the irreducibility against this specific augmentation. A second counter-model establishes the analogous result for axiom A15, a load-bearing universality clause for Spinoza's Proposition XIV. Bennett's diagnosis receives its first kernel-checked counter-model against the Della-Rocca PSR-substance reconstruction (the non-derivability claim itself a meta-logical consequence of kernel consistency); stronger PSR variants and the broader narrative claim against the full Section I + II + A1--A7 register remain open as future mechanical projects.","abs_url":"https://arxiv.org/abs/2605.02331","pdf_url":"https://arxiv.org/pdf/2605.02331v2","primary_category":"cs.LO","published":"2026-05-04","tags":["formalization","logic"],"lean_note":"Formalizes Spinoza's Ethica Pars I in Lean 4 with counter-models for PSR-reducibility of Propositions V and XIV.","repo":"https://github.com/Nakammura/spinoza-ethica-lean","thumb":"thumbs/2605.02331.jpg"},{"id":"2605.02113","title":"A Shallow Embedding of Datalog in Lean","authors":["Ramy Shahin"],"abstract":"Datalog is a lightweight logic programming language, based on the logic of Horn clauses. Lean, on the other hand, is a proof assistant system and language based on the Calculus of Inductive Constructions (CIC). Datalog is more constrained and less expressive than Lean but has a long history of established deduction algorithms. Writing definitions and queries in the Datalog fragment of Lean would be more succinct and understandable than writing them in Lean itself. This paper outlines the design and implementation of a shallow embedding of Datalog as a Domain Specific Language (DSL) on top of Lean. Bidirectional interoperability between the Datalog DSL and Lean is a primary goal of this design. In addition to rules and facts, backward chaining queries are automatically translated into theorems with tactic-based proofs. The paper also includes three simple examples of how the DSL can be used.","abs_url":"https://arxiv.org/abs/2605.02113","pdf_url":"https://arxiv.org/pdf/2605.02113v1","primary_category":"cs.SE","published":"2026-05-04","tags":["tooling","logic"],"lean_note":"Implements a shallow embedding of Datalog as a DSL in Lean, translating backward-chaining queries into theorems with tactic-based proofs.","repo":"https://github.com/qualgebra/LDatalog","thumb":"thumbs/2605.02113.jpg"},{"id":"2605.02064","title":"Gaps in Multiplicative Sidon Sets","authors":["Wouter van Doorn","Pietro Monticone","Quanyu Tang"],"abstract":"For a positive integer $n$, let $g(n)$ denote the infimum of all real numbers $L$ such that there exists a multiplicative Sidon set $A\\subseteq\\{1,2,\\dots,n\\}$ that intersects every interval $[x,x+L]\\subseteq[1,n]$. Sárközy asked for estimates on $g(n)$, and he in particular asked whether one has $g(n)\\le\\sqrt n$ for every $n\\in\\mathbb{N}$. We first show that this estimate does indeed hold, with a proof that was autonomously discovered and formally verified in Lean by Aristotle. Next, we improve the upper bound further and, with $ρ= \\frac{13-\\sqrt{69}}{10} < 0.47$, prove that $g(n)\\ll_{\\varepsilon} n^{ρ+\\varepsilon}$ for every $\\varepsilon > 0$.","abs_url":"https://arxiv.org/abs/2605.02064","pdf_url":"https://arxiv.org/pdf/2605.02064v1","primary_category":"math.NT","published":"2026-05-03","tags":["formalization","number-theory","ai-agents"],"lean_note":"An upper bound g(n)<=floor(sqrt(n)) for gaps in multiplicative Sidon sets was autonomously discovered and formally verified in Lean by the Aristotle agent.","thumb":"thumbs/2605.02064.jpg"},{"id":"2605.01660","title":"Testing, Credible Compilation, and Verification in the Axon Verified Compiler in Lean and Claude Code","authors":["Martin Rinard"],"abstract":"This paper presents the use of testing, credible compilation/translation validation, verification, and audits in the Axon compiler. Axon comes with fully machine checked proofs that guarantee the correctness of the generated code. All code and proofs were written in Lean by Claude Code, with the correctness proofs eliminating any need to audit or examine any verified code. I present a development process for using these validation techniques, evaluate the use of this process during the development of the compiler, and discuss implications for other development efforts.","abs_url":"https://arxiv.org/abs/2605.01660","pdf_url":"https://arxiv.org/pdf/2605.01660v2","primary_category":"cs.PL","published":"2026-05-03","tags":["verification","ai-agents","tooling"],"lean_note":"Axon, a verified optimizing compiler with fully machine-checked correctness proofs, was written entirely in Lean by Claude Code.","thumb":"thumbs/2605.01660.jpg"},{"id":"2605.01028","title":"Stokes' Theorem for Smooth Singular Cubes in Lean 4: True Pullback, Bridges to mathlib4, and Chain-Level d^2=0","authors":["David B. Hulak","Arthur F. Ramos","Ruy J. G. B. de Queiroz"],"abstract":"We present a sorry-free Lean 4/mathlib4 formalization of Stokes' theorem for smooth singular cubes in arbitrary dimension, using true differential-form pullback via the Frechet derivative. The development also includes a bridge to mathlib4's abstract extDeriv, chain-level Stokes extended by Z-linearity, d^2=0 for singular cubical chains, box Stokes for axis-aligned cubes, dimensional specializations, and a structured comparison with Harrison's HOL Light formalization.","abs_url":"https://arxiv.org/abs/2605.01028","pdf_url":"https://arxiv.org/pdf/2605.01028v1","primary_category":"cs.LO","published":"2026-05-01","tags":["formalization","mathlib","analysis"],"lean_note":"Sorry-free Lean 4/mathlib4 formalization of Stokes' theorem for smooth singular cubes with true pullback.","repo":"https://github.com/d0d1/lean-stokes-theorem","thumb":"thumbs/2605.01028.jpg"},{"id":"2605.00677","title":"Evaluating the Architectural Reasoning Capabilities of LLM Provers via the Obfuscated Natural Number Game","authors":["Lixing Li"],"abstract":"While Large Language Models have achieved notable success on formal mathematics benchmarks such as MiniF2F, it remains unclear whether these results stem from genuine logical reasoning or semantic pattern matching against pre-training data. This paper identifies Architectural Reasoning: the ability to synthesize formal proofs using exclusively local axioms and definitions within an alien math domain, as the necessary ability for future automated theorem discovery AI. We use the Obfuscated Natural Number Game, a benchmark to evaluate Architectural Reasoning. By renaming identifiers in the Natural Number Game in Lean 4, we created a zero-knowledge, closed environment. We evaluate state-of-the-art models, finding a universal latency tax where obfuscation increases inference time. The results also reveal a divergence in robustness: while general models (Claude-Sonnet-4.5, GPT-4o) suffer performance degradation, reasoning models (DeepSeek-R1, GPT-5, DeepSeek-Prover-V2) maintain the same accuracy despite the absence of semantic cues. These findings provide a quantitative metric for assessing the true capacity for mathematical reasoning.","abs_url":"https://arxiv.org/abs/2605.00677","pdf_url":"https://arxiv.org/pdf/2605.00677v1","primary_category":"cs.LG","published":"2026-05-01","tags":["theorem-proving-ml","benchmark"],"lean_note":"Evaluates LLM provers on an obfuscated Natural Number Game in Lean 4 to probe architectural reasoning.","repo":"https://github.com/lllx125/Obfuscated-NNG","thumb":"thumbs/2605.00677.jpg"},{"id":"2605.00674","title":"Beyond Benchmarks: MathArena as an Evaluation Platform for Mathematics with LLMs","authors":["Jasper Dekoninck","Nikola Jovanović","Tim Gehrunger","Kári Rögnvaldsson","Ivo Petrov","Chenhao Sun","Martin Vechev"],"abstract":"Large language models (LLMs) are becoming increasingly capable mathematical collaborators, but static benchmarks are no longer sufficient for evaluating progress: they are often narrow in scope, quickly saturated, and rarely updated. This makes it hard to compare models reliably and track progress over time. Instead, we need evaluation platforms: continuously maintained systems that run, aggregate, and analyze evaluations across many benchmarks to give a comprehensive picture of model performance within a broad domain. In this work, we build on the original MathArena benchmark by substantially broadening its scope from final-answer olympiad problems to a continuously maintained evaluation platform for mathematical reasoning with LLMs. MathArena now covers a much wider range of tasks, including proof-based competitions, research-level arXiv problems, and formal proof generation in Lean. Additionally, we maintain a clear evaluation protocol for all models and regularly design new benchmarks as model capabilities improve to ensure that MathArena remains challenging. Notably, the strongest model, GPT-5.5, now reaches 98% on the 2026 USA Math Olympiad and 74% on research-level questions, showing that frontier models can now comfortably solve extremely challenging mathematical problems. This highlights the importance of continuously maintained evaluation platforms like MathArena to track the rapid progress of LLMs in mathematical reasoning.","abs_url":"https://arxiv.org/abs/2605.00674","pdf_url":"https://arxiv.org/pdf/2605.00674v2","primary_category":"cs.CL","published":"2026-05-01","tags":["benchmark","theorem-proving-ml","meta-study"],"lean_note":"MathArena evaluation platform includes formal proof generation in Lean as one of its mathematical-reasoning task categories for LLMs.","thumb":"thumbs/2605.00674.jpg"},{"id":"2605.00660","title":"Hamilton decompositions of the directed 7-torus at odd modulus via root-flat certificates and a prefix-count construction","authors":["SangHyun Park"],"abstract":"We prove that the directed seven-dimensional equal-side torus D_7(m) = Cay((Z/mZ)^7, {e_0, e_1, ..., e_6}) admits a directed Hamilton decomposition for every odd integer m >= 3. The proof has two main contributions. First, we introduce the root-flat certificate: a named verification framework in which a Hamilton decomposition of D_n(m) follows from three local conditions on a single root flat -- row Latinness, layer bijectivity, and primitive return maps. This abstraction was used informally in the earlier odd D_5(m) construction; here it appears as a definition and a theorem, providing a common verification interface for prime-dimensional base cases. Second, for every odd m >= 7, we give a uniform prefix-coordinate construction: one-layer prefix maps, a symbol-count criterion, and explicit 7x7 count matrices produce all seven Hamilton factors without a finite search. The remaining moduli m = 3 and m = 5 are exactly the boundary where the prefix-count method provably cannot work; they are handled by finite root-flat certificates whose validity is checked in Lean 4. A Lean 4 formalization verifies the Cayley statement, with the symbolic branch and the finite boundary certificates checked in the same development.","abs_url":"https://arxiv.org/abs/2605.00660","pdf_url":"https://arxiv.org/pdf/2605.00660v1","primary_category":"math.CO","published":"2026-05-01","tags":["formalization","combinatorics"],"lean_note":"A Lean 4 formalization verifies the directed 7-torus Hamilton decomposition with symbolic and boundary certificates.","repo":"https://github.com/aria1th/Torus-Hamilton-Decomposition-Program","thumb":"thumbs/2605.00660.jpg"},{"id":"2605.00081","title":"Alignment Contracts for Agentic Security Systems","authors":["Isaac David","Marco Guarnieri","Arthur Gervais"],"abstract":"Agentic security systems increasingly combine LLM planners with tools that can discover, validate, and report vulnerabilities. This creates an asymmetric control problem: the system should retain strong offensive capability inside an authorized engagement, while the same capabilities must be denied outside scope. Existing guardrails provide useful policy controls, but they do not make this boundary a first-class formal contract over observable effects. We introduce alignment contracts, a framework for specifying and enforcing behavioral constraints over observable effect traces. A contract defines scope, allowed and forbidden effects, resource budgets, and disclosure policies. We give the language finite-trace semantics, characterize satisfaction as a safety property with finite violation witnesses, develop refinement and one-way composition rules for modular contract engineering, and show that admissibility checking is decidable. We instantiate the framework for web-focused agentic security workflows and show how the same structure extends to other effect profiles. Under an explicit Effect Observability Assumption, where all $\\SigmaEff$-effects are mediated, the soundness theorem quantifies over the agent model and gives guarantees for mediated $\\SigmaEff$-effects, including enforcement soundness for monitor-realized traces. We also state an assumption-lifted adaptation result and formalize limits through undecidability transfer and observability-boundary theorems. A Lean 4 artifact checks the formal core theorems used by the paper.","abs_url":"https://arxiv.org/abs/2605.00081","pdf_url":"https://arxiv.org/pdf/2605.00081v1","primary_category":"cs.CR","published":"2026-04-30","tags":["verification","ai-agents"],"lean_note":"A Lean 4 artifact checks the formal core theorems behind alignment contracts for agentic security systems.","thumb":"thumbs/2605.00081.jpg"},{"id":"2604.27362","title":"A note on the parameter $\\ell$ in Buchbinder--Feldman's deterministic submodular matroid algorithm","authors":["Shisheng Li"],"abstract":"Buchbinder and Feldman recently gave a deterministic $(1-1/e-\\varepsilon)$-approximation for maximizing a non-negative monotone submodular function subject to a matroid constraint, with query complexity $\\widetilde{O}_\\varepsilon(nr)$. Their algorithm uses an integer parameter $\\ell$, which Buchbinder and Feldman fix to $\\ell = 1 + \\lceil 1/\\varepsilon \\rceil$ via a loose bound on $(1+1/\\ell)^{-\\ell}$. We point out two purely elementary refinements. First, the classical Pólya--Szegő inequality $(1+1/\\ell)^{-\\ell} \\le e^{-1}(1+1/(2\\ell))$ replaces the loose step in their proof and permits $\\ell = \\lceil 1/(2e\\varepsilon) \\rceil$, shrinking the hidden constant in $\\widetilde{O}_\\varepsilon(nr)$ by a factor $\\approx 2^{0.816/\\varepsilon}$. Second, an alternating-series tail bound for $\\log(1+t)$ yields the asymptotically sharp inequality $(1+1/\\ell)^{-\\ell} \\le e^{-1}\\exp(1/(2\\ell) - 1/(3\\ell^2) + 1/(4\\ell^3))$, matching the true expansion of $(1+1/\\ell)^{-\\ell}$ through order $\\ell^{-3}$ and translating into $\\ell_\\star = 1/(2e\\varepsilon) - 5/12 + O(\\varepsilon)$. The asymptotic class $\\widetilde{O}_\\varepsilon(nr)$ of the query complexity is unchanged in either case; only the implicit constant in $\\varepsilon$ is improved. All inequalities in this note are formalized and machine-checked in Lean 4 against Mathlib.","abs_url":"https://arxiv.org/abs/2604.27362","pdf_url":"https://arxiv.org/pdf/2604.27362v1","primary_category":"cs.DS","published":"2026-04-30","tags":["formalization","mathlib","combinatorics"],"lean_note":"All inequalities in the note are machine-checked in Lean 4 against Mathlib.","repo":"https://github.com/daizisheng/bf24-note","thumb":"thumbs/2604.27362.jpg"},{"id":"2604.27140","title":"Hamilton decompositions of the directed 5-torus for odd modulus","authors":["SangHyun Park"],"abstract":"We prove that the directed five-dimensional torus $D_5(m) = \\operatorname{Cay}((\\mathbb{Z}_m)^5, \\{e_0, e_1, e_2, e_3, e_4\\})$ has a Hamilton decomposition for every odd integer $m \\geq 3$. This is the first higher-dimensional case in which the return-map method requires a genuine zero-set selector rather than an odometer-type correction. The construction assigns the five outgoing generators by a cyclic layer schedule with one non-constant layer determined by a zero-set Latin table; an explicit finite exact-cover certificate proves that this layer is a matching. By cyclic symmetry, Hamiltonicity of all color classes reduces to a single normalized return map. For $m \\geq 5$, an explicit first-return calculation on the section $p = 2$ gives one induced cycle whose excursion lengths sum to $m^4$. The remaining modulus $m = 3$ is settled by a printed finite cycle certificate. A companion Lean 4 formalization provides an independent machine verification of the Cayley statement and the finite certificates; source, audit scripts, and ancillary search code are available at https://github.com/aria1th/Torus-Hamilton-Decomposition-Program.","abs_url":"https://arxiv.org/abs/2604.27140","pdf_url":"https://arxiv.org/pdf/2604.27140v1","primary_category":"math.CO","published":"2026-04-29","tags":["formalization","combinatorics"],"lean_note":"A companion Lean 4 formalization independently verifies the Cayley statement and finite certificates.","repo":"https://github.com/aria1th/Torus-Hamilton-Decomposition-Program","thumb":"thumbs/2604.27140.jpg"},{"id":"2604.25962","title":"Coherent Rollout Oracles for Finite-Horizon Sequential Decision Problems","authors":["Nishant Shukla"],"abstract":"Coherent quantum rollout for sequential decision problems requires a unitary simulator: randomness must live in explicit quantum registers, and basis-state selectors must be mapped to actions reversibly. With branch-dependent valid actions, this mapping is totalized coherent rank-select over an entangled $N$-bit validity mask: return the position of the $r$-th valid bit, or a sentinel if $r$ is out of range. We give the first reversible-circuit complexity analysis of this primitive. For selector width $w = \\lceil \\log_2(N+1) \\rceil$, rank-select admits an $O(Nw)$-gate low-ancilla bounded-span scan, proved gate-optimal in its model, and an $O(N\\log w)$-gate low-ancilla blocked construction when long-range gates are available; across all bounded-fan-in layouts, the unconditional gate lower bound is $Ω(N)$. Composing rank-select with reversible transition and predicate-evaluation circuits gives an explicit polynomial-size coherent rollout oracle for finite-horizon planning problems satisfying these primitive assumptions. The resulting oracle satisfies the access model of the best-arm pipeline of Wang et al., yielding $\\widetilde{O}(\\sqrt{k}/\\varepsilon)$ coherent oracle calls against the standard classical $Ω(k/\\varepsilon^2)$ arm-pull lower bound. We give a bounded-influence lifting theorem that extends this lower-bound construction from a base configuration to an exponential family of configurations. We instantiate the construction on SIR epidemic intervention, with a stochastic placement-game sanity check, and machine-check the main results in Lean 4. Code and proofs: https://github.com/BinRoot/b01t/tree/main/demos/rollout.","abs_url":"https://arxiv.org/abs/2604.25962","pdf_url":"https://arxiv.org/pdf/2604.25962v1","primary_category":"quant-ph","published":"2026-04-28","tags":["verification","quantum"],"lean_note":"Machine-checks the main reversible-circuit results in Lean 4.","repo":"https://github.com/BinRoot/b01t","thumb":"thumbs/2604.25962.jpg"},{"id":"2604.25878","title":"Prime-Field PINI: Machine-Checked Composition Theorems for Post-Quantum NTT Masking","authors":["Ray Iskander","Khaled Kirah"],"abstract":"This is Paper 6 of a series of formally-verified analyses of masked NTT hardware for post-quantum cryptography; Paper 1 [1] established structural dependency analysis of the QANARY platform, and Paper 2 [2] quantified security margins under partial NTT masking. Boolean masking composition is well-understood through NI, SNI, and PINI. Arithmetic masking over $\\mathbb{Z}_q$ for prime $q$, the foundation of NTT-based post-quantum cryptography, has lacked an analogous theory. We prove, to our knowledge, the first machine-checked composition theorems for arithmetic masking over prime fields. Our key insight is the renewal argument: when a fresh random mask is applied between two pipeline stages, the intermediate wire becomes perfectly uniform regardless of Stage 1's security parameter. For two PF-PINI gadgets with parameters $k_1$ and $k_2$, the composed two-stage pipeline with fresh masking satisfies PF-PINI($k_2$), Stage 1's multiplicity is completely erased from the composed output. Without fresh masking, intermediate wires have multiplicity up to $k_1$, creating a necessary condition for differential power analysis. We formalize both theorems in Lean 4 with 18 machine-checked proofs and zero sorry stubs. We formally bridge the algebraic and hardware-faithful arithmetic models of Barrett reduction, and instantiate the theorems to formally diagnose Microsoft's Adams Bridge PQC accelerator: its absence of fresh inter-stage masking leaves Barrett output wires non-uniform under the first-order probing model, the same architectural flaw that two independent empirical analyses [3, 4] and our own prior structural analysis [1] identified. Computational evidence further suggests the 1-Bit Barrier is universal across Barrett and Montgomery reductions.","abs_url":"https://arxiv.org/abs/2604.25878","pdf_url":"https://arxiv.org/pdf/2604.25878v2","primary_category":"cs.CR","published":"2026-04-28","tags":["verification","cryptography"],"lean_note":"Formalizes prime-field masking composition theorems in Lean 4 (18 proofs, zero sorry).","thumb":"thumbs/2604.25878.jpg"},{"id":"2604.25419","title":"JURY-RL: Votes Propose, Proofs Dispose for Label-Free RLVR","authors":["Xinjie Chen","Biao Fu","Jing Wu","Guoxin Chen","Xinggao Liu","Dayiheng Liu","Minpeng Liao"],"abstract":"Reinforcement learning with verifiable rewards (RLVR) enhances the reasoning of large language models (LLMs), but standard RLVR often depends on human-annotated answers or carefully curated reward specifications. In machine-checkable domains, label-free alternatives such as majority voting or LLM-as-a-judge remove annotation cost but can introduce false positives that destabilize training. We introduce JURY-RL, a label-free RLVR framework that decouples answer proposal from reward disposal: votes from model rollouts propose a candidate answer, and a formal verifier determines whether that candidate can receive positive reward. Concretely, only rollouts matching the plurality-voted answer are rewarded when that answer is successfully verified in Lean. When verification is inconclusive, we invoke ResZero (Residual-Zero), a fallback reward that discards the unverified plurality proposal and redistributes a zero-mean, variance-preserving signal over the residual answers. This design maintains a stable optimization gradient without reinforcing unverifiable consensus. Across three backbone models trained on mathematical data, JURY-RL consistently outperforms other label-free baselines on mathematical reasoning benchmarks and transfers competitively to code generation and general benchmarks. It attains pass@1 performance comparable to supervised ground-truth training, with superior generalization demonstrated by higher pass@k and response diversity.","abs_url":"https://arxiv.org/abs/2604.25419","pdf_url":"https://arxiv.org/pdf/2604.25419v1","primary_category":"cs.AI","published":"2026-04-28","tags":["theorem-proving-ml","verification"],"lean_note":"Rewards only majority-voted answers that a Lean formal verifier confirms, using Lean verification to gate label-free RLVR training.","repo":"https://github.com/ruixin31/Spurious_Rewards","thumb":"thumbs/2604.25419.jpg"},{"id":"2604.25246","title":"Chebyshev quotients, Demazure multiplicities, and Dyck-path models","authors":["Rekha Biswal","Ken Ono","Jujian Zhang"],"abstract":"We study Chebyshev quotients that arise in the representation theory of Lie algebras, specifically within the theory of Demazure flags for fusion products of $\\mathfrak{sl}_2[t]$-modules. Using a recent formula that expresses numerical Demazure multiplicities as coefficients of such quotients, we prove a general eventual non-negativity theorem for the same rational functions that compute these multiplicities: each quotient either terminates or has strictly positive coefficients for sufficiently large degrees, which we in turn interpret in terms of matchings and bounded walks. In several natural infinite families, these are unsigned bounded Dyck path models, giving both a structural explanation for the observed positivity phenomenon and concrete combinatorial models for key families of Demazure multiplicities. The theorems in this paper were autonomously produced and formalized in Lean/Mathlib by AxiomProver from natural-language statements.","abs_url":"https://arxiv.org/abs/2604.25246","pdf_url":"https://arxiv.org/pdf/2604.25246v2","primary_category":"math.RT","published":"2026-04-28","tags":["autoformalization","mathlib","algebra"],"lean_note":"Theorems autonomously produced and formalized in Lean/Mathlib by AxiomProver.","repo":"https://github.com/AxiomMath/Biswal","thumb":"thumbs/2604.25246.jpg"},{"id":"2604.25028","title":"Null Measurability at the Symmetrization Interface in VC Learning","authors":["Dhruv Gupta"],"abstract":"Recent work revisiting measurability in the fundamental theorem of statistical learning imposes Borel measurability of ghost-gap suprema. We show that, at the one-sided ghost-gap interface actually used by the standard symmetrization proof, this requirement is stronger than necessary. For any Borel-parameterized concept class on a Polish domain, the bad event \"there exists a hypothesis whose ghost empirical error exceeds its training empirical error by at least ε/2\" is analytic. By Choquet capacitability, it is therefore measurable in the completion of every finite Borel measure. We then construct a concept class whose bad event is null-measurable but not Borel, giving a strict separation from the Borel supremum condition. Finally, we prove closure under patching, fixed and countable interpolation, and fiber-product amalgamation, showing that the weaker regularity level is stable under natural concept-class constructors. In the realizable setting, where targets belong to the class and are measurable, these results weaken the measurability hypothesis needed by the symmetrization route from finite VC dimension to PAC learnability. The main results and the descriptive-set-theoretic infrastructure used by them are formalized in Lean 4.","abs_url":"https://arxiv.org/abs/2604.25028","pdf_url":"https://arxiv.org/pdf/2604.25028v1","primary_category":"cs.LG","published":"2026-04-27","tags":["formalization","ml-theory"],"lean_note":"Formalizes the descriptive-set-theory infrastructure and main results in Lean 4.","repo":"https://github.com/Zetetic-Dhruv/formal-learning-theory-kernel","thumb":"thumbs/2604.25028.jpg"},{"id":"2604.24670","title":"Machine-Checked Cardinality Bounds for Masked Barrett Reduction: A 1-Bit Side-Channel Leakage Barrier in Post-Quantum Cryptographic Hardware","authors":["Ray Iskander","Khaled Kirah"],"abstract":"Barrett reduction is the nonlinear core of every practical NTT-based post-quantum cryptography implementation. Existing composition frameworks (ISW, t-SNI, PINI, DOM) address Boolean masking over GF(2); none provides a machine-checked characterization of Barrett's leakage under first-order arithmetic masking and the first-order probing model over prime fields. Building on our prior series, QANARY [15], partial-NTT-masking margins [14], algebraic foundations [16], and butterfly composition [18], we close this gap. We prove a trichotomy: for any $q > 0$ and shift $s$, the Barrett internal wire map $f_x(m) = ((x + 2^s - m) \\bmod 2^s) \\bmod q$ has preimage cardinality in $\\{0, 1, 2\\}$, never more. We call this the 1-Bit Barrier: max-multiplicity 2 implies at most 1 bit of min-entropy loss per internal wire, universal over all moduli. The count-zero cases, unreachable output values, reveal that actual leakage is often strictly less than 1 bit, making the bound conservative. We introduce PF-PINI (Prime-Field PINI): Barrett satisfies PF-PINI(2); the Cooley-Tukey butterfly satisfies PF-PINI(1). We observe (not yet proved) that with fresh inter-stage masking, the composed pipeline has max-multiplicity $\\max(k_1, k_2)$, so the 1-Bit Barrier propagates. The trichotomy, the PF-PINI instantiations, and cardinality results are machine-checked in Lean 4 with Mathlib: 12 proved results, zero sorry, universal over all $q > 0$ (the min-entropy bound follows by standard definitions). Adams Bridge lacks fresh inter-stage masking, violating PF-PINI composition and explaining why Papers 1 [15] and 2 [14] found vulnerabilities. NIST IR 8547 recommends formal methods for PQC implementation validation. The 1-Bit Barrier provides the first universal machine-checked cardinality bound for masked Barrett reduction in ML-KEM (FIPS 203) and ML-DSA (FIPS 204), with a corresponding 1-bit leakage interpretation.","abs_url":"https://arxiv.org/abs/2604.24670","pdf_url":"https://arxiv.org/pdf/2604.24670v2","primary_category":"cs.CR","published":"2026-04-27","tags":["verification","cryptography"],"lean_note":"Machine-checks cardinality/leakage bounds for masked Barrett reduction in Lean 4 with Mathlib.","thumb":"thumbs/2604.24670.jpg"},{"id":"2604.24195","title":"ZFLean: a framework for set-level mathematics in Lean","authors":["Vincent Trélat"],"abstract":"We present ZFLean, a Lean 4 library for doing core mathematics inside a model of ZFC with the ergonomics expected of typed Mathlib developments. Building on Mathlib's ZFC model, we contribute a relational calculus for sets with rewriting hints and small predictable tactics, canonical set-theoretic constructions -- Booleans, naturals, integers, sums/option -- and bridges between ZFC objects and Lean's native types enabling mixed set-level/typed proofs. The layer reduces boilerplate for extensional reasoning while remaining compatible with vanilla Mathlib. We discuss library organization and usage patterns that lower the friction of set-theoretic formalization in a dependently typed assistant. We demonstrate typical use of the framework with a case study exercising our constructions and relational calculus through a proof of an isomorphism theorem on curried functions.","abs_url":"https://arxiv.org/abs/2604.24195","pdf_url":"https://arxiv.org/pdf/2604.24195v1","primary_category":"cs.LO","published":"2026-04-27","tags":["tooling","mathlib","logic"],"lean_note":"Presents ZFLean, a Lean 4 library for set-level (ZFC) mathematics on top of Mathlib.","thumb":"thumbs/2604.24195.jpg"},{"id":"2604.24797","title":"The Network Structure of Mathlib","authors":["Xinze Li","Nanyun Peng","Simone Severini","Patrick Shafto"],"abstract":"The ongoing development of Lean 4's Mathlib has produced a macroscopic structural complexity that interweaves logical, mathematical, and infrastructural dependencies. We present a network analysis of this library, extracting its dependency structure into a multilayer graph of 308,129 declarations, 8.4 million edges, and 7,563 modules. By introducing graph decompositions that isolate explicit edges from those synthesized by the compiler or driven by proofs, we quantify the structural properties of formalized mathematics. Our analysis reveals three findings. First, taxonomies designed by humans diverge from logical structures, exhibiting a 50.9% coupling across namespaces. Second, developers utilize a median of 1.6% of the imported scope. Third, formalization compresses semantic hierarchies, with network centrality capturing language infrastructure rather than mathematical relevance.","abs_url":"https://arxiv.org/abs/2604.24797","pdf_url":"https://arxiv.org/pdf/2604.24797v2","primary_category":"cs.LO","published":"2026-04-26","tags":["mathlib","meta-study"],"lean_note":"Network analysis of Lean 4's Mathlib as a multilayer dependency graph (308k declarations).","repo":"https://github.com/MathNetwork/mathlib-network","thumb":"thumbs/2604.24797.jpg"},{"id":"2604.23712","title":"OptProver: Bridging Olympiad and Optimization through Continual Training in Formal Theorem Proving","authors":["Chenyi Li","Yanchen Nie","Zhenyu Ming","Gong Zhang","Kun Yuan","Zaiwen Wen"],"abstract":"Recent advances in formal theorem proving have focused on Olympiad-level mathematics, leaving undergraduate domains largely unexplored. Optimization, fundamental to machine learning, operations research, and scientific computing, remains underserved by existing provers. Its reliance on domain-specific formalisms (convexity, optimality conditions, and algorithmic analysis) creates significant distribution shift, making naive domain transfer ineffective. We present OptProver, a trained model that achieves robust transfer from Olympiad to undergraduate optimization. Starting from a strong Olympiad-level prover, our pipeline mitigates distribution shift through two key innovations. First, we employ large-scale optimization-focused data curation via expert iteration. Second, we introduce a specialized preference learning objective that integrates perplexity-weighted optimization with a mechanism to penalize valid but non-progressing proof steps. This not only addresses distribution shifts but also guides the search toward efficient trajectories. To enable rigorous evaluation, we construct a novel benchmark in Lean 4 focused on optimization. On this benchmark, OptProver achieves state-of-the-art Pass@1 and Pass@32 among comparably sized models while maintaining competitive performance on general theorem-proving tasks, demonstrating effective domain transfer without catastrophic forgetting.","abs_url":"https://arxiv.org/abs/2604.23712","pdf_url":"https://arxiv.org/pdf/2604.23712v2","primary_category":"cs.LG","published":"2026-04-26","tags":["theorem-proving-ml","benchmark"],"lean_note":"Builds a Lean 4 optimization theorem-proving benchmark and trains OptProver on it.","thumb":"thumbs/2604.23712.jpg"},{"id":"2604.23698","title":"Benchmarking Testing in Automated Theorem Proving","authors":["Jongyoon Kim","Hojae Han","Seung-won Hwang"],"abstract":"Recent advances in large language models (LLMs) have shown promise in formal theorem proving, yet evaluating semantic correctness remains challenging. Existing evaluations rely on indirect proxies such as lexical overlap with human-annotated proof, or expensive manual inspection. Inspired by the shift from lexical comparison to test-based evaluation in code generation, we propose T , a framework that evaluates the semantic correctness of formal theorems: a generated theorem is considered correct only if all dependent successor theorems compile successfully, analogous to integration testing. We construct a benchmark from 5 real-world Lean 4 repositories, comprising 2,206 problems paired with 41 successor theorems on average, automatically extracted without human effort. Experiments demonstrate that while state-of-the-art models achieve high compilation success, they perform significantly worse under our semantic metric. The best model, Claude-Sonnet-4.5, achieves only 38.9% Testing Accuracy on the full set, given both natural language proof and successor theorems as context, revealing a critical gap in current theorem generation capabilities.","abs_url":"https://arxiv.org/abs/2604.23698","pdf_url":"https://arxiv.org/pdf/2604.23698v1","primary_category":"cs.CL","published":"2026-04-26","tags":["benchmark","theorem-proving-ml"],"lean_note":"Constructs a semantic-correctness benchmark from 5 real-world Lean 4 repositories.","repo":"https://github.com/ldilab/T2","thumb":"thumbs/2604.23698.jpg"},{"id":"2604.23468","title":"Progress in Formalizing Sphere Packing in Dimension 8","authors":["Sidharth Hariharan","Christopher Birkbeck","Seewoo Lee","Ho Kiu Gareth Ma","Bhavik Mehta","Auguste Poiroux","Maryna Viazovska"],"abstract":"In 2016, Viazovska famously solved the sphere packing problem in dimension $8$, using modular forms to construct a 'magic' function satisfying optimality conditions determined by Cohn and Elkies in 2003. In March 2024, Hariharan and Viazovska launched a project to formalize this solution and related mathematical facts in the Lean Theorem Prover. A significant milestone was achieved in February 2026: the result was formally verified, with the final stages of the verification done by Math, Inc.'s autoformalization model 'Gauss'. We discuss the techniques used to achieve this milestone, reflect on the unique collaboration between humans and Gauss, and discuss project objectives that remain.","abs_url":"https://arxiv.org/abs/2604.23468","pdf_url":"https://arxiv.org/pdf/2604.23468v3","primary_category":"math.MG","published":"2026-04-25","tags":["formalization","mathlib","number-theory"],"lean_note":"Reports formal verification of 8-dimensional sphere packing in the Lean theorem prover.","repo":"https://github.com/thefundamentaltheor3m/Sphere-Packing-Lean","thumb":"thumbs/2604.23468.jpg"},{"id":"2604.23211","title":"Formalizing $A_1^{(1)}$ Curve Neighborhoods in Lean 4","authors":["Yihe Huang","Sizhe Cui","Jiaqi Wang","Jujian Zhang"],"abstract":"Combinatorial curve neighborhoods are somewhat foundational when setting up the quantum Schubert calculus for affine flag manifolds. In the specific case of type $A_1^{(1)}$, you can encode these neighborhoods entirely within the moment graph of the infinite dihedral group $D_\\infty$. Building on the framework developed by Mihalcea and Norton, this paper presents a complete, axiom-free formalization of these combinatorial curve neighborhoods in Lean 4. Rather than just wrapping mathematical statements, we formalized $D_\\infty$ directly as a Coxeter system to explicitly compute length functions and degree maps. Reachable sets are defined through edge chains bounded by specific degrees, and we ultimately characterize the curve neighborhood by the maximal vertices inside these sets. The core effort here lies in formally verifying the explicit combinatorial formulas for curve neighborhoods of arbitrary elements. Interestingly, by restricting our search space to finite sets, we also managed to extract a fully computable version of these neighborhoods.","abs_url":"https://arxiv.org/abs/2604.23211","pdf_url":"https://arxiv.org/pdf/2604.23211v1","primary_category":"math.CO","published":"2026-04-25","tags":["formalization","combinatorics"],"lean_note":"Axiom-free Lean 4 formalization of type-A_1^(1) combinatorial curve neighborhoods.","thumb":"thumbs/2604.23211.jpg"},{"id":"2604.23135","title":"Characterizing Paraphrase-Induced Failures in Lean 4 Autoformalization","authors":["William Feng","Ethan Lou","Aryan Sharma"],"abstract":"Lean 4 autoformalization has become increasingly popular in recent years, with frontier language models and open-weight autoformalizers now producing valid formalizations of mathematical theorems. However, these evaluations often rely on single canonical phrasings of theorems and rarely probe whether outputs are robust to natural variation in inputs, while prior work has shown that semantically equivalent paraphrases often induce divergent formal outputs. We study the structure of these divergences in Lean 4 by applying deterministic paraphrase rules to datasets of undergraduate and Olympiad-level math problems. Across four frontier models and three open-weight autoformalizers, we find that paraphrase sensitivity is dominated by failures at the code-generation layer, and that these failures are typed differently by dataset. Furthermore, these patterns generalize to open-weight models, showing that state-of-the-art autoformalizers still struggle to generate valid Lean code. Our results provide a failure-mode taxonomy for autoformalization and motivate training-time interventions targeted at specific compilation failures.","abs_url":"https://arxiv.org/abs/2604.23135","pdf_url":"https://arxiv.org/pdf/2604.23135v2","primary_category":"cs.LG","published":"2026-04-25","tags":["autoformalization","theorem-proving-ml"],"lean_note":"Studies paraphrase-induced failure modes of Lean 4 autoformalizers.","thumb":"thumbs/2604.23135.jpg"},{"id":"2604.23002","title":"FormalScience: Scalable Human-in-the-Loop Autoformalisation of Science with Agentic Code Generation in Lean","authors":["Jordan Meadows","Lan Zhang","Andre Freitas"],"abstract":"Formalising informal mathematical reasoning into formally verifiable code is a significant challenge for large language models. In scientific fields such as physics, domain-specific machinery (\\textit{e.g.} Dirac notation, vector calculus) imposes additional formalisation challenges that modern LLMs and agentic approaches have yet to tackle. To aid autoformalisation in scientific domains, we present FormalScience; a domain-agnostic human-in-the-loop agentic pipeline that enables a single domain expert (without deep formal language experience) to produce \\textit{syntactically correct} and \\textit{semantically aligned} formal proofs of informal reasoning for low economic cost. Applying FormalScience to physics, we construct FormalPhysics, a dataset of 200 university-level (LaTeX) physics problems and solutions (primarily quantum mechanics and electromagnetism), along with their Lean4 formal representations. Compared to existing formal math benchmarks, FormalPhysics achieves perfect formal validity and exhibits greater statement complexity. We evaluate open-source models and proprietary systems on a statement autoformalisation task on our dataset via zero-shot prompting, self-refinement with error feedback, and a novel multi-stage agentic approach, and explore autoformalisation limitations in modern LLM-based approaches. We provide the first systematic characterisation of semantic drift in physics autoformalisation in terms of concepts such as notational collapse and abstraction elevation which reveals what formal language verifies when full semantic preservation is unattainable. We release the codebase together with an interactive UI-based FormalScience system which facilitates autoformalisation and theorem proving in scientific domains beyond physics.https://github.com/jmeadows17/formal-science","abs_url":"https://arxiv.org/abs/2604.23002","pdf_url":"https://arxiv.org/pdf/2604.23002v1","primary_category":"cs.AI","published":"2026-04-24","tags":["autoformalization","dataset","ai-agents","physics"],"lean_note":"A human-in-the-loop agentic pipeline autoformalises informal physics reasoning into compiling Lean 4 / Mathlib code, producing the 200-problem FormalPhysics dataset.","repo":"https://github.com/jmeadows17/formal-science","thumb":"thumbs/2604.23002.jpg"},{"id":"2604.22602","title":"PASS: A Provenanced Access Subaccount System for Blockchain Wallets","authors":["Jay Yu","Shunfan Zhou","Hang Yin","Brian Seong"],"abstract":"Blockchain wallets conventionally follow an ownership model where possession of a private key grants unilateral control. However, this assumption is brittle for emerging settings such as AI agent wallets, organizational custody, and enterprise payroll, where multiple actors must coordinate without exposing secrets or leaking internal activity. We present PASS, a Provenanced Access Subaccount System that replaces role-based or identity-based control with provenance-based control: assets can only be used by subaccounts that can trace custody back to a valid deposit. A simple Inbox-Outbox mechanism ensures all external actions have verifiable lineage, while internal transfers remain private and indistinguishable from ordinary EOAs. We formalize PASS in Lean 4 and prove core invariants, including privacy of internal transfers, asset accessibility, and provenance integrity. We implement a prototype with enclave backends on AWS Nitro Enclaves and dstack Intel TDX, integrate with WalletConnect, and benchmark throughput across wallet operations. These results show that provenance-based wallets are both implementable and efficient. PASS bridges today's gap between strict self-custody and flexible shared access, advancing the design space for practical, privacy-preserving custody.","abs_url":"https://arxiv.org/abs/2604.22602","pdf_url":"https://arxiv.org/pdf/2604.22602v1","primary_category":"cs.CR","published":"2026-04-24","tags":["verification"],"lean_note":"Formalizes the PASS wallet protocol in Lean 4 and proves core privacy/provenance invariants.","thumb":"thumbs/2604.22602.jpg"},{"id":"2604.22531","title":"The Chase in Lean -- Crafting a Formal Library for Existential Rule Research","authors":["Lukas Gerlach"],"abstract":"The chase is a sound, complete, but possibly non-terminating algorithm for reasoning with existential rules (aka. tuple-generating dependencies), a highly expressive knowledge representation language. Although the procedure appears simple, research on theoretical properties and optimization for practical implementations has grown to a point where verifying correctness and reproducing proofs becomes challenging and intuition can sometimes be misleading. Lean is a purely functional programming language and interactive theorem prover whose community actively develops formal libraries for mathematics (Mathlib) and computer science (CSLib). In this work, we present our own endeavor of crafting a Lean framework around existential rules and the chase. We discuss design decisions concerning the nuances of chase definitions commonly found in the literature and show how these translate into Lean. To illustrate the framework's capabilities using known results, we show that the result of a chase is a universal model and outline the formalization for proving that without so-called \"alternative matches\" it is even a core. Beyond existing literature, we unify sufficient chase termination conditions in the likeness of Model-Faithful Acyclicity (MFA) into a common framework while also adding support for constants in rules.","abs_url":"https://arxiv.org/abs/2604.22531","pdf_url":"https://arxiv.org/pdf/2604.22531v1","primary_category":"cs.LO","published":"2026-04-24","tags":["formalization","tooling","logic"],"lean_note":"Crafts a Lean formal library for the chase algorithm over existential rules.","repo":"https://github.com/monsterkrampe/Existential-Rules-in-Lean","thumb":"thumbs/2604.22531.jpg"},{"id":"2604.22519","title":"Ablation and the Meno: Tools for Empirical Metamathematics","authors":["Zhengqin Fan","Simon DeDeo"],"abstract":"We present the results from Meno, a simple autoformalizer that proves theorems in Lean by systematically exploring the space of both formal and informal proofs, and tactic ablation, a new method for exploring mathematical creativity under constraint. We show these tools in action on simple theorems found in Terrence Tao's Analysis I, selectively ablating solution paths associated with non-constructive proofs, and analyze the properties of the resulting population using Goedel Prover embeddings. Among other things, our analysis of this novel population reveals that they lie on low (one or two) dimensional submanifolds of the much higher-dimensional representation space, and far away from their corresponding human constructions.","abs_url":"https://arxiv.org/abs/2604.22519","pdf_url":"https://arxiv.org/pdf/2604.22519v1","primary_category":"cs.LO","published":"2026-04-24","tags":["autoformalization","theorem-proving-ml","ai-agents","analysis"],"lean_note":"Meno, an MCP-orchestrated LLM autoformalizer, proves Lean 4 theorems and uses Lean metaprogramming to ablate tactics like Classical.choice.","repo":"https://github.com/oOo0oOo/lean-lsp-mcp","thumb":"thumbs/2604.22519.jpg"},{"id":"2604.21228","title":"Lean-certified four-point HRT results for three lattice points and one off-lattice point","authors":["Vignon Oussa"],"abstract":"We record a Lean-certified theorem package for the four-point Heil--Ramanathan--Topiwala configuration \\[ Λ=\\{0,a,b,ν\\}\\subset \\R^2, \\qquad \\Lzero=\\Z a+\\Z b, \\qquad ν=r a+s b, \\] with $a$ and $b$ linearly independent. The principal certified theorem states that if $|\\symp(a,b)|>1$ and $1,r,s$ are linearly independent over $\\Q$, then for every nonzero $f\\in L^2(\\R)$ the four vectors \\[ f,\\qquad π(a)f,\\qquad π(b)f,\\qquad π(ν)f \\] are linearly independent. A second certified theorem treats the rational-coordinate case $r,s\\in \\Q$, where the configuration lies in a finer full-rank lattice and linear independence follows from Linnell's theorem. The paper is written in standard mathematical prose. An appendix records the precise Lean certification ledger and the explicit analytic inputs used by the formal development and a download link is provided.","abs_url":"https://arxiv.org/abs/2604.21228","pdf_url":"https://arxiv.org/pdf/2604.21228v1","primary_category":"math.FA","published":"2026-04-23","tags":["formalization","analysis"],"lean_note":"Provides a Lean-certified theorem package proving linear independence of four-point HRT configurations with three lattice and one off-lattice point.","thumb":"thumbs/2604.21228.jpg"},{"id":"2604.21187","title":"Doubly Saturated Ramsey Graphs: A Case Study in Computer-Assisted Mathematical Discovery","authors":["Benjamin Przybocki","John Mackey","Marijn J. H. Heule","Bernardo Subercaseaux"],"abstract":"Ramsey-good graphs are graphs that contain neither a clique of size $s$ nor an independent set of size $t$. We study doubly saturated Ramsey-good graphs, defined as Ramsey-good graphs in which the addition or removal of any edge necessarily creates an $s$-clique or a $t$-independent set. We present a method combining SAT solving with bespoke LLM-generated code to discover infinite families of such graphs, answering a question of Grinstead and Roberts from 1982. In addition, we use LLMs to generate and formalize correctness proofs in Lean. This case study highlights the potential of integrating automated reasoning, large language models, and formal verification to accelerate mathematical discovery. We argue that such tool-driven workflows will play an increasingly central role in experimental mathematics.","abs_url":"https://arxiv.org/abs/2604.21187","pdf_url":"https://arxiv.org/pdf/2604.21187v1","primary_category":"math.CO","published":"2026-04-23","tags":["formalization","combinatorics"],"lean_note":"Uses LLMs to generate and formalize correctness proofs in Lean for discovered infinite families of doubly saturated Ramsey-good graphs.","thumb":"thumbs/2604.21187.jpg"},{"id":"2604.20793","title":"Fresh Masking Makes NTT Pipelines Composable: Machine-Checked Proofs for Arithmetic Masking in PQC Hardware","authors":["Ray Iskander","Khaled Kirah"],"abstract":"Post-quantum cryptographic (PQC) accelerators for ML-KEM (FIPS 203) and ML-DSA (FIPS 204) rely on pipelined Number Theoretic Transform (NTT) stages over $\\mathbb{Z}_q$. Our prior work established structural dependency analysis at scale [1] and quantified the security margin of partial NTT masking [2]. Whether per-stage arithmetic masking guarantees pipeline-level security had no prior machine-checked answer for the r-bearing case: composition frameworks (ISW, t-SNI, PINI, DOM) were formalized exclusively for Boolean masking over $\\mathrm{GF}(2)$; no proof assistant artifact addresses the NTT butterfly over $\\mathbb{Z}_q$. We present three machine-checked results in Lean 4 with Mathlib, all zero sorry. First, we close a stated limitation of prior work: value-independence implies constant marginal distribution under fresh randomness (via an algebraic MutualInfoZero proxy). Second, butterfly per-context uniformity: for any Cooley-Tukey butterfly with fresh output mask over $\\mathbb{Z}/q\\mathbb{Z}$ ($q > 0$), each output wire has exactly one mask value producing each output, a uniform marginal independent of secrets, universal over all moduli, twiddle factors, and inputs. Third, a k-stage NTT pipeline with fresh per-stage masking satisfies per-context uniformity at every stage under the ISW first-order probing model. We document a named warning: pointwise value-independence is false for butterfly outputs. The Adams Bridge accelerator (CHIPS Alliance Caliptra) fails the fresh masking hypothesis, masking active only in INTT round 0, architecturally explaining its structural insecurity. Artifact: nine theorems, 1,738 build jobs, zero sorry. Composition for nonlinear gadgets (Barrett) is addressed in forthcoming manuscripts proving Barrett's PF-PINI(2) satisfaction (one-bit barrier) [3] and k-stage composition for PF-PINI gadgets under fresh-mask renewal [4].","abs_url":"https://arxiv.org/abs/2604.20793","pdf_url":"https://arxiv.org/pdf/2604.20793v2","primary_category":"cs.CR","published":"2026-04-22","tags":["verification","cryptography"],"lean_note":"Three machine-checked Lean 4 + Mathlib results on fresh masking in NTT pipelines (zero sorry).","thumb":"thumbs/2604.20793.jpg"},{"id":"2604.20778","title":"Modularity, Extensions and Connectivity in Infinite Matroids","authors":["Mattias Ehatamm","Peter Nelson","Fernanda Rivera Omana"],"abstract":"We generalize the well-studied notion of a modular pair of a finite matroid to arbitrary families of sets in infinite matroids, and use it to develop the theory of infinite matroids in several as-yet-unexplored areas. Our results include a complete theory of single-element extensions, a description of the relationship between quotients and projections, a proof that matroids for which every flat is modular must be finitary, and two new perspectives on the infinite matroid connectivity parameter λ. In most cases, existing theory for finite matroids either fails completely or does not extend in obvious ways, and as a result we develop multiple new techniques for reasoning about infinite matroids, including establishing well-behaved infinite analogues of nullity, local connectivity and skewness. We also point to an online repository containing formalized proofs of all our results using the lean4 proof assistant","abs_url":"https://arxiv.org/abs/2604.20778","pdf_url":"https://arxiv.org/pdf/2604.20778v1","primary_category":"math.CO","published":"2026-04-22","tags":["formalization","combinatorics"],"lean_note":"Provides a Lean4-formalized repository of all main results on modularity, extensions, and connectivity in infinite matroids.","repo":"https://github.com/apnelson1/Matroid","thumb":"thumbs/2604.20778.jpg"},{"id":"2604.20209","title":"Scaling Self-Play with Self-Guidance","authors":["Luke Bailey","Kaiyue Wen","Kefan Dong","Tatsunori Hashimoto","Tengyu Ma"],"abstract":"LLM self-play algorithms are notable in that, in principle, nothing bounds their learning: a Conjecturer model creates problems for a Solver, and both improve together. However, in practice, existing LLM self-play methods do not scale well with large amounts of compute, instead hitting learning plateaus. We argue this is because over long training runs, the Conjecturer learns to hack its reward, collapsing to artificially complex problems that do not help the Solver improve. To overcome this, we introduce Self-Guided Self-Play (SGS), a self-play algorithm in which the language model itself guides the Conjecturer away from degeneracy. In SGS, the model takes on three roles: Solver, Conjecturer, and a Guide that scores synthetic problems by their relevance to unsolved target problems and how clean and natural they are, providing supervision against Conjecturer collapse. Our core hypothesis is that language models can assess whether a subproblem is useful for achieving a goal. We evaluate the scaling properties of SGS by running training for significantly longer than prior works and by fitting scaling laws to cumulative solve rate curves. Applying SGS to formal theorem proving in Lean4, we find that it surpasses the asymptotic solve rate of our strongest RL baseline in fewer than 80 rounds of self-play and enables a 7B parameter model, after 200 rounds of self-play, to solve more problems than a 671B parameter model pass@4.","abs_url":"https://arxiv.org/abs/2604.20209","pdf_url":"https://arxiv.org/pdf/2604.20209v1","primary_category":"cs.LG","published":"2026-04-22","tags":["theorem-proving-ml"],"lean_note":"Applies the Self-Guided Self-Play algorithm to Lean4 formal theorem proving, with a Guide model preventing conjecturer collapse.","repo":"https://github.com/LukeBailey181/sgs","thumb":"thumbs/2604.20209.jpg"},{"id":"2604.22844","title":"Operational Inexpressibility at the Step-Duplicating Primitive Recursor Orientation Boundary","authors":["Moses Rahnama"],"abstract":"We identify a structural property of term-rewriting proof systems called operational inexpressibility: no derivation depends on a specified input dimension and also constrains the target question. The canonical instance is direct aggregation on the primitive recursion duplicator $F(x,y,Z)\\to x$, $F(x,y,S(n))\\to G(y,F(x,y,n))$, where the step argument $y$ is duplicated on the right. Under any direct whole-term measure the recursor's mass profile coincides with that of a true circular reference; the boundary operator's channel-preservation axiom and the dependency-pair soundness license separate them. Sound responses split into construction methods (polynomial interpretations, path orderings) extending the proof language, and confession methods (dependency pairs, counter-projection, size-change termination, argument filtering) projecting away the unincorporable dimension under external license; all four share a projection rank and certified-forgetting interface. Arts-Giesl soundness is $Π^0_2$-combinatorial, formalizable in $\\mathrm{I}Σ_1$, with an artifact-facing $ω^3$ termination measure inside $\\mathrm{RCA}_0$, far below the $\\varepsilon_0$-scale of classical Gödelian reflection. The confessed burden grows quadratically across the canonical trace while residual proof work grows linearly. An architectural necessity theorem shows that any first-order step rule emitting a per-step record frame while preserving its generator must duplicate. A Layer-Crossing-Under-External-License (LCEL) schema places the confession in the Feferman-Beklemishev reflection family rather than the Lawvere-Yanofsky diagonal family, recovering the six-step structural identity with Gödel 1931 as a specialization. A witness-language hierarchy with minimal order $κ^{}$ identifies the boundary as $κ^{}(x)>0$.","abs_url":"https://arxiv.org/abs/2604.22844","pdf_url":"https://arxiv.org/pdf/2604.22844v3","primary_category":"cs.LO","published":"2026-04-21","tags":["formalization","verification","logic","type-theory"],"lean_note":"Ships unconditional formal equivalence theorems for a primitive-recursion duplicator as named Lean modules on a typed Arts-Giesl carrier.","repo":"https://github.com/MosesRahnama/The-Orientation-Boundary","thumb":"thumbs/2604.22844.jpg"},{"id":"2604.19459","title":"Do LLMs Game Formalization? Evaluating Faithfulness in Logical Reasoning","authors":["Kyuhee Kim","Auguste Poiroux","Antoine Bosselut"],"abstract":"Formal verification guarantees proof validity but not formalization faithfulness. For natural-language logical reasoning, where models construct axiom systems from scratch without library constraints, this gap between valid proofs and faithful translations is especially acute. We investigate whether frontier models exploit this gap when generating Lean 4 proofs, a behavior we term formalization gaming. We evaluate GPT-5 and DeepSeek-R1 on 303 first-order logic problems (203 from FOLIO, 100 from Multi-LogiEval), comparing unified generation against a two-stage pipeline that separates formalization from proving. Despite compilation rates of 87-99%, we find no evidence of systematic gaming in unified generation: models prefer reporting failure over forcing proofs, even under prompting designed to encourage it. However, unfaithfulness that evades our detection signals may still occur. The two-stage pipeline reveals two distinct modes of unfaithfulness: GPT-5 fabricates axioms during proof generation, a reactive fallback detectable via cross-stage comparison, while DeepSeek-R1 mistranslates premises during formalization, producing internally consistent outputs that evade detection entirely. These findings show that high compilation rates or accuracies should not be equated with faithful reasoning. Code and data are available at https://github.com/koreankiwi99/formalization-gaming.","abs_url":"https://arxiv.org/abs/2604.19459","pdf_url":"https://arxiv.org/pdf/2604.19459v1","primary_category":"cs.AI","published":"2026-04-21","tags":["theorem-proving-ml","autoformalization"],"lean_note":"Evaluates whether frontier LLMs 'game' formalization when generating Lean 4 proofs.","repo":"https://github.com/koreankiwi99/formalization-gaming","thumb":"thumbs/2604.19459.jpg"},{"id":"2604.19000","title":"Decompose, Structure, and Repair: A Neuro-Symbolic Framework for Autoformalization via Operator Trees","authors":["Xiaoyang Liu","Zineng Dong","Yifan Bai","Yantao Li","Yuntian Liu","Tao Luo"],"abstract":"Statement autoformalization acts as a critical bridge between human mathematics and formal mathematics by translating natural language problems into formal language. While prior works have focused on data synthesis and diverse training paradigms to optimize end-to-end Large Language Models (LLMs), they typically treat formal code as flat sequences, neglecting the hierarchical logic inherent in mathematical statements. In this work, we introduce Decompose, Structure, and Repair (DSR), a neuro-symbolic framework that restructures autoformalization into a modular pipeline. DSR decomposes statements into logical components and maps them to structured operator trees, leveraging this topological blueprint to precisely localize and repair errors via sub-tree refinement. Furthermore, we introduce PRIME, a benchmark of 156 undergraduate and graduate-level theorems selected from canonical textbooks and expertly annotated in Lean 4. Experimental results demonstrate that DSR establishes a new state-of-the-art, consistently outperforming baselines under equivalent computational budgets. The datasets, model, and code are available at https://github.com/XiaoyangLiu-sjtu/DSR.","abs_url":"https://arxiv.org/abs/2604.19000","pdf_url":"https://arxiv.org/pdf/2604.19000v2","primary_category":"cs.LG","published":"2026-04-21","tags":["autoformalization","benchmark"],"lean_note":"Introduces the PRIME benchmark of 156 theorems annotated in Lean 4 for autoformalization.","repo":"https://github.com/XiaoyangLiu-sjtu/DSR","thumb":"thumbs/2604.19000.jpg"},{"id":"2604.18882","title":"Formally Verified Patent Analysis via Dependent Type Theory: Machine-Checkable Certificates from a Hybrid AI + Lean 4 Pipeline","authors":["George Koomullil"],"abstract":"We present a formally verified framework for patent analysis as a hybrid AI + Lean 4 pipeline. The DAG-coverage core (Algorithm 1b) is fully machine-verified once bounded match scores are fixed. Freedom-to-operate, claim-construction sensitivity, cross-claim consistency, and doctrine-of-equivalents analyses are formalized at the specification level with kernel-checked candidate certificates. Existing patent-analysis approaches rely on manual expert analysis (slow, non-scalable) or ML/NLP methods (probabilistic, opaque, non-compositional). To our knowledge, this is the first framework that applies interactive theorem proving based on dependent type theory to intellectual property analysis. Claims are encoded as DAGs in Lean 4, match strengths as elements of a verified complete lattice, and confidence scores propagate through dependencies via proven-correct monotone functions. We formalize five IP use cases (patent-to-product mapping, freedom-to-operate, claim construction sensitivity, cross-claim consistency, doctrine of equivalents) via six algorithms. Structural lemmas, the coverage-core generator, and the closed-path identity coverage = W_cov are machine-verified in Lean 4. Higher-level theorems for the other use cases remain informal proof sketches, and their proof-generation functions are architecturally mitigated (untrusted generators whose outputs are kernel-checked and sorry-free axiom-audited). Guarantees are conditional on the ML layer: they certify mathematical correctness of computations downstream of ML scores, not the accuracy of the scores themselves. A case study on a synthetic memory-module claim demonstrates weighted coverage and construction-sensitivity analysis. Validation against adjudicated cases is future work.","abs_url":"https://arxiv.org/abs/2604.18882","pdf_url":"https://arxiv.org/pdf/2604.18882v1","primary_category":"cs.AI","published":"2026-04-20","tags":["verification","ai-agents"],"lean_note":"Hybrid AI + Lean 4 pipeline emitting kernel-checked certificates for patent analysis.","thumb":"thumbs/2604.18882.jpg"},{"id":"2604.18869","title":"Global Product Intersection Sets in Semigroups","authors":["Wouter van Doorn","Pietro Monticone","Quanyu Tang"],"abstract":"For a family $(A_q)_{q\\in Q}$ of subsets of a semigroup, the product intersection set records those exponents $h \\in \\mathbb{N}$ for which the $h$-fold product set of the intersection, $(\\bigcap_q A_q)^h$, is equal to $\\bigcap_q A_q^h$, the intersection of the product sets. Nathanson recently asked which subsets of $\\mathbb{N}$ can occur as a product intersection set, both for arbitrary and for decreasing families $(A_q)_{q\\in Q}$. We solve both problems by giving a complete classification. In particular, when $|Q| \\ge 2$, we show that in either case any subset $X \\subseteq \\mathbb{N}$ with $1 \\in X$ occurs as a product intersection set. Both classifications were autonomously discovered and formally verified in Lean by Aristotle, a formal reasoning agent developed by Harmonic.","abs_url":"https://arxiv.org/abs/2604.18869","pdf_url":"https://arxiv.org/pdf/2604.18869v2","primary_category":"math.CO","published":"2026-04-20","tags":["formalization","ai-agents","combinatorics","algebra"],"lean_note":"Both classifications of product intersection sets in semigroups were autonomously discovered and formally verified in Lean by Harmonic's Aristotle agent.","thumb":"thumbs/2604.18869.jpg"},{"id":"2604.18717","title":"From Finite Enumeration to Universal Proof: Ring-Theoretic Foundations for PQC Hardware Masking Verification","authors":["Ray Iskander","Khaled Kirah"],"abstract":"Formal verification of masking in post-quantum cryptographic (PQC) hardware relies on SMT solvers over finite domains. Our prior work established structural dependency analysis at scale [1] and quantified the security margin of partial NTT masking [2]. QANARY, our structural dependency analysis framework, verified 1.17 million cells across 30 modules of the Adams Bridge ML-DSA/ML-KEM accelerator [3, 4], but its core soundness result (Theorem 3.9.1) was machine-checked only at $q = 5$ via $2^{25}$ Boolean wire functions. This left portability to ML-KEM ($q = 3{,}329$, FIPS 203 [5]) and ML-DSA ($q = 8{,}380{,}417$, FIPS 204 [6]) as an open gap. NIST IR 8547 [7] (March 2025) motivates closing such gaps. We present the first machine-checked universal proof of the $r$-free sub-theorem of Theorem 3.9.1: for every $q > 0$, every wire function, and every pair of secrets, value-independence implies identical marginal distributions. The proof, in Lean 4 [8] with Mathlib [9], requires five lines versus $2^{25}$ finite evaluations. It is sorry-free, reducing the trusted base from {Z3 [10], CVC5 [11], Python} to the Lean 4 kernel. We provide nine theorems (T1--T6, T1', T3') covering reparametrization, bijectivity, overflow bounds, RNG bias, and a universal non-tightness counterexample for all $q \\geq 2$. The results establish commutative ring axioms of $\\mathbb{Z}/q\\mathbb{Z}$ as the natural abstraction layer for arithmetic masking verification.","abs_url":"https://arxiv.org/abs/2604.18717","pdf_url":"https://arxiv.org/pdf/2604.18717v2","primary_category":"cs.CR","published":"2026-04-20","tags":["verification","cryptography"],"lean_note":"First machine-checked universal proof of the QANARY r-free sub-theorem in Lean 4 with Mathlib.","thumb":"thumbs/2604.18717.jpg"},{"id":"2604.18216","title":"A Counterexample to EFX $n \\ge 3$ Agents, $m \\ge n + 5$ Items, Submodular Valuations via SAT-Solving","authors":["Hannaneh Akrami","Alexander Mayorov","Kurt Mehlhorn","Shreyas Srinivas","Christoph Weidenbach"],"abstract":"The existence of EFX allocations is a central open problem in discrete fair division. An allocation is EFX (envy-free up to any good) if no agent envies another agent after the removal of any single good from the other agent's bundle. We resolve this longstanding question by providing the \\textbf{first-ever counterexample} to the existence of EFX allocations for agents with monotone valuations, which in turn immediately implies a counterexample for submodular valuations. Specifically, we show that EFX allocations need not exist for instances with $n \\ge 3$ agents and $m \\ge n+5$ goods. In contrast, we prove that every instance with three agents and seven goods admits an EFX allocation. Both results are obtained via SAT solving. We encode the negation of EFX existence as a SAT instance: satisfiability yields a counterexample, while unsatisfiability establishes universal existence. The correctness of the encoding is formally verified in Lean. Finally, we establish positive guarantees for fair allocations with three agents and an arbitrary number of goods. Although EFX allocations may fail to exist, we prove that every instance with three agents and monotone valuations admits at least one of two natural relaxations of EFX: tEFX, or EF1 and EEFX.","abs_url":"https://arxiv.org/abs/2604.18216","pdf_url":"https://arxiv.org/pdf/2604.18216v3","primary_category":"cs.GT","published":"2026-04-20","tags":["verification","logic","combinatorics"],"lean_note":"Formally verifies in Lean (on Mathlib) the EFX fair-division theory and the correctness of the SAT/SMT encoding used to derive the counterexample.","repo":"https://zenodo.org/records/18637095","thumb":"thumbs/2604.18216.jpg"},{"id":"2604.17229","title":"Yanasse: Finding New Proofs from Deep Vision's Analogies, Part 1","authors":["Alexandre Linhares"],"abstract":"Project Yanasse presents a method for discovering new proofs of theorems in one area of mathematics by transferring proof strategy patterns (e.g., Lean 4 tactic invocation patterns) from a structurally distant area. The system extracts tactic usage distributions across 27 top-level areas of Mathlib (217,133 proof states), computes z-scores to identify tactics that are heavily used in a source area but rare or absent in a target area, matches source and target proof states via GPU-accelerated NP-hard analogy (running on a MacBook Air via Apple's MPS backend), and then asks an AI reasoning agent to semantically adapt--not symbol-substitute--the source tactics invocation pattern to the target theorem. In this first part of the study, the method is applied to the pair Probability -> Representation Theory, producing 4 Lean-verified new proofs out of 10 attempts (40%). The proofs compile with zero sorry declarations. The key finding is that tactic schemas decompose into a head (domain-gated, rarely transfers) and a modifier (domain-general, often transfers): filter upwards's head fails in representation theory (no Filter structure), but its [LIST] with ω modifier transfers cleanly as ext1 + simp [LIST] + rfl. Crucially, the underlying matching engine--deep vision lib.py--is entirely domain independent: the same optimization code for an NP-hard matching that matches chess positions by analogy matches Lean proof states by analogy, without knowing which domain it is processing. Only a relation extractor is domain-specific.","abs_url":"https://arxiv.org/abs/2604.17229","pdf_url":"https://arxiv.org/pdf/2604.17229v1","primary_category":"cs.AI","published":"2026-04-19","tags":["theorem-proving-ml","mathlib"],"lean_note":"Transfers Mathlib tactic-invocation patterns across areas to find new Lean-verified proofs.","thumb":"thumbs/2604.17229.jpg"},{"id":"2604.16898","title":"From Swap Axioms to Weighted Geometric Means: A Characterization of AMMs","authors":["Björn Assmann","Ulan Degenbaev"],"abstract":"Many automated market makers can be understood through the geometry of their trading orbits, the sets of states reachable from one another through swaps. In prominent designs, this geometry is captured by a simple closed-form invariant such as the constant product $xy$ in Uniswap or a weighted geometric mean $x^w y^{1-w}$ in Balancer. This paper explains why these forms arise by deriving them from three basic assumptions: validity invariance (swaps preserve the validity of states), Pareto efficiency (no state on an orbit weakly dominates another), and unit invariance (changing measurement units does not change the mechanism). Together, these force every trading orbit of a two-asset AMM to be a level set of a weighted geometric mean $x^w y^{1-w}$. Applied pairwise, the axioms extend the classification to $n$-asset pools: orbits are level sets of $\\prod_i x_i^{w_i}$ with positive weights $w_i$ summing to $1$. Imposing token-relabeling symmetry then pins down the weights, recovering the constant-product form $xy$ in the two-asset case and $\\prod_i x_i$ in general. The main text provides an intuitive proof sketch and discusses fees and liquidity operations. Complete proofs and a machine-checked Lean 4 formalization accompany the paper.","abs_url":"https://arxiv.org/abs/2604.16898","pdf_url":"https://arxiv.org/pdf/2604.16898v1","primary_category":"cs.DC","published":"2026-04-18","tags":["formalization","economics"],"lean_note":"A machine-checked Lean 4 formalization accompanies the AMM characterization proofs.","repo":"https://github.com/bjoernek/amm-axioms-lean","thumb":"thumbs/2604.16898.jpg"},{"id":"2604.16584","title":"Certified Program Synthesis with a Multi-Modal Verifier","authors":["Yueyang Feng","Dipesh Kafle","Vladimir Gladshtein","Vitaly Kurin","George Pîrlea","Qiyuan Zhao","Peter Müller","Ilya Sergey"],"abstract":"Certified program synthesis (aka vericoding) is the process of automatically generating a program, its formal specification, and a machine-checkable proof of their alignment from a natural-language description. Two challenges make vericoding difficult. First, specifications synthesised from natural language are often either too weak to be meaningful or too strong to be implementable, yet existing approaches lack systematic means to detect such defects. Second, the landscape of program verifiers is fragmented: each tool supports a particular reasoning mode -- auto-active (e.g., Dafny, Verus) or interactive (e.g., Coq, Lean) -- with its own trade-off between automation and expressivity. This forces every synthesis methodology to be tailored to a single verification paradigm, limiting the class of tasks it can handle effectively. We overcome both challenges by structuring the certified synthesis workflow around a multi-modal verifier -- a single tool combining dynamic validation, automated proofs, and interactive proof scripting in one foundational framework. We realise this idea in LeetProof, an agentic pipeline built on Velvet, a multi-modal verifier embedded in Lean. Multi-modality enables LeetProof to validate generated specifications via randomised property-based testing before any code is synthesised, decompose the synthesis task into sub-problems guided by verification conditions, and delegate residual proof obligations to frontier AI provers specialised for Lean. We evaluate LeetProof on benchmarks derived from prior work on certified synthesis. Our specification validation uncovers defects in existing reference benchmarks, and LeetProof's staged pipeline achieves a significantly higher rate of fully certified solutions than a single-mode baseline at the same budget -- consistently across two frontier LLM backends.","abs_url":"https://arxiv.org/abs/2604.16584","pdf_url":"https://arxiv.org/pdf/2604.16584v1","primary_category":"cs.SE","published":"2026-04-17","tags":["verification","ai-agents"],"lean_note":"Builds the LeetProof certified-synthesis pipeline on Velvet, a multi-modal verifier embedded in Lean, delegating residual proof obligations to Lean-specialized AI provers.","thumb":"thumbs/2604.16584.jpg"},{"id":"2604.15839","title":"Discover and Prove: An Open-source Agentic Framework for Hard Mode Automated Theorem Proving in Lean 4","authors":["Chengwu Liu","Yichun Yin","Ye Yuan","Jiaxuan Xie","Botao Li","Siqi Li","Jianhao Shen","Yan Xu","Lifeng Shang","Ming Zhang"],"abstract":"Most ATP benchmarks embed the final answer within the formal statement -- a convention we call \"Easy Mode\" -- a design that simplifies the task relative to what human competitors face and may lead to optimistic estimates of model capability. We call the stricter, more realistic setting \"Hard Mode\": the system must independently discover the answer before constructing a formal proof. To enable Hard Mode research, we make two contributions. First, we release MiniF2F-Hard and FIMO-Hard, expert-reannotated Hard Mode variants of two widely-used ATP benchmarks. Second, we introduce Discover And Prove (DAP), an agentic framework that uses LLM natural-language reasoning with explicit self-reflection to discover answers, then rewrites Hard Mode statements into Easy Mode ones for existing ATP provers. DAP sets the state of the art: on CombiBench it raises solved problems from 7 (previous SOTA, Pass@16) to 10; on PutnamBench it is the first system to formally prove 36 theorems in Hard Mode -- while simultaneously revealing that state-of-the-art LLMs exceed 80% answer accuracy on the same problems where formal provers manage under 10%, exposing a substantial gap that Hard Mode benchmarks are uniquely suited to measure.","abs_url":"https://arxiv.org/abs/2604.15839","pdf_url":"https://arxiv.org/pdf/2604.15839v1","primary_category":"cs.AI","published":"2026-04-17","tags":["theorem-proving-ml","benchmark"],"lean_note":"Open-source agentic framework for Hard Mode automated theorem proving in Lean 4.","repo":"https://github.com/liuchengwucn/discover-and-prove","thumb":"thumbs/2604.15839.jpg"},{"id":"2604.16538","title":"Understanding Tool-Augmented Agents for Lean Formalization: A Factorial Analysis","authors":["Ke Zhang","Patricio Gallardo","Maziar Raissi","Sudhir Murthy"],"abstract":"Automatic translation of natural language mathematics into faithful Lean 4 code is hindered by the fundamental dissonance between informal set-theoretic intuition and strict formal type theory. This gap often causes LLMs to hallucinate non-existent library definitions, resulting in code that fails to compile or lacks semantic fidelity. In this work, we investigate the effectiveness of tool-augmented agents for this task through a systematic factorial analysis of three distinct tool categories: Fine-tuned Model Querying (accessing expert drafts), Knowledge Search (retrieving symbol definitions), and Compiler Feedback (verifying code via a Lean REPL). We first benchmark the agent against one-shot baselines, demonstrating large gains in both compilation success and semantic equivalence. We then use the factorial decomposition to quantify the impact of each category, isolating the marginal contribution of each tool type to overall performance.","abs_url":"https://arxiv.org/abs/2604.16538","pdf_url":"https://arxiv.org/pdf/2604.16538v1","primary_category":"cs.SE","published":"2026-04-16","tags":["theorem-proving-ml","autoformalization"],"lean_note":"Factorial study of tool-augmented agents (incl. a Lean REPL) for Lean 4 formalization.","thumb":"thumbs/2604.16538.jpg"},{"id":"2604.15305","title":"Erdős's diameter conjecture for separated distances fails in high dimensions","authors":["Boon Suan Ho"],"abstract":"Erdős asked whether every $n$-point set in Euclidean space whose $\\binom{n}{2}$ pairwise distances are mutually at least $1$ apart must have diameter at least $(1+o(1))n^2$. We disprove this statement by constructing for every prime power $q$ a set $\\mathcal X_q\\subset \\mathbb R^{q^2+q}$ of $n=q+1$ points such that all pairwise distances in $\\mathcal X_q$ are mutually at least $1$ apart, while $$\\operatorname{diam}(\\mathcal X_q)\\le\\Bigl(1-\\frac{1}{π^2}+o(1)\\Bigr)n^2.$$ The proof is fully formalized in Lean 4.","abs_url":"https://arxiv.org/abs/2604.15305","pdf_url":"https://arxiv.org/pdf/2604.15305v1","primary_category":"math.CO","published":"2026-04-16","tags":["formalization","combinatorics"],"lean_note":"Disproof of an Erdős conjecture, fully formalized in Lean 4.","repo":"https://github.com/boonsuan/erdos670","thumb":"thumbs/2604.15305.jpg"},{"id":"2604.14912","title":"Formalizing Wu-Ritt Method in Lean 4","authors":["Yuxuan Xiao","Hao Shen","Junyu Guo","Dingkang Wang","Lihong Zhi"],"abstract":"We formalize the Wu-Ritt characteristic set method for the triangular decomposition of polynomial systems in the Lean 4 theorem prover. Our development includes the core algebraic notions of the method, such as polynomial initials, orders, pseudo-division, pseudo-remainders with respect to a polynomial or a triangular set, and standard and weak ascending sets. On this basis, we formalize algorithms for computing basic sets, characteristic sets, and zero decompositions, and prove their termination and correctness. In particular, we formalize the well-ordering principle relating a polynomial system to its characteristic set and verify that zero decomposition expresses the zero set of the original system as a union of zero sets of triangular sets away from the zeros of the corresponding initials. This work provides a machine-checked verification of Wu-Ritt's method in Lean 4 and establishes a foundation for certified polynomial system solving and geometric theorem proving.","abs_url":"https://arxiv.org/abs/2604.14912","pdf_url":"https://arxiv.org/pdf/2604.14912v1","primary_category":"math.AC","published":"2026-04-16","tags":["formalization","algebra","tooling"],"lean_note":"Formalizes the Wu-Ritt characteristic-set method in Lean 4 with termination/correctness proofs.","thumb":"thumbs/2604.14912.jpg"},{"id":"2604.14512","title":"CBCL: Safe Self-Extending Agent Communication","authors":["Hugo O'Connor"],"abstract":"Agent communication languages (ACLs) enable heterogeneous agents to share knowledge and coordinate across diverse domains. This diversity demands extensibility, but expressive extension mechanisms can push the input language beyond the complexity classes where full validation is tractable. We present CBCL (Common Business Communication Language), an agent communication language that constrains all messages, including runtime language extensions, to the deterministic context-free language (DCFL) class. CBCL allows agents to define, transmit, and adopt domain-specific \"dialect\" extensions as first-class messages; three safety invariants (R1--R3), machine-checked in Lean 4 and enforced in a Rust reference implementation, prevent unbounded expansion, applying declared resource limits, and preserving core vocabulary. We formalize the language and its safety properties in Lean 4, implement a reference parser and dialect engine in Rust with property-based and differential tests, and extract a verified parser binary. Our results demonstrate that homoiconic protocol design, where extension definitions share the same representation as ordinary messages, can be made provably safe. As autonomous agents increasingly extend their own communication capabilities, formally bounding what they can express to each other is a precondition for oversight.","abs_url":"https://arxiv.org/abs/2604.14512","pdf_url":"https://arxiv.org/pdf/2604.14512v1","primary_category":"cs.CR","published":"2026-04-16","tags":["verification","ai-agents"],"lean_note":"Three CBCL safety invariants machine-checked in Lean 4 (with a verified Rust parser).","repo":"https://codeberg.org/anuna/cbcl-rs","thumb":"thumbs/2604.14512.jpg"},{"id":"2604.13515","title":"SFT-GRPO Data Overlap as a Post-Training Hyperparameter for Autoformalization","authors":["Xiaole Su","Kasey Zhang","Andy Lyu"],"abstract":"Supervised fine-tuning (SFT) followed by Group Relative Policy Optimization (GRPO) is a common post-training recipe. We conduct a controlled ablation over SFT-GRPO data overlap, evaluating Qwen3-8B (thinking disabled) post-trained for Lean 4 autoformalization under six conditions that differ solely in training recipe: a base model, SFT-only, GRPO-only, and three SFT+GRPO configurations where 0 percent, 30 percent, or 100 percent of the GRPO prompts coincide with the SFT corpus. Keeping SFT and GRPO data disjoint consistently outperforms full overlap at zero additional compute cost. Evaluating on Gaokao-Formal and PutnamBench under both compile pass at k and semantic pass at k assessed by an LLM judge, we find that lower overlap is monotonically associated with higher compilation and semantic accuracy. At 0 percent overlap, GRPO yields a 10.4 percentage point semantic gain over SFT alone on Gaokao, while at 100 percent overlap both metrics remain flat, rendering the GRPO stage effectively redundant. We further show that dual-metric evaluation reveals compile semantic gaps exceeding 30 percentage points for the highest compiling models, a disparity invisible under compile-only benchmarking. To our knowledge, this is the first controlled investigation of SFT-GRPO data overlap as a post-training hyperparameter, demonstrating how model behavior varies based on the degree of data sharing between training stages.","abs_url":"https://arxiv.org/abs/2604.13515","pdf_url":"https://arxiv.org/pdf/2604.13515v1","primary_category":"cs.LG","published":"2026-04-15","tags":["theorem-proving-ml","autoformalization"],"lean_note":"Ablates SFT-GRPO data overlap for a Lean 4 autoformalization model.","thumb":"thumbs/2604.13515.jpg"},{"id":"2604.13514","title":"Automated Tactics for Polynomial Reasoning in Lean 4","authors":["Hao Shen","Junyu Guo","Junqi Liu","Lihong Zhi"],"abstract":"Applying Gröbner basis theory to concrete problems in Lean 4 remains difficult since the current formalization of multivariate polynomials is based on a non-computable representation and is therefore not suitable for efficient symbolic computation. As a result, computing Gröbner bases directly inside Lean is impractical for realistic examples. To address this issue, we propose a certificate-based approach that combines external computer algebra systems, such as SageMath or SymPy, with formal verification in Lean 4. Our approach uses a computable representation of multivariate polynomials in Lean to import and verify externally generated Gröbner basis computations. The external solver carries out the main algebraic computations, while the returned results are verified inside Lean. Based on this method, we develop automated tactics that transfer polynomial data between Lean and the external system and certify the returned results. These tactics support tasks such as remainder verification, Gröbner basis checking, ideal equality, and ideal or radical membership. This work provides a practical way to integrate external symbolic computation into Lean 4 while preserving the reliability of formal proof.","abs_url":"https://arxiv.org/abs/2604.13514","pdf_url":"https://arxiv.org/pdf/2604.13514v1","primary_category":"cs.LO","published":"2026-04-15","tags":["tactic-automation","tooling"],"lean_note":"Certificate-based Lean 4 tactics that import and verify external Gröbner-basis computations.","repo":"https://github.com/WuProver/GroebnerTactic","thumb":"thumbs/2604.13514.jpg"},{"id":"2604.16507","title":"Deep Vision: A Formal Proof of Wolstenholmes Theorem in Lean 4","authors":["Alexandre Linhares"],"abstract":"We present a formal verification of Wolstenholme's theorem -- $\\binom{2p}{p} \\equiv 2 \\pmod{p^3}$ for prime $p \\geq 5$ -- in Lean~4 with Mathlib. The proof proceeds by expanding the shifted factorial product $\\prod_{k=1}^{p-1}(p+k)$ to second order in $p$, identifying the quadratic coefficient as the second elementary symmetric product, and showing its divisibility by $p$ via power sum vanishing in $\\mathbb{Z}/p\\mathbb{Z}$. The formalization comprises nine lemmas across approximately 800 lines of Lean, with zero \\texttt{sorry} declarations. To our knowledge, this is the first formal verification of Wolstenholme's theorem in Lean~4. The proof was discovered through a collaboration between a relational analogy engine for theorem proving and human-directed formalization.","abs_url":"https://arxiv.org/abs/2604.16507","pdf_url":"https://arxiv.org/pdf/2604.16507v1","primary_category":"cs.LO","published":"2026-04-14","tags":["formalization","number-theory"],"lean_note":"First Lean 4 + Mathlib formal proof of Wolstenholme's theorem (~800 lines, zero sorry).","repo":"https://github.com/ARGO-LABORATORY/Wolstenholme_1862","thumb":"thumbs/2604.16507.jpg"},{"id":"2604.13259","title":"Global attractors and fast-slow reduction for finite-state actor-critic mean dynamics","authors":["Vladyslav Prytula"],"abstract":"When a learning algorithm reshapes the data distribution it trains on, the long-run behavior depends on the joint evolution of the policy, the value estimate, and the data distribution. We study finite-state actor-critic mean dynamics on the enlarged phase space $(θ,w,μ)$, where $θ$ is the actor parameter, $w$ is an auxiliary critic state, and $μ$ is a state-law variable (the distribution over states induced by the current policy). The state-law coordinate follows the exact controlled-Markov equation $δ\\dotμ= Q_θ^*μ$. Under a softmax actor with box confinement (a smooth proxy for parameter clipping), a uniformly coercive linear critic equation, and a Lipschitz generator family $θ\\mapsto Q_θ$, we prove that for each $δ> 0$ the resulting autonomous semiflow possesses a compact global attractor. Under a uniform exponential-mixing assumption, we prove that the invariant-law map $θ\\mapsto μ_θ$ is Lipschitz and that the reduced invariant-law system on $(θ,w)$ is well posed. Under an additional pathwise exponential-stability estimate for the non-autonomous fast state equation, we show that the exact flow tracks the reduced flow on every finite time interval up to the initial layer, and that the exact attractors converge upper semicontinuously to the lifted reduced attractor as $δ\\to 0$. We also give a concrete finite-state reference-state minorization condition implying the pathwise hypothesis. All results are formalized in Lean 4 without custom axioms.","abs_url":"https://arxiv.org/abs/2604.13259","pdf_url":"https://arxiv.org/pdf/2604.13259v1","primary_category":"math.DS","published":"2026-04-14","tags":["formalization","ml-theory"],"lean_note":"Formalizes global-attractor / fast-slow reduction results in Lean 4 without custom axioms.","thumb":"thumbs/2604.13259.jpg"},{"id":"2604.13238","title":"A quadratic form generalization of rational dinv","authors":["Yifeng Huang"],"abstract":"We introduce a quadratic form $Q$ on the space of functions on the gap poset $G$ of the numerical semigroup $\\langle a,b\\rangle$. We prove combinatorially that when evaluated on the indicator function of an upward closed subset $D$, this quadratic form precisely recovers the Gorsky--Mazin $\\mathtt{dinv}$ statistic of $D$, viewed as a Young subdiagram of $G$. Furthermore, we prove Theorem~1.2 that when evaluated on a pair of subdiagrams of $G$, the symmetric bilinear form associated with $Q$ is equal to a novel cross-$\\mathtt{dinv}$ statistic, which is nonnegative. Combining these, we prove the inequality \\[ Q(\\mathbf{n})\\geq \\dfrac{1}{|G|}\\,\\|\\mathbf{n}\\|_\\infty^2\\] if $\\mathbf{n}$ is a real-valued decreasing function on $G$, showing an effective positive definiteness of $Q$ on the corresponding cone. Theorem~1.2, the main engine of the paper, was autoformalized in Lean/Mathlib by AxiomProver.","abs_url":"https://arxiv.org/abs/2604.13238","pdf_url":"https://arxiv.org/pdf/2604.13238v1","primary_category":"math.CO","published":"2026-04-14","tags":["autoformalization","mathlib","combinatorics"],"lean_note":"Main theorem autoformalized in Lean/Mathlib by AxiomProver.","repo":"https://github.com/AxiomMath/quadratic-dinv","thumb":"thumbs/2604.13238.jpg"},{"id":"2604.12981","title":"Recursive Completion in Higher K-Models: Front-Seed Semantics, Proof-Relevant Witnesses, and the K-Infinity Model","authors":["Daniel O. Martinez-Rivillas","Arthur F. Ramos","Ruy J. G. B. de Queiroz"],"abstract":"Martinez-Rivillas and de Queiroz gave extensional Kan semantics for the untyped lambda-calculus and later constructed the concrete K-infinity homotopy-model. The two main mathematical results of the present paper are these. First, we show that a smaller front-seed coherence package (WL, WR) together with an inner-right-front pentagon contraction already suffices to recover the associator comparison, semantic pentagon, and bridge theorems used in the later semantic arguments. Second, we prove explicit global reify, reflect, and application formulas for K-infinity, with exact coordinatewise identities at every finite stage. We also record two structural clarifications: the recursive all-dimensional continuation of the explicit low-dimensional tower is obtained by a finite packaging phase followed by a uniform equality-generated recursion; and, on a deliberately fixed forward witness language for the classical separation span, the canonical identity-type higher tower on K-infinity forces all higher non-connection once the two witness classes land at distinct points. The paper is fully formalized in Lean 4, and the project sources contain no local uses of sorry, admit, or axiom.","abs_url":"https://arxiv.org/abs/2604.12981","pdf_url":"https://arxiv.org/pdf/2604.12981v1","primary_category":"cs.LO","published":"2026-04-14","tags":["formalization","type-theory"],"lean_note":"Fully formalizes higher K-model semantics in Lean 4 (no sorry/admit/axiom).","repo":"https://github.com/Arthur742Ramos/HigherLambdaModel","thumb":"thumbs/2604.12981.jpg"},{"id":"2604.10638","title":"Timing, Entry, and Revenue in Clock-Based Platform Markets","authors":["Thomas Pitz","Vinicius Ferraz"],"abstract":"On platforms where time-to-contract is itself payoff-relevant--Aalsmeer's flower auctions, ride-hailing dispatch, on-demand-labor matching--the textbook revenue equivalence between Dutch and first-price formats holds the trading outcome fixed. Once participation is endogenous and both sides bear waiting costs, the trading format directly shapes who enters, market thickness, volume, and platform revenue. The platform's ranking of the descending clock against immediate and batched posted-price benchmarks is decided by two estimable primitives on each side of the market: an earnings gap and a timing gap. A bidirectional four-case classification identifies when the descending clock dominates at every level of waiting costs, only above a floor, only below a ceiling, or not at all; the last case is unconditional -- when the descending clock charges no more per trade and contracts no faster than the posted-price benchmark, it cannot win. No format admits a universal ranking. The local verdict propagates through endogenous entry, and cross-side complementarity amplifies shared local advantages into joint dominance. A conditional revenue theorem converts entry and volume gains into a platform-revenue ranking. In calibrated parameterizations the revenue-ranking switching boundary lies near $p_0/\\bar v\\approx 1$, inside the empirical range for ride-hailing platforms. A measurement protocol provides explicit nonparametric estimators for the six reduced-form objects and a test statistic for the dominance condition, and a Lean~4 formalization audits the algebraic and order-theoretic content. In markets where goods or services cannot wait, the speed of the trading mechanism is a primitive of market design.","abs_url":"https://arxiv.org/abs/2604.10638","pdf_url":"https://arxiv.org/pdf/2604.10638v2","primary_category":"econ.TH","published":"2026-04-12","tags":["formalization","economics"],"lean_note":"A Lean 4 formalization audits the algebraic and order-theoretic content of the market model.","repo":"https://github.com/vferraz/dutch-auctions-matching-markets","thumb":"thumbs/2604.10638.jpg"},{"id":"2604.10392","title":"Intent-aligned Formal Specification Synthesis via Traceable Refinement","authors":["Zhe Ye","Aidan Z. H. Yang","Huangyuan Su","Zhenyu Liao","Samuel Tenka","Zhizhen Qin","Udaya Ghai","Dawn Song","Soonho Kong"],"abstract":"Large language models are increasingly used to generate code from natural language, but ensuring correctness remains challenging. Formal verification offers a principled way to obtain such guarantees by proving that a program satisfies a formal specification. However, specifications are frequently missing in real-world codebases, and writing high-quality specifications remains expensive and expertise-intensive. We present VeriSpecGen, a traceable refinement framework that synthesizes intent-aligned specifications in Lean through requirement-level attribution and localized repair. VeriSpecGen decomposes natural language into atomic requirements and generates requirement-targeted tests with explicit traceability maps to validate generated specifications. When validation fails, traceability maps attribute failures to specific requirements, enabling targeted clause-level repairs. VeriSpecGen achieve 86.6% on VERINA SpecGen task using Claude Opus 4.5, improving over baselines by up to 31.8 points across different model families and scales. Beyond inference-time gains, we generate 343K training examples from VeriSpecGen refinement trajectories and demonstrate that training on these trajectories substantially improves specification synthesis by 62-106% relative and transfers gains to general reasoning abilities.","abs_url":"https://arxiv.org/abs/2604.10392","pdf_url":"https://arxiv.org/pdf/2604.10392v1","primary_category":"cs.LG","published":"2026-04-12","tags":["autoformalization","verification","tooling","dataset"],"lean_note":"Synthesizes intent-aligned formal specifications in Lean, validating them with Lean-checked requirement-targeted tests and traceable clause-level repair.","thumb":"thumbs/2604.10392.jpg"},{"id":"2604.09808","title":"A formal proof of the Ramanujan--Nagell theorem in Lean 4","authors":["Barinder S. Banwait"],"abstract":"We present a complete formalization, in the Lean interactive theorem prover with the Mathlib library, of the Ramanujan--Nagell theorem: the only integer solutions to the Diophantine equation $x^2 + 7 = 2^n$ are $(n,x) \\in \\{(3,\\pm1),(4,\\pm3),(5,\\pm5),(7,\\pm11),(15,\\pm181)\\}$. The formalization includes all dependencies, notably the computation of the ring of integers of the quadratic field $\\mathbb{Q}(\\sqrt{-7})$, its class number, and unit group. We describe the proof strategy, the architecture of the formalization, and the challenges encountered in bridging the gap between textbook proofs and their machine-checked counterparts, with particular attention to the algebraic number theory infrastructure required.","abs_url":"https://arxiv.org/abs/2604.09808","pdf_url":"https://arxiv.org/pdf/2604.09808v2","primary_category":"math.NT","published":"2026-04-10","tags":["formalization","mathlib","number-theory"],"lean_note":"Complete Lean + Mathlib formalization of the Ramanujan-Nagell theorem and its number-theory dependencies.","repo":"https://github.com/sacerdot/QuadraticIntegers","thumb":"thumbs/2604.09808.jpg"},{"id":"2604.08485","title":"Formalizing building-up constructions of self-dual codes through isotropic lines in Lean","authors":["Jae-Hyun Baek","Jon-Lark Kim"],"abstract":"The purpose of this paper is two-fold. First we show that Kim's building-up construction of binary self-dual codes is equivalent to Chinburg-Zhang's Hilbert symbol construction. Second we introduce a $q$-ary version of Chinburg-Zhang's construction in order to construct $q$-ary self-dual codes efficiently. For the latter, we study self-dual codes over split finite fields \\(\\F_q\\) with \\(q \\equiv 1 \\pmod{4}\\) through three complementary viewpoints: the building-up construction, the binary arithmetic reduction of Chinburg--Zhang, and the hyperbolic geometry of the Euclidean plane. The condition that \\(-1\\) be a square is the common algebraic input linking these viewpoints: in the binary case it underlies the Lagrangian reduction picture, while in the split \\(q\\)-ary case it produces the isotropic line governing the correction terms in the extension formulas. As an application of our efficient form of generator matrices, we construct optimal self-dual codes from the split boxed construction, including self-dual \\([6,3,4]\\) and \\([8,4,4]\\) codes over \\(\\GF{5}\\), MDS self-dual \\([8,4,5]\\) and \\([10,5,6]\\) codes over \\(\\GF{13}\\), and a self-dual \\([12,6,6]\\) code over \\(\\GF{13}\\). These structural statements are accompanied by a Lean~4 formalization of the algebraic core.","abs_url":"https://arxiv.org/abs/2604.08485","pdf_url":"https://arxiv.org/pdf/2604.08485v1","primary_category":"cs.IT","published":"2026-04-09","tags":["formalization","algebra"],"lean_note":"A Lean 4 formalization of the algebraic core of the self-dual-code constructions.","thumb":"thumbs/2604.08485.jpg"},{"id":"2604.08388","title":"Awakening the Sleeping Agent: Lean-Specific Agentic Data Reactivates General Tool Use in Goedel Prover","authors":["Jui-Hui Chung","Hongzhou Lin","Lai Jiang","Shange Tang","Chi Jin"],"abstract":"Heavy supervised fine-tuning on a target domain can strongly suppress capabilities that were present in the base model. We study this phenomenon in formal mathematics using Goedel-Prover-V2, an open-source model heavily trained on 1.8 million formal-math examples. After domain specialization, the model almost completely loses its ability to produce valid tool calls, even when explicitly instructed to use tools, dropping from 89.4% function-calling accuracy in the base model to nearly 0%. We ask whether this agentic collapse is permanent or instead reversible. To answer this question, we fine-tune the specialized model on a small amount of Lean-specific tool-use data. Remarkably, as few as 100 agentic traces are sufficient to restore strong tool-calling behavior. Importantly, this recovery is not the result of reward hacking or benchmark-specific optimization: the recovery data is entirely drawn from the Lean setting, where the model uses natural-language queries to search the Mathlib library for relevant theorems and lemmas, yet the regained capability transfers well beyond that domain. In particular, these same 100 Lean-specific traces improve performance on the Berkeley Function Calling Leaderboard from near zero to 83.8%, approaching the base model's 89.4% despite the mismatch in task distribution and protocol. The recovered capability is also practically useful in-domain. On ProofNet, pass@32 improves from 21.51% to 25.81%. Together, these results show that heavy domain supervised fine-tuning can suppress general tool-use ability without permanently erasing it, and that a small amount of domain-specific agentic data can awaken dormant tool-use capabilities.","abs_url":"https://arxiv.org/abs/2604.08388","pdf_url":"https://arxiv.org/pdf/2604.08388v1","primary_category":"cs.AI","published":"2026-04-09","tags":["theorem-proving-ml"],"lean_note":"Shows 100 Lean-specific agentic traces restore tool use in the Goedel-Prover model.","thumb":"thumbs/2604.08388.jpg"},{"id":"2605.21492","title":"The Attribution Impossibility: No Feature Ranking Is Faithful, Stable, and Complete Under Collinearity","authors":["Drake Caraker","Bryan Arnold","David Rhoads"],"abstract":"No feature ranking can be simultaneously faithful, stable, and complete when features are collinear. For collinear pairs, ranking reduces to a coin flip. We prove this impossibility, quantify it for four model classes, resolve it via ensemble averaging (DASH), and machine-verify it with 305 Lean 4 theorems. We characterize the complete attribution design space: exactly two families of methods exist -- faithful-complete methods (unstable, with rankings that flip up to 50% of the time) and ensemble methods like DASH (stable, reporting ties for symmetric features) -- and no method lies outside this dichotomy. The impossibility is quantitative: the attribution ratio diverges as 1/(1-rho^2) for gradient boosting, is infinite for Lasso, and converges for random forests. DASH (Diversified Aggregation of SHAP) is provably Pareto-optimal among unbiased aggregations, achieving the Cramer-Rao variance bound with a tight ensemble size formula. In a survey of 77 public datasets, 68% exhibit attribution instability. Switching to conditional SHAP does not escape the impossibility when features have equal causal effects. The framework includes practical diagnostics -- a Z-test workflow and single-model screening tool -- and has direct consequences for fairness auditing: SHAP-based proxy discrimination audits are provably unreliable under collinearity. The design space theorem, diagnostics, and impossibility are mechanically verified in Lean 4 (305 theorems from 16 axioms, 0 sorry) -- to our knowledge, the first formally verified impossibility in explainable AI.","abs_url":"https://arxiv.org/abs/2605.21492","pdf_url":"https://arxiv.org/pdf/2605.21492v1","primary_category":"cs.LG","published":"2026-04-08","tags":["verification","ml-theory"],"lean_note":"Machine-verifies an explainable-AI impossibility result with 305 Lean 4 theorems.","repo":"https://github.com/DrakeCaraker/dash-impossibility-lean","thumb":"thumbs/2605.21492.jpg"},{"id":"2604.07626","title":"Token-Sensitive Enclosure Semantics for Measurement-Bearing Expressions","authors":["David B. Hulak","Arthur F. Ramos","Ruy J. G. B. de Queiroz"],"abstract":"Token identity is semantic information for measurement-bearing expressions. Intervals, dimension tags, and token-erased syntax can say what values a measured leaf may take, but they cannot say whether two occurrences name the same observation or two fresh observations. We give a small formal semantics in which each measured leaf carries an interval of possible exact values and an opaque observation-event token. Here \"token\" means an identity for a measurement event, not a lexical token of the source syntax. The denotation of an expression is its warranted enclosure: the set of exact values still justified by hidden-value environments that assign one value to each observation token and respect the declared intervals. Over this semantics, e -> e' is a claim-tightening judgment, equivalently enclosure containment Encl(e') subseteq Encl(e), while interchangeability is equality of enclosures. The distinction is visible in cancellation, background subtraction, and self-division: reusing one token gives interchangeability with the expected simplified expression, while using distinct tokens gives only one-way containment. We prove that provenance-blind summaries of the kind studied here, preserving intervals, dimension tags, and token-erased syntax, are insufficient to recover the correct rewrite class. The formal results are mechanized in Lean 4 with no sorry or admit placeholders.","abs_url":"https://arxiv.org/abs/2604.07626","pdf_url":"https://arxiv.org/pdf/2604.07626v2","primary_category":"cs.LO","published":"2026-04-08","tags":["formalization","type-theory"],"lean_note":"Mechanizes token-sensitive enclosure semantics in Lean 4 (no sorry/admit).","repo":"https://github.com/d0d1/measurement-provenance-semantics","thumb":"thumbs/2604.07626.jpg"},{"id":"2604.07349","title":"Descent Before Hardness: Orbit-Gap Obstructions in Exact Certification","authors":["Tristan Simas"],"abstract":"Exact certification has a quotient: states are equivalent when they have the same correct outputs. A tractability proxy must first define a predicate on this quotient before ordinary hardness or algorithmic questions arise. Raw syntactic proxies can fail at that earlier step, because correctness-preserving presentation moves may change the statistics they inspect while preserving the exact-certification problem. Orbit gaps are the complete obstruction. An orbit gap occurs when one closure orbit contains both positive and negative presentations of a target. Exact closure-invariant classification is possible if and only if the positive and negative orbit hulls are disjoint. When the hulls are disjoint, the closure hull is the least exact classifier. With computable orbit representatives, this hull classifier becomes a quotient-level algorithm. These are predicate-level results: they establish when a proxy defines a property of the certification problem at all, a precondition logically prior to class lower bounds on the resulting recovery task and deliberately not a substitute for them. The structural transfer applies to every fixed correctness relation, independent of whether that relation is polynomial-time accessible. In the direct finite-local regime, where local routing tests are computed from raw pairwise syntax, three binary-pairwise proxy families and one offset-normalization witness exhibit same-orbit disagreement. Positive results arise from quotient-preserving normalizations, computable orbit catalogues whose descended predicates compose under Boolean operations, and predicates defined directly on the correctness quotient. The result complements the Rice-analog line of Borchert, Stephan, Hemaspaandra, and Rothe. All numbered results are mechanized in Lean 4; the supplementary ledger maps each claim to its formal identifier.","abs_url":"https://arxiv.org/abs/2604.07349","pdf_url":"https://arxiv.org/pdf/2604.07349v10","primary_category":"cs.CC","published":"2026-04-08","tags":["formalization","logic"],"lean_note":"All numbered results mechanized in Lean 4, with a ledger mapping claims to formal identifiers.","repo":"https://doi.org/10.5281/zenodo.19457896","thumb":"thumbs/2604.07349.jpg"},{"id":"2604.06436","title":"The Defense Trilemma: Why Prompt Injection Defense Wrappers Fail?","authors":["Manish Bhatt","Sarthak Munshi","Vineeth Sai Narajala","Idan Habler","Ammar Al-Kahfah","Ken Huang","Joel Webb","Blake Gatto","Md Tamjidul Hoque"],"abstract":"We prove that no continuous, utility-preserving wrapper defense-a function $D: X\\to X$ that preprocesses inputs before the model sees them-can make all outputs strictly safe for a language model with connected prompt space, and we characterize exactly where every such defense must fail. We establish three results under successively stronger hypotheses: boundary fixation-the defense must leave some threshold-level inputs unchanged; an $ε$-robust constraint-under Lipschitz regularity, a positive-measure band around fixed boundary points remains near-threshold; and a persistent unsafe region under a transversality condition, a positive-measure subset of inputs remains strictly unsafe. These constitute a defense trilemma: continuity, utility preservation, and completeness cannot coexist. We prove parallel discrete results requiring no topology, and extend to multi-turn interactions, stochastic defenses, and capacity-parity settings. The results do not preclude training-time alignment, architectural changes, or defenses that sacrifice utility. The full theory is mechanically verified in Lean 4 and validated empirically on three LLMs.","abs_url":"https://arxiv.org/abs/2604.06436","pdf_url":"https://arxiv.org/pdf/2604.06436v3","primary_category":"cs.CR","published":"2026-04-07","tags":["verification","ai-agents"],"lean_note":"The prompt-injection defense trilemma is mechanically verified in Lean 4.","repo":"https://github.com/mbhatt1/stuff","thumb":"thumbs/2604.06436.jpg"},{"id":"2604.06401","title":"ProofSketcher: Hybrid LLM + Lightweight Proof Checker for Reliable Math/Logic Reasoning","authors":["Kranthi Kommuru","Kunal Khanvilkar","Gaurav Parekh"],"abstract":"The large language models (LLMs) might produce a persuasive argument within mathematical and logical fields, although such argument often includes some minor missteps, including the entire omission of side conditions, invalid inference patterns, or appeals to a lemma that cannot be derived logically out of the context being discussed. These omissions are infamously hard to notice solely out of the text, as even the misconstrued construction still may seem mostly accurate. Conversely, interactive theorem provers like Lean and Coq have rigorous reliability by ensuring that syntactic and semantic statements only accept statements that can pass all the syntactic and semantic steps in the program which is a small trusted kernel of the language type-checks with. Despite the fact that this technique provides strong guarantees, it comes at quite a heavy price: the evidence must be completely formalized, and the evidence user or a auxiliary search program must provide an avalanche of low-level information. This paper presents a hybrid pipeline where an LLM generates a typed proof sketch in a compact DSL and a lightweight trusted kernel expands the sketch into explicit proof obligations.","abs_url":"https://arxiv.org/abs/2604.06401","pdf_url":"https://arxiv.org/pdf/2604.06401v1","primary_category":"cs.AI","published":"2026-04-07","tags":["theorem-proving-ml","benchmark","ai-agents"],"lean_note":"LLM-generated typed proof sketches are expanded by a trusted kernel and counted solved only when a Lean kernel accepts a replayable proof on miniF2F, LeanDojo, and ProofNet.","thumb":"thumbs/2604.06401.jpg"},{"id":"2604.05984","title":"Formalization of De Giorgi--Nash--Moser Theory in Lean","authors":["Scott Armstrong","Julia Kempe"],"abstract":"We present a formalization in Lean of the core interior De Giorgi--Nash--Moser theory for uniformly elliptic divergence-form equations with bounded measurable coefficients. The formalized results include local boundedness of weak subsolutions, the weak Harnack inequality for positive weak supersolutions, Moser's Harnack inequality for positive weak solutions, and interior Hölder regularity. This is, to our knowledge, the first machine-checked formalization of a major theorem in modern PDE theory. The development also required substantial new infrastructure for Sobolev spaces on bounded domains, weak solutions of elliptic equations, and quantitative regularity estimates. More broadly, it suggests that large-scale autoformalization of hard analysis in Lean is now within reach.","abs_url":"https://arxiv.org/abs/2604.05984","pdf_url":"https://arxiv.org/pdf/2604.05984v1","primary_category":"math.AP","published":"2026-04-07","tags":["formalization","analysis"],"lean_note":"Formalizes the interior De Giorgi-Nash-Moser theory for uniformly elliptic divergence-form equations, building new Sobolev-space and weak-solution infrastructure in Lean.","repo":"https://github.com/scottnarmstrong/DeGiorgi","thumb":"thumbs/2604.05984.jpg"},{"id":"2604.05238","title":"A Prime-Generated Formalization of Nagata's Factoriality Theorem in Lean 4","authors":["Arthur F. Ramos","Ruy J. G. B. de Queiroz","Anjolina G. de Oliveira"],"abstract":"We present a Lean 4 Mathlib formalization of Nagata's factoriality theorem: if R is a noetherian domain and S <= R is a prime-generated submonoid such that S^{-1}R is a UFD, then R itself is a UFD. The prime-generated hypothesis -- every element of S is a finite product of primes belonging to S -- replaces a superficially cleaner but degenerate prime-or-unit condition that the formalization effort exposed. The development packages the theorem both for the concrete type Localization S and through abstract IsLocalization formulations. As applications, we formalize two Nagata-based proofs that R[X] is a UFD whenever R is a noetherian UFD: one via Laurent-polynomial localization at powers of X, and one via localization at the constant primes and identification with Frac(R)[X]. Reusing the same package, we also obtain the iterated polynomial corollary R[X][Y]. No public formalization of this result is known to us in Lean, Coq, or Isabelle.","abs_url":"https://arxiv.org/abs/2604.05238","pdf_url":"https://arxiv.org/pdf/2604.05238v1","primary_category":"math.AC","published":"2026-04-06","tags":["formalization","mathlib","algebra"],"lean_note":"Lean 4 + Mathlib formalization of Nagata's factoriality theorem.","repo":"https://github.com/Arthur742Ramos/NagataFactoriality","thumb":"thumbs/2604.05238.jpg"},{"id":"2604.03884","title":"Formalizing CHSH Rigidity in Lean 4","authors":["Tianrun Zhao","Nengkun Yu"],"abstract":"Violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality certifies genuine quantum correlations. In this work, we formalize in Lean 4 the rigidity theorem -- any strategy achieving near-optimal CHSH value must be locally isometric to the canonical qubit strategy. In the course of formalization, we identified a gap in the argument of McKague, Yang, and Scarani (arXiv:1203.2976).","abs_url":"https://arxiv.org/abs/2604.03884","pdf_url":"https://arxiv.org/pdf/2604.03884v1","primary_category":"quant-ph","published":"2026-04-04","tags":["formalization","quantum"],"lean_note":"Formalizes the CHSH rigidity theorem in Lean 4 (and finds a gap in a prior proof).","thumb":"thumbs/2604.03884.jpg"},{"id":"2604.03789","title":"Automated Conjecture Resolution with Formal Verification","authors":["Haocheng Ju","Guoxiong Gao","Jiedong Jiang","Bin Wu","Zeming Sun","Shurui Liu","Leheng Chen","Yutong Wang","Yuefeng Wang","Zichen Wang","Wanyi He","Peihao Wu","Liang Xiao","Ruochuan Liu","Bryan Dai","Bin Dong"],"abstract":"Recent advances in large language models have significantly improved their ability to perform mathematical reasoning, extending from elementary problem solving to increasingly capable performance on research-level problems. However, reliably solving and verifying such problems remains challenging due to the inherent ambiguity of natural language reasoning. In this paper, we propose an automated framework that integrates natural language reasoning with formal verification to tackle research-level mathematical problems. Our framework consists of two components: an informal reasoning agent, Rethlas, and a formal verification agent, Archon. Rethlas combines reasoning primitives with our theorem search engine, Matlas, to explore solution strategies and construct candidate proofs. Archon, equipped with LeanSearch, translates informal arguments into formalized Lean 4 projects through task decomposition, iterative refinement, and automated proof synthesis, ensuring machine-checkable correctness. Using this framework, we resolve an open problem in commutative algebra and formally verify the resulting proof in Lean 4 with essentially no human involvement. Additional case studies illustrate the capabilities of Rethlas in informal mathematical reasoning and discovery, as well as the ability of Archon to formalize research-level proofs in Lean 4. Our experiments demonstrate that strong theorem retrieval tools enable the discovery and application of cross-domain mathematical techniques, while the formal agent can autonomously fill nontrivial gaps in informal arguments. More broadly, our work illustrates a promising paradigm for mathematical research in which informal and formal reasoning systems, equipped with theorem retrieval tools, operate in tandem to produce verifiable results, reduce human effort, and support human-AI collaborative mathematical research.","abs_url":"https://arxiv.org/abs/2604.03789","pdf_url":"https://arxiv.org/pdf/2604.03789v2","primary_category":"cs.LG","published":"2026-04-04","tags":["theorem-proving-ml","ai-agents"],"lean_note":"Archon agent formalizes research-level proofs into Lean 4 projects via LeanSearch.","repo":"https://github.com/frenzymath/Anderson-Conjecture","thumb":"thumbs/2604.03789.jpg"},{"id":"2604.03539","title":"CB-VER: A Stable Foundation for Modular Control Plane Verification","authors":["Dexin Zhang","Timothy Alberdingk Thijm","David Walker","Aarti Gupta"],"abstract":"Network operators are often interested in verifying \\emph{eventually-stable properties} of network control planes: properties of control plane states that hold eventually, and hold forever thereafter, provided the operating environment remains unchanged. Examples include eventually-stable reachability, access control, or path length properties. In this work, we introduce \\textsc{CB-Ver}, a new framework for verifying such properties, based on the key idea of a \\emph{converges-before graph} (CB-graph for short). When a user provides interfaces for each network component, \\textsc{CB-Ver} checks the necessary component-by-component requirements in parallel using an SMT solver. In addition, the tool automatically synthesizes a CB-graph and checks whether it connects all nodes in a network -- if it does, the interfaces are valid and users can check whether additional eventually-stable properties are implied. Moreover, the CB-graph can then be used to determine fault tolerance properties of the network. We formalize our verification algorithm in the Lean theorem proving environment and prove its soundness. We evaluate the performance of \\textsc{CB-Ver} on a range of benchmarks that demonstrate its ability to verify expressive properties in reasonable time. Finally, we demonstrate it is possible to automatically generate suitable interfaces by turning the problem around: Given a CB-graph, we use an off-the-shelf Constrained Horn Clause (CHC) solver to synthesize interfaces for every network component that together ensure the given correctness property.","abs_url":"https://arxiv.org/abs/2604.03539","pdf_url":"https://arxiv.org/pdf/2604.03539v2","primary_category":"cs.NI","published":"2026-04-04","tags":["verification","formalization"],"lean_note":"The converges-before-graph verification algorithm for eventually-stable control-plane properties is formalized in Lean and proved sound.","repo":"https://github.com/dz7903/cbgraphs-benchmarks","thumb":"thumbs/2604.03539.jpg"},{"id":"2604.03071","title":"Automatic Textbook Formalization","authors":["Fabian Gloeckle","Ahmad Rammal","Charles Arnal","Remi Munos","Vivien Cabannes","Gabriel Synnaeve","Amaury Hayat"],"abstract":"We present a case study where an automatic AI system formalizes a textbook with more than 500 pages of graduate-level algebraic combinatorics to Lean. The resulting formalization represents a new milestone in textbook formalization scale and proficiency, moving from early results in undergraduate topology and restructuring of existing library content to a full standalone formalization of a graduate textbook. The formalization comprises 130K lines of code and 5900 Lean declarations and was conducted within one week by a total of 30K Claude 4.5 Opus agents collaborating in parallel on a shared code base via version control, simultaneously setting a record in multi-agent software engineering with usable results. The inference cost matches or undercuts what we estimate as the salaries required for a team of human experts, and we expect there is still the potential for large efficiencies to be made without the need for better models. We make our code, the resulting Lean code base and a side-by-side blueprint website available open-source.","abs_url":"https://arxiv.org/abs/2604.03071","pdf_url":"https://arxiv.org/pdf/2604.03071v1","primary_category":"cs.AI","published":"2026-04-03","tags":["autoformalization","formalization","ai-agents","combinatorics"],"lean_note":"About 30K parallel Claude agents autonomously formalized a 500-page algebraic combinatorics textbook into Lean, producing 130K lines and 5900 declarations.","thumb":"thumbs/2604.03071.jpg"},{"id":"2604.02598","title":"Explorable Theorems: Making Written Theorems Explorable by Grounding Them in Formal Representations","authors":["Hita Kambhamettu","Will Crichton","Sean Welleck","Harrison Goldstein","Andrew Head"],"abstract":"LLM-generated explanations can make technical content more accessible, but there is a ceiling on what they can support interactively. Because LLM outputs are static text, they cannot be executed or stepped through. We argue that grounding explanations in a formalized representation enables interactive affordances beyond what static text supports. We instantiate this idea for mathematical proof comprehension with explorable theorems, a system that uses LLMs to translate a theorem and its written proof into Lean, a programming language for machine-checked proofs, and links the written proof with the Lean code. Readers can work through the proof at a step-level granularity, test custom examples or counterexamples, and trace the logical dependencies bridging each step. Each worked-out step is produced by executing the Lean proof on that example and extracting its intermediate state. A user study ($n = 16$) shows potential advantages of this approach: in a proof-reading task, participants who had access to the provided explorability features gave better, more correct, and more detailed answers to comprehension questions, demonstrating a stronger overall understanding of the underlying mathematics.","abs_url":"https://arxiv.org/abs/2604.02598","pdf_url":"https://arxiv.org/pdf/2604.02598v2","primary_category":"cs.HC","published":"2026-04-03","tags":["autoformalization","tooling","theorem-proving-ml"],"lean_note":"Uses LLMs to translate a written theorem and proof into Lean, then executes the Lean proof on examples to extract intermediate states for interactive exploration.","thumb":"thumbs/2604.02598.jpg"},{"id":"2604.01571","title":"Bipartite Exact Matching in P","authors":["Yuefeng Du"],"abstract":"The Exact Matching problem asks whether a bipartite graph with edges colored red and blue admits a perfect matching with exactly $t$ red edges. Introduced by Papadimitriou and Yannakakis in 1982, the problem has resisted deterministic polynomial-time algorithms for over four decades, despite admitting a randomized solution via the Schwartz-Zippel lemma since 1987. We establish the Affine-Slice Nonvanishing Theorem (ASNC) for all bipartite braces: a Vandermonde-weighted determinant polynomial is nonzero whenever the exact-$t$ fiber is nonempty. This yields a deterministic $O(n^6)$ algorithm for Exact Matching on all bipartite graphs via the tight-cut decomposition into brace blocks. The proof proceeds by structural induction on McCuaig's brace decomposition. We handle the McCuaig exceptional families via a parity-resolved cylindric-network positivity argument, the replacement determinant algebra, and the narrow-extension cases (KA, $J3 \\to D1$). For the superfluous-edge step, we introduce two closure tools: a matching-induced Two-extra Hall theorem that resolves the rank-$(m-2)$ branch via projective-collapse contradiction, and a distinguished-state $q$-circuit lemma that eliminates the rank-$(m-1)$ branch entirely by showing that any minimal dependent set containing the superfluous state forces rank $m-2$. A Lean 4 formalization accompanies the paper. The formalization reduces the main theorem to eight explicit hypotheses corresponding to results proved here and in McCuaig (2001), with all algebraic tools, the induction skeleton, and the combinatorial infrastructure fully machine-checked.","abs_url":"https://arxiv.org/abs/2604.01571","pdf_url":"https://arxiv.org/pdf/2604.01571v3","primary_category":"cs.DM","published":"2026-04-02","tags":["formalization","combinatorics"],"lean_note":"A Lean 4 formalization reduces the main matching theorem to eight machine-checked hypotheses.","thumb":"thumbs/2604.01571.jpg"},{"id":"2604.01483","title":"Type-Checked Compliance: Deterministic Guardrails for Agentic Financial Systems Using Lean 4 Theorem Proving","authors":["Devakh Rashie","Veda Rashi"],"abstract":"The rapid evolution of autonomous, agentic artificial intelligence within financial services has introduced an existential architectural crisis: large language models (LLMs) are probabilistic, non-deterministic systems operating in domains that demand absolute, mathematically verifiable compliance guarantees. Existing guardrail solutions -- including NVIDIA NeMo Guardrails and Guardrails AI -- rely on probabilistic classifiers and syntactic validators that are fundamentally inadequate for enforcing complex multi-variable regulatory constraints mandated by the SEC, FINRA, and OCC. This paper presents the Lean-Agent Protocol, a formal-verification-based AI guardrail platform that leverages the Aristotle neural-symbolic model developed by Harmonic AI to auto-formalize institutional policies into Lean 4 code. Every proposed agentic action is treated as a mathematical conjecture: execution is permitted if and only if the Lean 4 kernel proves that the action satisfies pre-compiled regulatory axioms. This architecture provides cryptographic-level compliance certainty at microsecond latency, directly satisfying SEC Rule 15c3-5, OCC Bulletin 2011-12, FINRA Rule 3110, and CFPB explainability mandates. A three-phase implementation roadmap from shadow verification through enterprise-scale deployment is provided.","abs_url":"https://arxiv.org/abs/2604.01483","pdf_url":"https://arxiv.org/pdf/2604.01483v1","primary_category":"cs.LO","published":"2026-04-01","tags":["verification","ai-agents"],"lean_note":"Auto-formalizes regulatory policy into Lean 4 so the kernel gates each agent action.","repo":"https://github.com/arkanemystic/lean-agent-protocol","thumb":"thumbs/2604.01483.jpg"},{"id":"2604.01443","title":"All Substitution Is Local","authors":["Nidhish Shah","Shaurjya Mandal","Asfandyar Azhar"],"abstract":"When does consulting one information source raise the value of another, and when does it diminish it? We study this question for Bayesian decision-makers facing finite actions. The interaction decomposes into two opposing forces: a complement force, measuring how one source moves beliefs to where the other becomes more useful, and a substitute force, measuring how much the current decision is resolved. Their balance obeys a localization principle: substitution requires an observation to cross a decision boundary, though crossing alone does not guarantee it. Whenever posteriors remain inside the current decision region, the substitute force vanishes, and sources are guaranteed to complement each other, even when one source cannot, on its own, change the decision. The results hold for arbitrarily correlated sources and are formalized in Lean 4. Substitution is confined to the thin boundaries where decisions change. Everywhere else, information cooperates. Code and proofs: https://github.com/nidhishs/all-substitution-is-local.","abs_url":"https://arxiv.org/abs/2604.01443","pdf_url":"https://arxiv.org/pdf/2604.01443v1","primary_category":"econ.TH","published":"2026-04-01","tags":["formalization","economics"],"lean_note":"The information-substitution results are formalized in Lean 4.","repo":"https://github.com/nidhishs/all-substitution-is-local","thumb":"thumbs/2604.01443.jpg"},{"id":"2604.01334","title":"The Non-Linearity Perturbation Threshold: Width Scaling and Landscape Bifurcations in Deep Learning","authors":["Michael Alexander"],"abstract":"We study how the optimization landscape of a neural network deforms as a non-linear activation is introduced through a smooth homotopy. Working first in an abstract local setting - a smooth one-parameter family of objective functions together with a critical branch that loses non-degeneracy through a simple Hessian kernel - we show via Lyapunov-Schmidt reduction that the local transition is controlled by the classical codimension-one normal forms (transcritical or pitchfork) and that the associated topology change is governed by Morse-theoretic handle attachment. We then move beyond the abstract framework and verify these assumptions for a concrete two-layer architecture. We prove that bilinear overparameterization creates an (m-1)d-dimensional Hessian kernel at the linear endpoint, which Tikhonov regularization lifts to a floor alpha > 0; the activation homotopy softens this floor, yielding an explicit bifurcation point lambda* approximately equal to alpha/|lambda_1'(0)|. We derive the eigenvalue-softening rate as a functional of activation derivatives and data moments, and prove that the near-pitchfork normal form (|g_aa/g_aaa| much less than 1) is a structural consequence of sigma''(0)=0 for tanh-like activations. The bifurcation point scales as lambda* proportional to alpha m with network width, connecting the framework to the NTK regime: at large m the landscape reorganization is pushed past lambda=1 and the linearized picture prevails. The foundational algebraic theorems have been formally verified in the Lean 4 theorem prover, and theoretical predictions computed for widths m in {3, 5, 10, 20, 50, 100} exhibit quantitative agreement with the abstract framework.","abs_url":"https://arxiv.org/abs/2604.01334","pdf_url":"https://arxiv.org/pdf/2604.01334v1","primary_category":"math.OC","published":"2026-04-01","tags":["formalization","ml-theory"],"lean_note":"The foundational algebraic theorems are formally verified in the Lean 4 theorem prover.","repo":"https://github.com/pacedproton/deeplearning-symmetry-breaking","thumb":"thumbs/2604.01334.jpg"},{"id":"2603.29970","title":"ABC implies that Ramanujan's tau function misses almost all primes","authors":["David Kurniadi Angdinata","Evan Chen","Chris Cummins","Ben Eltschig","Dejan Grubisic","Leopold Haller","Letong Hong","Andranik Kurghinyan","Kenny Lau","Hugh Leather","Seewoo Lee","Simon Mahns","Aram H. Markosyan","Rithikesh Muddana","Ken Ono","Manooshree Patel","Gaurang Pendharkar","Vedant Rathi","Alex Schneidman","Volker Seeker","Shubho Sengupta","Ishan Sinha","Jimmy Xin","Jujian Zhang"],"abstract":"Lehmer conjectured that Ramanujan's tau-function never vanishes. In a related direction, a folklore conjecture asserts that infinitely many primes arise as absolute values of Ramanujan's tau-function. Recently, Xiong showed that these prime values form a subset of the primes with density at most $2/11$. Assuming the $abc$ Conjecture, we prove the stronger upper bound \\[ S(X):=\\#\\{\\ell\\le X:\\ \\ell\\ \\text{prime and } |τ(n)|=\\ell \\text{ for some } n\\ge 1\\} = O(X^{13/22}), \\] which implies that Ramanujan's tau-function misses a density 1 subset of the primes. We give a heuristic suggesting that $S(X)$ should nevertheless be infinite, with predicted order of magnitude \\[ S(X)\\asymp \\frac{C X^{\\frac{1}{11}}}{(\\log X)^2}. \\] The main engine in this note was formalized and produced automatically in Lean/Mathlib by AxiomProver from a natural-language statement of the problem.","abs_url":"https://arxiv.org/abs/2603.29970","pdf_url":"https://arxiv.org/pdf/2603.29970v3","primary_category":"math.NT","published":"2026-03-31","tags":["autoformalization","number-theory"],"lean_note":"The main engine was formalized automatically in Lean/Mathlib by AxiomProver from a natural-language statement.","repo":"https://github.com/AxiomMath/ramanujan-tau-misses-primes","thumb":"thumbs/2603.29970.jpg"},{"id":"2603.29088","title":"WybeCoder: Verified Imperative Code Generation","authors":["Fabian Gloeckle","Mantas Baksys","Darius Feher","Kunhao Zheng","Amaury Hayat","Sean B. Holden","Gabriel Synnaeve","Peter O'Hearn"],"abstract":"Recent progress in large language models (LLMs) has substantially advanced automatic code generation and formal theorem proving, yet software verification has not seen comparable gains. To address this gap, we propose WybeCoder, an agentic code verification framework that enables prove-as-you-generate development, in which code, invariants, and proofs co-evolve. WybeCoder builds on a recent framework that combines automatic verification condition generation and SMT solving with interactive proofs in Lean. To enable systematic evaluation, we translate two benchmarks for functional verification in Lean, Verina and Clever, into equivalent imperative code specifications. On complex algorithms such as Heapsort, we observe consistent performance improvements as we scale our approach, synthesizing dozens of valid invariants and dispatching dozens of subgoals, ultimately producing hundreds of lines of verified code and overcoming plateaus reported in previous work. Our best system solves 74% of Verina tasks and 62% of Clever tasks at moderate compute budgets, substantially surpassing previous evaluations and paving the way for the automated construction of large-scale datasets of verified imperative code.","abs_url":"https://arxiv.org/abs/2603.29088","pdf_url":"https://arxiv.org/pdf/2603.29088v2","primary_category":"cs.SE","published":"2026-03-31","tags":["verification","theorem-proving-ml","ai-agents","benchmark"],"lean_note":"Builds on a framework combining VC generation, SMT solving, and interactive Lean proofs to co-evolve verified imperative code, invariants, and proofs.","thumb":"thumbs/2603.29088.jpg"},{"id":"2603.28636","title":"Optimal bounds for an Erdős problem on matching integers to distinct multiples","authors":["Wouter van Doorn","Yanyang Li","Quanyu Tang"],"abstract":"Let $f(m)$ be the largest integer such that for every set $A = \\{a_1 < \\cdots < a_m\\}$ of $m$ positive integers and every open interval $I$ of length $2a_m$, there exist at least $f(m)$ disjoint pairs $(a, b)$ with $a \\in A$ dividing $b \\in I$. Solving a problem of Erdős, we determine $f(m)$ exactly, and show $$ f(m)=\\min\\bigl(m,\\lceil 2\\sqrt{m}\\,\\rceil\\bigr) $$ for all $m$. The proof was obtained through an AI-assisted workflow: the proof strategy was first proposed by ChatGPT, and the detailed argument was subsequently made fully rigorous and formally verified in Lean by Aristotle. The exposition and final proofs presented here are entirely human-written. [This paper solves Problem #650 on Bloom's website \"Erdős problems\".]","abs_url":"https://arxiv.org/abs/2603.28636","pdf_url":"https://arxiv.org/pdf/2603.28636v1","primary_category":"math.CO","published":"2026-03-30","tags":["formalization","number-theory","combinatorics"],"lean_note":"The argument resolving an Erdős divisibility problem was made fully rigorous and formally verified in Lean by the Aristotle system.","repo":"https://github.com/QuanyuTang/erdos-problem-650","thumb":"thumbs/2603.28636.jpg"},{"id":"2603.28406","title":"Physics as Code: From Scans to Theorems with ITP APIs in $SU(5)$ Model Building","authors":["Sven Krippendorf","Joseph Tooby-Smith"],"abstract":"A recurring challenge in theoretical physics is to make reliable global statements about bounded but combinatorially large model spaces. Exhaustive scans quickly become opaque or impractical, while statistical exploration does not by itself provide theorem-backed guarantees. This motivates workflows in which the model-building problem itself is formalized inside an interactive theorem prover (ITP). In this paper we develop an API-based methodology for formalizing such bounded model-building questions inside Lean, an interactive theorem prover. The central step is to represent the relevant charge spectra, predicates, and reduction moves as reusable ITP definitions, and then to derive the classification from proved reduction theorems rather than from an ad hoc scan. We demonstrate the strategy in a concrete $SU(5)$ case study motivated by F-theory model building with additional Abelian symmetries. At the charge-spectrum layer, we classify bounded spectra that admit a top-quark Yukawa coupling, avoid a selected set of dangerous operators, and satisfy a minimal charge-spectrum completeness condition. Our main result shows that every such spectrum in the bounded search space arises from finitely many minimal top-Yukawa witnesses together with controlled completions and certified closure steps. This classification represents a formally verified description of the full viable class in the charge-spectrum setting studied here. The development is implemented inside PhysLib as reusable infrastructure rather than as a one-off verification script. It provides a proof of principle for how interactive theorem provers can turn combinatorially difficult model-building problems into correctness-first, reusable workflows, and we discuss how the resulting certified classification can serve as reliable input for downstream analyses.","abs_url":"https://arxiv.org/abs/2603.28406","pdf_url":"https://arxiv.org/pdf/2603.28406v1","primary_category":"hep-th","published":"2026-03-30","tags":["formalization","verification","physics"],"lean_note":"Formalizes bounded SU(5) charge-spectrum model-building classification inside Lean as reusable PhysLib definitions and proved reduction theorems.","thumb":"thumbs/2603.28406.jpg"},{"id":"2603.27220","title":"Cohesion-Sensitive Power Indices: Representation Results for Banzhaf and Shapley Values","authors":["Thomas Pitz","Vinicius Ferraz"],"abstract":"In many applications of cooperative game theory -- from corporate governance and cartel formation to parliamentary voting -- not all winning coalitions are feasible. Ideological distances, institutional constraints, or pre-electoral agreements may render certain coalitions implausible. Classical power indices ignore this and weight all winning coalitions equally. We introduce cohesion structures to quantify coalition feasibility and axiomatically characterize two families of cohesion-sensitive power indices, represented as expected marginal contributions under Luce-type distributions. In the Banzhaf branch, coalition weights are a power transformation of cohesion; in the Shapley branch, additional axioms separate size from cohesion, recovering the classical size weights with cohesion acting within each size class. All results have been mechanically verified in Lean 4 with Mathlib. We illustrate the framework on the German Bundestag and the French Assemblée Nationale, where cordon sanitaire and double cordon scenarios produce sharp, interpretable power shifts.","abs_url":"https://arxiv.org/abs/2603.27220","pdf_url":"https://arxiv.org/pdf/2603.27220v1","primary_category":"econ.TH","published":"2026-03-28","tags":["formalization","economics"],"lean_note":"Axiomatic characterizations of cohesion-sensitive power indices are mechanically verified in Lean 4 with Mathlib.","repo":"https://github.com/vferraz/cohesion-power-indices","thumb":"thumbs/2603.27220.jpg"},{"id":"2603.27202","title":"Sal: Multi-modal Verification of Replicated Data Types","authors":["Pranav Ramesh","Vimala Soundarapandian","KC Sivaramakrishnan"],"abstract":"Designing correct replicated data types (RDTs) is challenging because replicas evolve independently and must be merged while preserving application intent. A promising approach is correct-by-construction development in a proof-oriented programming language such as F*, Dafny and Lean, where desired correctness guarantees are specified and checked as the RDTs are implemented. Recent work Neem proposes the use of replication-aware linearizability (RA linearizability) as the correctness condition for state-based CRDTs and mergeable replicated data types (MRDTs), with automation in the SMT-aided, proof-oriented programming language F*. However, SMT-centric workflows can be opaque when automation fails to discharge a verification condition (VC), and they enlarge the trusted computing base (TCB). We present Sal, a multi-modal workflow to design and verify state-based CRDTs and MRDTs in Lean. Sal combines (i) kernel-checkable automation with proof reconstruction, (ii) SMT-aided automation when needed, and (iii) AI-assisted interactive theorem proving for remaining proof obligations. When a verification condition is shown to be invalid, we leverage Lean's property-based testing to automatically generate and visualize counterexamples, helping developers debug incorrect specifications or implementations. We report on our experience verifying a suite of 13 CRDTs and MRDTs with Sal: 69% of verification conditions are discharged by kernel-verified automation without SMT, and counterexamples automatically expose subtle bugs such as the well-known enable-wins flag anomaly. The codebase for Sal is open-sourced, and is available at \\href{https://github.com/fplaunchpad/sal}{https://github.com/fplaunchpad/sal}","abs_url":"https://arxiv.org/abs/2603.27202","pdf_url":"https://arxiv.org/pdf/2603.27202v1","primary_category":"cs.PL","published":"2026-03-28","tags":["verification","tactic-automation","ai-agents"],"lean_note":"Designs and verifies state-based CRDTs and MRDTs in Lean using staged kernel-checked tactics, SMT fallback, AI-assisted proving, and Plausible counterexample generation.","repo":"https://github.com/fplaunchpad/sal","thumb":"thumbs/2603.27202.jpg"},{"id":"2603.27007","title":"Pairwise Independence of Representation, Classification, and Composition in Finite Extensional Magmas","authors":["Stefano Palmieri"],"abstract":"Nontrivial combinatory algebras with S and K must be infinite. Associativity is incompatible with combining a classifier and a retraction pair in a finite extensional magma. These obstructions exclude several standard settings from the finite extensional framework studied here, most notably nontrivial finite S+K-style combinatory algebras and associative structures (semigroups, monoids, groups, rings) carrying both a classifier and a retraction pair. What algebraic structure exists in the remaining landscape: finite, non-associative, total? We identify three properties of finite extensional 2-pointed magmas: self-representation (R), the classifier dichotomy (D), and the Internal Composition Property (H). We prove they are pairwise independent. Lean-verified finite counterexamples at sizes 4 through 10 establish all six non-implications, four with provably tight bounds. The minimum coexistence witness has N = 5, which is optimal: ICP requires 3 pairwise distinct core elements, so N \\ge 5. The three-category decomposition induced by D is an isomorphism invariant, and the ICP is logically equivalent to the standard Compose+Inert axioms. All results are formalized in Lean 4 with zero sorry.","abs_url":"https://arxiv.org/abs/2603.27007","pdf_url":"https://arxiv.org/pdf/2603.27007v2","primary_category":"cs.LO","published":"2026-03-27","tags":["formalization","logic","algebra"],"lean_note":"Formalizes pairwise-independence results for finite extensional magmas in Lean 4 with verified finite counterexamples, zero sorry.","repo":"https://github.com/stefanopalmieri/finite-magma-independence","thumb":"thumbs/2603.27007.jpg"},{"id":"2603.26996","title":"FormalProofBench: Can Models Write Graduate Level Math Proofs That Are Formally Verified?","authors":["Nikil Ravi","Kexing Ying","Vasilii Nesterov","Rayan Krishnan","Elif Uskuplu","Bingyu Xia","Janitha Aswedige","Langston Nashold"],"abstract":"We present FormalProofBench, a private benchmark designed to evaluate whether AI models can produce formally verified mathematical proofs at the graduate level. Each task pairs a natural-language problem with a Lean~4 formal statement, and a model must output a Lean proof accepted by the Lean 4 checker. FormalProofBench targets advanced undergraduate and graduate mathematics, with problems drawn from qualifying exams and standard textbooks across topics including analysis, algebra, probability, and logic. We evaluate a range of frontier models with an agentic harness, and find that the best-performing foundation model achieves 33.5% accuracy, with performance dropping rapidly after that. In addition to the accuracy numbers, we also provide empirical analysis of tool-use, failure modes, cost and latency, thereby providing a thorough evaluation of the formal-theorem proving abilities of frontier models.","abs_url":"https://arxiv.org/abs/2603.26996","pdf_url":"https://arxiv.org/pdf/2603.26996v1","primary_category":"cs.AI","published":"2026-03-27","tags":["benchmark","theorem-proving-ml"],"lean_note":"FormalProofBench pairs natural-language problems with Lean 4 statements that models must prove for the Lean checker.","repo":"https://github.com/vals-ai/proof-bench","thumb":"thumbs/2603.26996.jpg"},{"id":"2603.26342","title":"Lean on Vampire Proofs (Short Paper)","authors":["Jonas Bodingbauer","Márton Hajdu","Laura Kovács","Axel Polaczek","Michael Rawson"],"abstract":"Vampire proves theorems completely automatically in first- and higher-order logic extended with theories. Proof checking is increasingly demanded to consolidate user trust in Vampires output. We describe ongoing efforts in reconstructing Vampire proofs as trusted proofs in Lean","abs_url":"https://arxiv.org/abs/2603.26342","pdf_url":"https://arxiv.org/pdf/2603.26342v1","primary_category":"cs.LO","published":"2026-03-27","tags":["tooling","verification","logic"],"lean_note":"Reconstructs Vampire's automatic first- and higher-order proofs as trusted, checkable Lean proofs to consolidate user trust.","repo":"https://github.com/vprover/vampire","thumb":"thumbs/2603.26342.jpg"},{"id":"2603.24823","title":"A formalization of the Gelfond-Schneider theorem","authors":["Michail Karatarakis","Freek Wiedijk"],"abstract":"We formalize Hilbert's Seventh Problem and its solution, the Gelfond-Schneider theorem, in the Lean 4 proof assistant. The theorem states that if $α$ and $β$ are algebraic numbers with $α\\neq 0,1$ and $β$ irrational, then $α^β$ is transcendental. Originally proven independently by Gelfond and Schneider in 1934, this result is a cornerstone of transcendental number theory, bridging algebraic number theory and complex analysis.","abs_url":"https://arxiv.org/abs/2603.24823","pdf_url":"https://arxiv.org/pdf/2603.24823v1","primary_category":"cs.LO","published":"2026-03-25","tags":["formalization","number-theory"],"lean_note":"Formalizes Hilbert's Seventh Problem and the Gelfond-Schneider theorem in Lean 4.","thumb":"thumbs/2603.24823.jpg"},{"id":"2603.24708","title":"Hamilton decompositions of the directed 3-torus: a return-map and odometer view","authors":["SangHyun Park"],"abstract":"We prove that the directed 3-torus D_3(m), or equivalently the Cartesian product of three directed m-cycles, admits a decomposition into three arc-disjoint directed Hamilton cycles for every integer m >= 3. The proof reduces Hamiltonicity to the m-step return maps on the layer section S=i+j+k=0. For odd m, five Kempe swaps of the canonical coloring produce return maps that are explicitly affine-conjugate to the standard 2-dimensional odometer. For even m, a sign-product invariant rules out Kempe-from-canonical constructions, and a different low-layer witness reduces after one further first-return map to a finite-defect clock-and-carry system. The remaining closure is a finite splice analysis, and the case m=4 is handled separately by a finite witness. A Lean 4 formalization accompanies the construction.","abs_url":"https://arxiv.org/abs/2603.24708","pdf_url":"https://arxiv.org/pdf/2603.24708v1","primary_category":"math.CO","published":"2026-03-25","tags":["formalization","combinatorics"],"lean_note":"A Lean 4 formalization accompanies the Hamilton decomposition construction for the directed 3-torus.","repo":"https://github.com/aria1th/Torus-Hamilton-Decomposition","thumb":"thumbs/2603.24708.jpg"},{"id":"2603.24465","title":"Mechanic: Sorrifier-Driven Formal Decomposition Workflow for Automated Theorem Proving","authors":["Ruichen Qiu","Yichuan Cao","Junqi Liu","Dakai Guo","Xiao-Shan Gao","Lihong Zhi","Ruyong Feng"],"abstract":"Recent advances in large language models (LLMs) and LLM-based agents have substantially improved the capabilities of automated theorem proving. However, for problems requiring complex mathematical reasoning, current systems rarely succeed on the first try and must repeatedly modify their proof strategies. Existing approaches for handling failed attempts typically either discard the entire proof and regenerate it from scratch or iteratively fix errors within the proof. The former is inefficient, as it may abandon mostly correct reasoning due to localized errors, while the latter, although preserving prior progress, leads to progressively longer contexts which progressively degrades the model's ability to attend to the remaining unresolved subproblems. To address this dilemma, we propose Mechanic, a novel agent system that employs a sorry-driven formal decomposition strategy. By leveraging the sorry placeholder in Lean to precisely isolate unresolved subgoals while preserving the surrounding verified proof structure, Mechanic extracts each failed subproblem into a clean, self-contained context and resolves it independently. This avoids both the waste of full regeneration and the excessive context length induced by repeated repairs. Experimental results on challenging mathematical competition benchmarks, including IMO 2025 and Putnam 2025, demonstrate that our agent achieves significant advantages in proving efficiency.","abs_url":"https://arxiv.org/abs/2603.24465","pdf_url":"https://arxiv.org/pdf/2603.24465v1","primary_category":"cs.CL","published":"2026-03-25","tags":["theorem-proving-ml","ai-agents","tactic-automation"],"lean_note":"Mechanic uses Lean's sorry placeholder to isolate unresolved subgoals while preserving verified proof structure, resolving each failed subproblem in clean context.","repo":"https://github.com/oOo0oOo/lean-lsp-mcp","thumb":"thumbs/2603.24465.jpg"},{"id":"2603.24372","title":"Improving Lean4 Autoformalization via Cycle Consistency Fine-tuning","authors":["Arsen Shebzukhov"],"abstract":"Autoformalization - automatically translating natural language mathematical texts into formal proof language such as Lean4 - can help accelerate AI-assisted mathematical research, be it via proof verification or proof search. I fine-tune Qwen3.5-2B with LoRA for natural language to Lean4 formalization on FineLeanCorpus and consider three training regimes: supervised fine-tuning (SFT) with curriculum learning (difficulty 1 to 10), SFT without curriculum ordering, and reinforcement learning using group relative policy optimization (GRPO) with a cycle consistency reward. Cycle consistency measures how well the meaning of a statement is preserved through a NL to Lean4 to NL' loop, computed as cosine similarity of off-the-shelf sentence embeddings. On an unseen subset of FineLeanCorpus (FLC) and on PutnamBench, RL substantially outperforms both SFT variants (mean cycle consistency 0.669 vs. 0.513 on FLC; 0.561 vs. 0.422 on PutnamBench), while increasing cross-entropy loss by only 0.011 nats, with minimal impact on formalization quality. Curriculum ordering provides no measurable benefit over shuffled training.","abs_url":"https://arxiv.org/abs/2603.24372","pdf_url":"https://arxiv.org/pdf/2603.24372v1","primary_category":"cs.CL","published":"2026-03-25","tags":["autoformalization","theorem-proving-ml"],"lean_note":"Fine-tunes a 2B model for NL-to-Lean4 autoformalization with a GRPO cycle-consistency reward.","thumb":"thumbs/2603.24372.jpg"},{"id":"2603.24330","title":"On the Quartic Invariant of Odd Degree Binary Forms","authors":["Ashvin Swaminathan"],"abstract":"We determine the squarefree part of the scalar factor that arises when the quartic invariant of the generic binary form $F$ of odd degree $2n+1$ is expressed as the discriminant of the unique quadratic covariant $(F,F)_{2n}$. This squarefree part is exactly $p$ when $n+2$ is a power of an odd prime $p$, and $1$ otherwise. Equivalently, for each prime $p$: $v_2(S(n))$ is always even, and for odd $p$, $v_p(S(n))$ is odd if and only if $n+2$ is a power of $p$. This generalizes the classical identity $\\operatorname{disc}(H(F))=-3\\cdot\\operatorname{disc}(F)$ for binary cubics, which dates back to the work of Cayley and Sylvester in the 1850s. The proof, which involves substantial explicit coefficient analysis and $p$-adic deformation arguments, was developed using an AI-assisted research workflow: the author's earlier partial attempts were completed through systematic collaboration with Claude Code (Anthropic) and Codex (OpenAI), and key arithmetic lemmas were formally verified in Lean~4 using Aristotle (Harmonic). We describe this workflow in detail as a case study in AI-assisted mathematical research. We also discuss representation-theoretic, geometric, and arithmetic interpretations of the quadratic covariant.","abs_url":"https://arxiv.org/abs/2603.24330","pdf_url":"https://arxiv.org/pdf/2603.24330v1","primary_category":"math.NT","published":"2026-03-25","tags":["formalization","number-theory"],"lean_note":"Key arithmetic lemmas about the quartic invariant of binary forms were formally verified in Lean 4 using Aristotle.","repo":"https://github.com/ashvin-swaminathan/quartic-invariant","thumb":"thumbs/2603.24330.jpg"},{"id":"2603.23928","title":"On the paucity of lattice triangles","authors":["David Kurniadi Angdinata","Evan Chen","Ken Ono","Jiaxin Zhang","Jujian Zhang"],"abstract":"A rational triangle $T$ (one whose angles are rational multiples of $π$) unfolds to a translation surface $(X_T,ω_T)$. The lattice triangle problem asks to classify those $T$ for which $(X_T,ω_T)$ is a Veech (lattice) surface, which means that the $\\operatorname{SL}_2(\\mathbb R)$-orbit of $(X_T,ω_T)$ is closed in its stratum (so its projection to moduli space is a Teichmüller curve). The most mysterious regime is the \"hard obtuse window\" (largest angle in $(π/2,2π/3]$), where it is conjectured that no lattice triangles exist. Using an arithmetic reformulation of the Mirzakhani-Wright rank obstruction, we prove a quantitative theorem that rules out all but a density 0 subset of the triangles in this window. The main engine in this paper was autoformalized by AxiomProver in Lean (using mathlib).","abs_url":"https://arxiv.org/abs/2603.23928","pdf_url":"https://arxiv.org/pdf/2603.23928v1","primary_category":"math.DS","published":"2026-03-25","tags":["autoformalization","geometry"],"lean_note":"AxiomProver autoformalized the paper's main engine in Lean using mathlib.","repo":"https://github.com/AxiomMath/lattice-triangle","thumb":"thumbs/2603.23928.jpg"},{"id":"2603.23696","title":"Semantics for 2D Rasterization","authors":["Bhargav Kulkarni","Henry Whiting","Pavel Panchekha"],"abstract":"Rasterization is the process of determining the color of every pixel drawn by an application. Powerful rasterization libraries like Skia, CoreGraphics, and Direct2D put exceptional effort into drawing, blending, and rendering efficiently. Yet applications are still hindered by the inefficient sequences of operations that they ask these libraries to perform. Even Google Chrome, a highly optimized program co-developed with the Skia rasterization library, still produces inefficient instruction sequences even on the top 100 most visited websites. The underlying reason for this inefficiency is that rasterization libraries have complex semantics and opaque and non-obvious execution models. To address this issue, we introduce $μ$Skia, a formal semantics for the Skia 2D graphics library, and mechanize this semantics in Lean. $μ$Skia covers language and graphics features like canvas state, the layer stack, blending, and color filters, and the semantics itself is split into three strata to separate concerns and enable extensibility. We then identify four patterns of sub-optimal Skia code produced by Google Chrome, and then write replacements for each pattern. $μ$Skia allows us to verify the replacements are correct, including identifying numerous tricky side conditions. We then develop a high-performance Skia optimizer that applies these patterns to speed up rasterization. On 99 Skia programs gathered from the top 100 websites, this optimizer yields a speedup of 18.7% over Skia's most modern GPU backend, while taking at most 32 $μ$s for optimization. The speedups persist across a variety of websites, Skia backends, and GPUs. To provide true, end-to-end verification, optimization traces produced by the optimizer are loaded back into the $μ$Skia semantics and translation validated in Lean.","abs_url":"https://arxiv.org/abs/2603.23696","pdf_url":"https://arxiv.org/pdf/2603.23696v1","primary_category":"cs.PL","published":"2026-03-24","tags":["verification","formalization"],"lean_note":"Mechanizes muSkia, a formal semantics for the Skia 2D graphics library, in Lean and uses it to verify and translation-validate optimizer rewrites.","thumb":"thumbs/2603.23696.jpg"},{"id":"2603.23364","title":"Canonical Byte-String Encoding for Finite-Ring Cryptosystems","authors":["Kyrylo Riabov","Serhii Kryvyi"],"abstract":"Ring-mapping protocols need a canonical byte-to-residue layer before any algebraic encryption step can begin. This paper isolates that layer and presents the base-m length codec, a canonical map from byte strings of length less than 2^64 to lists of residues modulo m. The encoder builds on and adapts an rANS-based system proposed by Duda. Decoding is exact for all moduli satisfying the paper's parameter bounds. Because the encoding carries the byte length in its fixed-width header, decoding is also tolerant to appended valid suffix digits. The paper is accompanied by a Rust implementation of the described protocol, a Lean 4 formalization of the abstract codec with machine-checked proofs, and performance benchmarks. The Lean 4 formalization establishes fixed-width prefix inversion and payload-state bounds below 2^64, stream-level roundtrip correctness, and that every emitted symbol is a valid residue modulo m. We conclude with a complexity analysis and a discussion of practical considerations arising in real-world use of the codec.","abs_url":"https://arxiv.org/abs/2603.23364","pdf_url":"https://arxiv.org/pdf/2603.23364v2","primary_category":"cs.CR","published":"2026-03-24","tags":["verification","cryptography"],"lean_note":"Provides a Lean 4 formalization of an rANS-based byte-to-residue codec with machine-checked roundtrip proofs.","repo":"https://github.com/KyrylR/phd-symmetric-cryptography","thumb":"thumbs/2603.23364.jpg"},{"id":"2603.23095","title":"Formalizing Pick's Theorem, efficiently","authors":["Michael Eisermann"],"abstract":"Pick's astonishing theorem explains how to obtain the area of any integer polygon by counting lattice points. It is a notoriously difficult challenge to translate the geometric statement and intuitive reasoning into a formal statement and rigorous proof. We transform the beautiful geometry into equally elegant algebra, and then implement the algebraic proof in Lean.","abs_url":"https://arxiv.org/abs/2603.23095","pdf_url":"https://arxiv.org/pdf/2603.23095v1","primary_category":"math.GT","published":"2026-03-24","tags":["formalization","geometry"],"lean_note":"Pick's theorem is reformulated as an algebraic identity over an ordered field and the resulting proof is implemented in Lean.","repo":"https://github.com/Palamedez314/PicksTheorem2025","thumb":"thumbs/2603.23095.jpg"},{"id":"2603.21065","title":"LongCat-Flash-Prover: Advancing Native Formal Reasoning via Agentic Tool-Integrated Reinforcement Learning","authors":["Jianing Wang","Jianfei Zhang","Qi Guo","Linsen Guo","Rumei Li","Chao Zhang","Chong Peng","Cunguang Wang","Dengchang Zhao","Jiarong Shi","Jingang Wang","Liulin Feng","Mengxia Shen","Qi Li","Shengnan An","Shun Wang","Wei Shi","Xiangyu Xi","Xiaoyu Li","Xuezhi Cao","Yi Lu","Yunke Zhao","Zhengyu Chen","Zhimin Lin","Wei Wang","Peng Pei","Xunliang Cai"],"abstract":"We introduce LongCat-Flash-Prover, a flagship 560-billion-parameter open-source Mixture-of- Experts (MoE) model that advances Native Formal Reasoning in Lean4 through agentic tool-integrated reasoning (TIR). We decompose the native formal reasoning task into three independent formal capabilities, i.e., auto-formalization, sketching, and proving. To facilitate these capabilities, we propose a Hybrid-Experts Iteration Framework to expand high-quality task trajectories, including generating a formal statement based on a given informal problem, producing a whole-proof directly from the statement, or a lemma-style sketch. During agentic RL, we present a Hierarchical Importance Sampling Policy Optimization (HisPO) algorithm, which aims to stabilize the MoE model training on such long-horizon tasks. It employs a gradient masking strategy that accounts for the policy staleness and the inherent train-inference engine discrepancies at both sequence and token levels. Additionally, we also incorporate theorem consistency and legality detection mechanisms to eliminate reward hacking issues. Extensive evaluations show that our LongCat-Flash-Prover sets a new state-of-the-art for open-weights models in both auto-formalization and theorem proving. Demonstrating remarkable sample efficiency, it achieves a 97.1% pass rate on MiniF2F-Test using only 72 inference budget per problem. On more challenging benchmarks, it solves 70.8% of ProverBench and 41.5% of PutnamBench with no more than 220 attempts per problem, significantly outperforming existing open-weights baselines.","abs_url":"https://arxiv.org/abs/2603.21065","pdf_url":"https://arxiv.org/pdf/2603.21065v1","primary_category":"cs.AI","published":"2026-03-22","tags":["theorem-proving-ml","autoformalization"],"lean_note":"LongCat-Flash-Prover, a 560B-parameter MoE, performs native formal reasoning in Lean4 via agentic tool-integrated RL.","repo":"https://github.com/meituan-longcat/LongCat-Flash-Prover","thumb":"thumbs/2603.21065.jpg"},{"id":"2603.20991","title":"Structural Sensitivity in Compressed Transformers: Relative Error Propagation and Layer Removal","authors":["Abhinaba Basu","Kumkum Basu","Koushik Deb"],"abstract":"Compressing transformer weights makes large language models cheaper to deploy. But each layer's compression introduces an error. These errors accumulate as the signal passes through later layers, and how they accumulate is not well understood. We measure this directly: at each layer, we take the ratio of output to input error, calling it rho. A value below one means the layer absorbs the error; above one means it grows. Computing rho on six transformers (117M to 8B parameters) yields three findings. (i) Errors at layer t scale downstream by the product of later rho values, predicting representation drift (Spearman r = -0.44, p < 10^-4). This explains why compressing early layers hurts more than late ones, and why depth-decreasing sparsity schedules outperform uniform ones. Across architecture families, however, model width and redundancy matter more than rho alone. (ii) Within a layer, naive pruning shows a ~600x spread in component sensitivity. Activation-aware pruning (Wanda) shrinks this to 3-7x; the ranking reverses across architectures, so fixed importance scores do not transfer. (iii) For depth pruning, ranking layers by how far rho is from one takes two forward passes. It beats ShortGPT's Block Influence with 1.6x lower perplexity at eight layers removed, and physical deletion delivers 1.22x wall-clock speed-up. A blend of the two criteria does best (perplexity 14.2, 60.0% downstream accuracy on LLaMA-2-7B). Twelve Lean 4 norm inequalities provide machine-checked per-matrix error bounds. The contraction profile thus gives a training-free instrument for two decisions: where to compress within layers, and which to remove.","abs_url":"https://arxiv.org/abs/2603.20991","pdf_url":"https://arxiv.org/pdf/2603.20991v2","primary_category":"cs.LG","published":"2026-03-22","tags":["formalization","ml-theory"],"lean_note":"Twelve Lean 4 norm inequalities provide machine-checked per-matrix error bounds for transformer compression.","thumb":"thumbs/2603.20991.jpg"},{"id":"2603.20655","title":"Exponential Family Discriminant Analysis: Generalizing LDA-Style Generative Classification to Non-Gaussian Models","authors":["Anish Lakkapragada"],"abstract":"We introduce Exponential Family Discriminant Analysis (EFDA), a unified generative framework that extends classical Linear Discriminant Analysis (LDA) beyond the Gaussian setting to any member of the exponential family. Under the assumption that each class-conditional density belongs to a common exponential family, EFDA derives closed-form maximum-likelihood estimators for all natural parameters and yields a decision rule that is linear in the sufficient statistic, recovering LDA as a special case and capturing nonlinear decision boundaries in the original feature space. We prove that EFDA is asymptotically calibrated and statistically efficient under correct specification, and we generalise it to $K \\geq 2$ classes and multivariate data. Through extensive simulation across five exponential-family distributions (Weibull, Gamma, Exponential, Poisson, Negative Binomial), EFDA matches the classification accuracy of LDA, QDA, and logistic regression while reducing Expected Calibration Error (ECE) by $2$-$6\\times$, a gap that is structural: it persists for all $n$ and across all class-imbalance levels, because misspecified models remain asymptotically miscalibrated. We further prove and empirically confirm that EFDA's log-odds estimator approaches the Cramér-Rao bound under correct specification, and is the only estimator in our comparison whose mean squared error converges to zero. Complete derivations are provided for nine distributions. Finally, we formally verify all four theoretical propositions in Lean 4, using Aristotle (Harmonic) and OpenGauss (Math, Inc.) as proof generators, with all outputs independently machine-checked by AXLE (Axiom).","abs_url":"https://arxiv.org/abs/2603.20655","pdf_url":"https://arxiv.org/pdf/2603.20655v2","primary_category":"cs.LG","published":"2026-03-21","tags":["formalization","ml-theory","probability"],"lean_note":"Formally verifies all four theoretical propositions of exponential-family discriminant analysis in Lean 4 via multiple proof generators.","repo":"https://github.com/anish-lakkapragada/EFDA","thumb":"thumbs/2603.20655.jpg"},{"id":"2603.20396","title":"Compression is all you need: Modeling Mathematics","authors":["Vitaly Aksenov","Eve Bodnia","Michael H. Freedman","Michael Mulligan"],"abstract":"Human mathematics (HM), the mathematics humans discover and value, is a vanishingly small subset of formal mathematics (FM), the totality of all valid deductions. We argue that HM is distinguished by its compressibility through hierarchically nested definitions, lemmas, and theorems. We model this with monoids. A mathematical deduction is a string of primitive symbols; a definition or theorem is a named substring or macro whose use compresses the string. In the free abelian monoid $A_n$, a logarithmically sparse macro set achieves exponential expansion of expressivity. In the free non-abelian monoid $F_n$, even a polynomially-dense macro set only yields linear expansion; superlinear expansion requires near-maximal density. We test these models against MathLib, a large Lean~4 library of mathematics that we take as a proxy for HM. Each element has a depth (layers of definitional nesting), a wrapped length (tokens in its definition), and an unwrapped length (primitive symbols after fully expanding all references). We find unwrapped length grows exponentially with both depth and wrapped length; wrapped length is approximately constant across all depths. These results are consistent with $A_n$ and inconsistent with $F_n$, supporting the thesis that HM occupies a polynomially-growing subset of the exponentially growing space FM. We discuss how compression, measured on the MathLib dependency graph, and a PageRank-style analysis of that graph can quantify mathematical interest and help direct automated reasoning toward the compressible regions where human mathematics lives.","abs_url":"https://arxiv.org/abs/2603.20396","pdf_url":"https://arxiv.org/pdf/2603.20396v1","primary_category":"cs.AI","published":"2026-03-20","tags":["meta-study","mathlib"],"lean_note":"Analyzes Mathlib's Lean 4 library as a proxy for human mathematics, measuring definitional depth and compression.","repo":"https://github.com/Aksenov239/lean-fun","thumb":"thumbs/2603.20396.jpg"},{"id":"2603.19514","title":"Learning to Disprove: Formal Counterexample Generation with Large Language Models","authors":["Zenan Li","Zhaoyu Li","Kaiyu Yang","Xiaoxing Ma","Zhendong Su"],"abstract":"Mathematical reasoning demands two critical, complementary skills: constructing rigorous proofs for true statements and discovering counterexamples that disprove false ones. However, current AI efforts in mathematics focus almost exclusively on proof construction, often neglecting the equally important task of finding counterexamples. In this paper, we address this gap by fine-tuning large language models (LLMs) to reason about and generate counterexamples. We formalize this task as formal counterexample generation, which requires LLMs not only to propose candidate counterexamples but also to produce formal proofs that can be automatically verified in the Lean 4 theorem prover. To enable effective learning, we introduce a symbolic mutation strategy that synthesizes diverse training data by systematically extracting theorems and discarding selected hypotheses, thereby producing diverse counterexample instances. Together with curated datasets, this strategy enables a multi-reward expert iteration framework that substantially enhances both the effectiveness and efficiency of training LLMs for counterexample generation and theorem proving. Experiments on three newly collected benchmarks validate the advantages of our approach, showing that the mutation strategy and training framework yield significant performance gains.","abs_url":"https://arxiv.org/abs/2603.19514","pdf_url":"https://arxiv.org/pdf/2603.19514v1","primary_category":"cs.AI","published":"2026-03-19","tags":["theorem-proving-ml","benchmark"],"lean_note":"Fine-tunes LLMs to generate formal counterexamples with proofs automatically verified in Lean 4.","thumb":"thumbs/2603.19514.jpg"},{"id":"2603.19329","title":"Goedel-Code-Prover: Hierarchical Proof Search for Open State-of-the-Art Code Verification","authors":["Zenan Li","Ziran Yang","Deyuan He","Haoyu Zhao","Andrew Zhao","Shange Tang","Kaiyu Yang","Aarti Gupta","Zhendong Su","Chi Jin"],"abstract":"Large language models (LLMs) can generate plausible code but offer limited guarantees of correctness. Formally verifying that implementations satisfy specifications requires constructing machine-checkable proofs, a task that remains beyond current automation. We propose a hierarchical proof search framework for automated code verification in Lean~4 that decomposes complex verification goals into structurally simpler subgoals before attempting tactic-level proving. Central to our approach is a principled decomposition score that combines constructive justification with structural effectiveness. Crucially, this score serves as both the training reward and the inference-time ranking criterion, ensuring strict alignment between optimization and deployment. We train Goedel-Code-Prover-8B, a single unified policy for both decomposition and completion, via supervised initialization followed by hybrid reinforcement learning, where a continuous decomposition reward drives planning exploration while supervised replay stabilizes proof generation. On three Lean-based code verification benchmarks comprising 427 tasks, our 8B-parameter model achieves a 62.0\\% prove success rate, a 2.6$\\times$ improvement over the strongest baseline, surpassing neural provers up to 84$\\times$ larger. We further observe consistent inference-time scaling: success rates improve monotonically with search iterations and sampling budget, with our trained model achieving greater efficiency than frontier off-the-shelf models of comparable scale.","abs_url":"https://arxiv.org/abs/2603.19329","pdf_url":"https://arxiv.org/pdf/2603.19329v2","primary_category":"cs.SE","published":"2026-03-18","tags":["theorem-proving-ml","verification"],"lean_note":"Hierarchical proof-search framework for automated code verification in Lean 4, training Goedel-Code-Prover-8B.","thumb":"thumbs/2603.19329.jpg"},{"id":"2603.17997","title":"The Ferrers bound for spanning trees in bipartite graphs","authors":["Boon Suan Ho"],"abstract":"We prove Ehrenborg's conjecture that every connected bipartite graph $G$ with parts of size $m$ and $n$ has at most $\\frac{1}{mn}\\prod_{v\\in V(G)} \\operatorname{deg}(v)$ spanning trees, and that equality holds if and only if $G$ is a Ferrers graph. The proof is fully formalized in Lean 4.","abs_url":"https://arxiv.org/abs/2603.17997","pdf_url":"https://arxiv.org/pdf/2603.17997v1","primary_category":"math.CO","published":"2026-03-18","tags":["formalization","combinatorics"],"lean_note":"Fully formalizes in Lean 4 the proof of Ehrenborg's conjecture on spanning trees in bipartite graphs.","repo":"https://github.com/cokernel/ferrers-bound-checking","thumb":"thumbs/2603.17997.jpg"},{"id":"2603.17457","title":"Synthetic Differential Geometry in Lean","authors":["Riccardo Brasca","Gabriella Clemente"],"abstract":"This article is about the formalization of synthetic differential geometry with the Lean proof assistant and the mathematical library mathlib. The main result we prove and formalize is a Taylor theorem for functions of several variables, where the series expansion is around an infinitesimal neighborhood. Most of our proofs are in fact new. Our investigations highlight the possibility of using mathlib to do constructive mathematics.","abs_url":"https://arxiv.org/abs/2603.17457","pdf_url":"https://arxiv.org/pdf/2603.17457v2","primary_category":"cs.LO","published":"2026-03-18","tags":["formalization","geometry"],"lean_note":"Formalizes synthetic differential geometry and a multivariable Taylor theorem in Lean with mathlib.","thumb":"thumbs/2603.17457.jpg"},{"id":"2604.16347","title":"Lean Atlas: An Integrated Proof Environment for Scalable Human-AI Collaborative Formalization","authors":["Banri Yanahama","Akiyoshi Sannai"],"abstract":"AI-driven autoformalization of mathematics is advancing rapidly. However, the type checker of a proof assistant guarantees only the logical correctness of proofs; it does not verify whether propositions and definitions faithfully capture their intended mathematical content. Consequently, AI-generated formal proofs can exhibit semantic hallucination-passing the type checker yet failing to express the intended mathematics. We propose a human-in-the-loop approach in which human scientists and AI collaboratively produce formal proofs, with humans responsible for the semantic verification of propositions and definitions. To realize this approach, we develop Lean Atlas, a Lean 4 tool that visualizes the dependency graph of a Lean 4 project as an interactive web viewer, enabling human scientists to grasp the overall structure of a formalization efficiently. Its core feature, Lean Compass, is an algorithm that, given a selected theorem set, automatically extracts the project-specific nodes whose semantic correctness can affect those target statements, thereby reducing the candidate set for semantic review in large-scale formalizations. We further define *aligned Lean code* as formalization code that has undergone human semantic verification, and propose it as a quality standard for AI-generated formalizations. We evaluate the tool on six Lean 4 formalization projects with different structural characteristics; proof-heavy projects (PrimeNumberTheoremAnd, Carleson, Brownian Motion) achieved 94-99% average node reduction, a 6-theorem milestone subset of FLT achieved 59.8%, mixed PhysLib 69.0%, and definition-heavy XMSS 27.3%. Lean Atlas is available as open-source software at https://github.com/NyxFoundation/lean-atlas .","abs_url":"https://arxiv.org/abs/2604.16347","pdf_url":"https://arxiv.org/pdf/2604.16347v1","primary_category":"cs.HC","published":"2026-03-16","tags":["tooling","ai-agents"],"lean_note":"Develops Lean Atlas, a Lean 4 tool visualizing project dependency graphs to support human semantic review of formalizations.","repo":"https://github.com/NyxFoundation/lean-atlas","thumb":"thumbs/2604.16347.jpg"},{"id":"2603.15929","title":"Semi-Autonomous Formalization of the Vlasov-Maxwell-Landau Equilibrium","authors":["Vasily Ilin"],"abstract":"We present a complete Lean 4 formalization of the equilibrium characterization in the Vlasov-Maxwell-Landau (VML) system, which describes the motion of charged plasma. The project demonstrates the full AI-assisted mathematical research loop: an AI reasoning model (Gemini DeepThink) generated the proof from a conjecture, an agentic coding tool (Claude Code) translated it into Lean from natural-language prompts, a specialized prover (Aristotle) closed 111 lemmas, and the Lean kernel verified the result. A single mathematician supervised the process over 10 days at a cost of \\$200, writing zero lines of code. The entire development process is public: all 229 human prompts, and 213 git commits are archived in the repository. We report detailed lessons on AI failure modes -- hypothesis creep, definition-alignment bugs, agent avoidance behaviors -- and on what worked: the abstract/concrete proof split, adversarial self-review, and the critical role of human review of key definitions and theorem statements. Notably, the formalization was completed before the final draft of the corresponding math paper was finished.","abs_url":"https://arxiv.org/abs/2603.15929","pdf_url":"https://arxiv.org/pdf/2603.15929v2","primary_category":"cs.AI","published":"2026-03-16","tags":["formalization","ai-agents","physics"],"lean_note":"Presents a semi-autonomous Lean 4 formalization of the Vlasov-Maxwell-Landau equilibrium via an AI research loop.","repo":"https://github.com/Vilin97/aristotle","thumb":"thumbs/2603.15929.jpg"},{"id":"2603.15770","title":"Formalization of QFT","authors":["Michael R. Douglas","Sarah Hoback","Anna Mei","Ron Nissim"],"abstract":"A foundational result in constructive quantum field theory is the construction of the free bosonic quantum field theory in four-dimensional Euclidean spacetime and the proof that it satisfies the Glimm-Jaffe axioms, a variant of the Osterwalder-Schrader axioms. We present a formalization of this result in the Lean 4 interactive theorem prover. The project is intended as a proof of concept that extended arguments in mathematical physics can be translated into machine-checked proofs using existing AI tools. We begin by introducing interactive theorem proving and constructive quantum field theory, then describe our formalization and the design decisions that shaped it. We also explain the methods we used, including coding assistants, and conclude by considering how AI assisted formalization may influence the future of theoretical physics. Our original release assumed three results, Minlos' theorem, the nuclear property of Schwartz space, and Goursat's theorem. In subsequent releases from our group and from contributors from the Lean community, these assumptions have been proven (or avoided), so that the OS/GJ axioms are now proven using only Lean and its library Mathlib.","abs_url":"https://arxiv.org/abs/2603.15770","pdf_url":"https://arxiv.org/pdf/2603.15770v1","primary_category":"hep-th","published":"2026-03-16","tags":["formalization","mathlib","physics"],"lean_note":"Formalizes free bosonic QFT and the Osterwalder-Schrader/Glimm-Jaffe axioms in Lean 4 with Mathlib.","repo":"https://github.com/cameronfreer/lean4-skills","thumb":"thumbs/2603.15770.jpg"},{"id":"2603.14689","title":"The Optimizer Quotient and the Certification Trilemma","authors":["Tristan Simas"],"abstract":"The optimizer quotient is the canonical object for exact decision-relevant information: it is the coarsest exact decision-preserving abstraction (Theorem 2.15). This paper proves that exact certification of this object's coordinate structure is subject to an impossibility trilemma: under $\\mathrm{P} \\neq \\mathrm{coNP}$, no certifier can be simultaneously sound, complete on all in-scope instances, and polynomial-budgeted (Theorem 7.1). The cost of this impossibility varies by regime: coNP (static), PP-hard (stochastic decisiveness), PSPACE-complete (sequential). Six structural restrictions collapse certification to polynomial time. The finite reduction and verification core is mechanized in Lean 4.","abs_url":"https://arxiv.org/abs/2603.14689","pdf_url":"https://arxiv.org/pdf/2603.14689v2","primary_category":"cs.CC","published":"2026-03-16","tags":["formalization","logic"],"lean_note":"Mechanizes the finite reduction and verification core of the certification trilemma in Lean 4.","repo":"https://doi.org/10.5281/zenodo.19057595","thumb":"thumbs/2603.14689.jpg"},{"id":"2603.14663","title":"Formalizing the Classical Isoperimetric Inequality in the Two-Dimensional Case","authors":["Miraj Samarakkody"],"abstract":"We present a formal verification of the classical isoperimetric inequality in the plane using the Lean 4 proof assistant and its mathematical library Mathlib. We follow Adolf Hurwitz's analytic approach to establish the inequality $L^2 \\ge 4πA$, which states that among all simple closed curves of a given perimeter $L$, the circle uniquely maximizes the enclosed area $A$. The formalization proceeds in two phases. In the first phase, we establish the Fourier-analytic foundations required by Hurwitz's approach: we formalize orthogonality relations for trigonometric functions over $[-π,π]$, Parseval's theorem for classical Fourier series, uniform convergence of Fourier partial sums via the Weierstrass M-test, term-by-term differentiability, and Wirtinger's inequality. In the second phase, we carry out Hurwitz's proof itself: working with simple closed $C^1$ curves given in arc-length parametrization, we reparametrize over $[0,2π]$, establish the shoelace area formula, apply integration by parts, invoke the AM--GM inequality, apply Wirtinger's inequality, and use the arc-length constraint to derive the bound $A \\le L^2/(4π)$. We discuss the key formalization challenges encountered, including the interchange of infinite sums and integrals, term-by-term differentiation, and the coordination of different indexing conventions within Mathlib. The complete formalization is available at https://github.com/mirajcs/IsoperimetricInequality","abs_url":"https://arxiv.org/abs/2603.14663","pdf_url":"https://arxiv.org/pdf/2603.14663v1","primary_category":"math.MG","published":"2026-03-15","tags":["formalization","analysis","geometry"],"lean_note":"Formally verifies the planar isoperimetric inequality via Hurwitz's Fourier-analytic approach in Lean 4 and Mathlib.","repo":"https://github.com/mirajcs/IsoperimetricInequality","thumb":"thumbs/2603.14663.jpg"},{"id":"2603.14038","title":"Machine-Verifying Toom-Cook Multiplication with Integer Evaluation Points","authors":["Srihari Nanniyur","Siddhartha Jayanti"],"abstract":"We present a machine-verified proof of the correctness of Toom-Cook multiplication with generalized integer evaluation points. Toom-Cook is a class of fast multiplication algorithms parameterized by a triple $(k_x, k_y, \\vec v)$ consisting of two positive integer split sizes $k_x, k_y$ and a vector $\\vec v$ of distinct evaluation points. As part of our proof, we verify that for any selection of $k_x+k_y-1$ distinct integer evaluation points, we can compute a threshold function $θ(k_x, k_y, \\vec v)$ such that, if the algorithm's base-case problem size is set above this threshold, then the algorithm's termination is guaranteed regardless of the values of the operands. The threshold formula, which we derive by obtaining upper bounds on the subproblem sizes produced by the Toom-Cook recurrence, does not depend on the operands; it depends only on $k_x$, $k_y$, $\\vec v$, and the base $b$ in which we operate. We write the proof in Lean 4, making use of the Mathlib library. We formalize the algorithm, our base case threshold formula, and our key lemma statements in Lean. We then use the AI theorem prover Aristotle to assist in completing the machine verification of the algorithm's correctness. This proof, through its synthesis of human input and AI assistance, demonstrates the considerable power of AI to automate the machine verification process.","abs_url":"https://arxiv.org/abs/2603.14038","pdf_url":"https://arxiv.org/pdf/2603.14038v1","primary_category":"cs.DS","published":"2026-03-14","tags":["verification","mathlib"],"lean_note":"Machine-verifies correctness of Toom-Cook multiplication in Lean 4 with Mathlib, using the Aristotle AI prover.","repo":"https://github.com/sriharinanniyur/lean-multiplication","thumb":"thumbs/2603.14038.jpg"},{"id":"2603.12744","title":"TaoBench: Do Automated Theorem Prover LLMs Generalize Beyond MathLib?","authors":["Alexander K Taylor","Junyi Zhang","Ethan Ji","Vigyan Sahai","Haikang Deng","Yuanzhou Chen","Yifan Yuan","Di Wu","Jia-Chen Gu","Kai-Wei Chang","Nanyun Peng","Amit Sahai","Wei Wang"],"abstract":"Automated theorem proving (ATP) benchmarks largely consist of problems formalized in MathLib, so current ATP training and evaluation are heavily biased toward MathLib's definitional framework. However, frontier mathematics is often exploratory and prototype-heavy, relying on bespoke constructions that deviate from standard libraries. In this work, we evaluate the robustness of current ATP systems when applied to a novel definitional framework, specifically examining the performance gap between standard library problems and bespoke mathematical constructions. We introduce TaoBench, an undergraduate-level benchmark derived from Terence Tao's Analysis I, which formalizes analysis by constructing core mathematical concepts from scratch, without relying on standard Mathlib definitions, as well as by mixing from-scratch and MathLib constructions. For fair evaluation, we build an agentic pipeline that automatically extracts a compilable, self-contained local environment for each problem. To isolate the effect of definitional frameworks, we additionally translate every problem into a mathematically equivalent Mathlib formulation, yielding paired TaoBench-Mathlib statements for direct comparison. While state-of-the-art ATP models perform capably within the MathLib framework, performance drops by an average of roughly 26% on the definitionally equivalent Tao formulation. This indicates that the main bottleneck is limited generalization across definitional frameworks rather than task difficulty. TaoBench thus highlights a gap between benchmark performance and applicability, and provides a concrete foundation for developing and testing provers better aligned with research mathematics.","abs_url":"https://arxiv.org/abs/2603.12744","pdf_url":"https://arxiv.org/pdf/2603.12744v1","primary_category":"cs.LG","published":"2026-03-13","tags":["benchmark","theorem-proving-ml"],"lean_note":"Introduces TaoBench evaluating Lean ATP models on bespoke versus Mathlib definitional frameworks.","thumb":"thumbs/2603.12744.jpg"},{"id":"2603.12349","title":"Budget-Sensitive Discovery Scoring: A Formally Verified Framework for Evaluating AI-Guided Scientific Selection","authors":["Abhinaba Basu","Pavan Chakraborty"],"abstract":"Scientific discovery increasingly relies on AI systems to select candidates for expensive experimental validation, yet no principled, budget-aware evaluation framework exists for comparing selection strategies -- a gap intensified by large language models (LLMs), which generate plausible scientific proposals without reliable downstream evaluation. We introduce the Budget-Sensitive Discovery Score (BSDS), a formally verified metric -- 20 theorems machine-checked by the Lean 4 proof assistant -- that jointly penalizes false discoveries (lambda-weighted FDR) and excessive abstention (gamma-weighted coverage gap) at each budget level. Its budget-averaged form, the Discovery Quality Score (DQS), provides a single summary statistic that no proposer can inflate by performing well at a cherry-picked budget. As a case study, we apply BSDS/DQS to: do LLMs add marginal value to an existing ML pipeline for drug discovery candidate selection? We evaluate 39 proposers -- 11 mechanistic variants, 14 zero-shot LLM configurations, and 14 few-shot LLM configurations -- using SMILES representations on MoleculeNet HIV (41,127 compounds, 3.5% active, 1,000 bootstrap replicates) under both random and scaffold splits. Three findings emerge. First, the simple RF-based Greedy-ML proposer achieves the best DQS (-0.046), outperforming all MLP variants and LLM configurations. Second, no LLM surpasses the Greedy-ML baseline under zero-shot or few-shot evaluation on HIV or Tox21, establishing that LLMs provide no marginal value over an existing trained classifier. Third, the proposer hierarchy generalizes across five MoleculeNet benchmarks spanning 0.18%-46.2% prevalence, a non-drug AV safety domain, and a 9x7 grid of penalty parameters (tau >= 0.636, mean tau = 0.863). The framework applies to any setting where candidates are selected under budget constraints and asymmetric error costs.","abs_url":"https://arxiv.org/abs/2603.12349","pdf_url":"https://arxiv.org/pdf/2603.12349v1","primary_category":"cs.LG","published":"2026-03-12","tags":["verification","ml-theory"],"lean_note":"Defines the Budget-Sensitive Discovery Score with 20 theorems machine-checked in Lean 4.","thumb":"thumbs/2603.12349.jpg"},{"id":"2603.12183","title":"Proof-Carrying Materials: Falsifiable Safety Certificates for Machine-Learned Interatomic Potentials","authors":["Abhinaba Basu","Pavan Chakraborty"],"abstract":"Machine-learned interatomic potentials (MLIPs) are deployed for high-throughput materials screening without formal reliability guarantees. We show that a single MLIP used as a stability filter misses 93% of density functional theory (DFT)-stable materials (recall 0.07) on a 25,000-material benchmark. Proof-Carrying Materials (PCM) closes this gap through three stages: adversarial falsification across compositional space, bootstrap envelope refinement with 95% confidence intervals, and Lean 4 formal certification. Auditing CHGNet, TensorNet and MACE reveals architecture-specific blind spots with near-zero pairwise error correlations (r <= 0.13; n = 5,000), confirmed by independent Quantum ESPRESSO validation (20/20 converged; median DFT/CHGNet force ratio 12x). A risk model trained on PCM-discovered features predicts failures on unseen materials (AUC-ROC = 0.938 +/- 0.004) and transfers across architectures (cross-MLIP AUC-ROC ~ 0.70; feature importance r = 0.877). In a thermoelectric screening case study, PCM-audited protocols discover 62 additional stable materials missed by single-MLIP screening - a 25% improvement in discovery yield.","abs_url":"https://arxiv.org/abs/2603.12183","pdf_url":"https://arxiv.org/pdf/2603.12183v2","primary_category":"cond-mat.mtrl-sci","published":"2026-03-12","tags":["verification","physics"],"lean_note":"Uses Lean 4 formal certification as the final stage of safety certificates for machine-learned interatomic potentials.","repo":"https://github.com/abhinaba/alloy_pcm","thumb":"thumbs/2603.12183.jpg"},{"id":"2603.11900","title":"Existence as Distinguishability: Quantum Mechanics from Finite Graded Equality","authors":["Julian G. Zilly"],"abstract":"We derive finite-dimensional quantum mechanics from a single ontological principle, that \\emph{existence is constituted by distinguishability}, together with two structural commitments: finite capacity $N$ (parametric input) and self-referential consistency (SRC, a closure schema with two equivalent forms, operational and information-theoretic). SRC unpacks into eight derived structural conditions; structural unambiguity (S5) completes the hierarchy, uniquely selecting the Born rule as the geometric/probabilistic closure. The graded distinguishability kernel $K(x,y) \\in [0,1]$ realises both axioms, with a state constituted by its $K$-profile against all others. For each $N \\geq 3$, the unique distinguishability space is $(\\mathbb{C} P^{N-1}, K)$ with $K(ψ,φ) = 1 - |\\langleψ|φ\\rangle|^2$, from which complex coefficients, the Born rule $p_k = |c_k|^2$, unitary dynamics, and tensor-product composition all follow. Indeterminism is forced by capacity overflow; alternatives (e.g. Bohmian mechanics) are classified rather than refuted. Standard QM is the $N \\to \\infty$ limit; finite $N$ is the only free parameter. The algebraic spine is machine-checked in Lean 4 modulo five imported classical theorems and the existence direction of Stone's theorem; the Appendix states the verification scope.","abs_url":"https://arxiv.org/abs/2603.11900","pdf_url":"https://arxiv.org/pdf/2603.11900v2","primary_category":"quant-ph","published":"2026-03-12","tags":["formalization","quantum","physics"],"lean_note":"Machine-checks in Lean 4 the algebraic spine of a derivation of finite-dimensional quantum mechanics modulo five imported theorems.","repo":"https://github.com/jzilly/QuantumRelational","thumb":"thumbs/2603.11900.jpg"},{"id":"2603.09198","title":"On the extension of inner derivations from dense ideals in Banach algebras","authors":["Hamid Shafieasl","Amir Mohammad Tavakkoli"],"abstract":"Let $A$ be a Banach algebra and $I$ a dense ideal in $A$. A natural question in the theory of operator algebras is whether the property that all derivations $D: A \\to I$ are inner (implemented by elements in $I$) implies that all derivations $D: A \\to A$ are inner (implemented by elements in $A$). We present a rigorous negative answer to this question. By utilizing the algebra of compact operators $A = K(H)$ and the dense ideal of finite-rank operators $I = F(H)$ on a separable infinite-dimensional Hilbert space $H$, we demonstrate that while every derivation into $F(H)$ is inner, there exist outer derivations on $K(H)$. Furthermore, we generalize this result to Schatten $p$-classes and discuss the cohomological implications and the role of approximate identities. Moreover, the main results and counterexamples presented in this paper have been formally verified using the Lean theorem prover.","abs_url":"https://arxiv.org/abs/2603.09198","pdf_url":"https://arxiv.org/pdf/2603.09198v2","primary_category":"math.FA","published":"2026-03-10","tags":["formalization","analysis","algebra"],"lean_note":"Formally verifies in the Lean theorem prover a negative answer on extending inner derivations in Banach algebras.","repo":"https://github.com/tavakkoliamirmohammad/inner-deriv-dense-lean","thumb":"thumbs/2603.09198.jpg"},{"id":"2603.08322","title":"Agentic Neurosymbolic Collaboration for Mathematical Discovery: A Case Study in Combinatorial Design","authors":["Hai Xia","Carla P. Gomes","Bart Selman","Stefan Szeider"],"abstract":"We study mathematical discovery through the lens of neurosymbolic reasoning, where an AI agent powered by a large language model (LLM), coupled with symbolic computation tools, and human strategic direction, jointly produced a new result in combinatorial design theory. The main result of this human-AI collaboration is a tight lower bound on the imbalance of Latin squares for the notoriously difficult case $n \\equiv 1 \\pmod{3}$. We reconstruct the discovery process from detailed interaction logs spanning multiple sessions over several days and identify the distinct cognitive contributions of each component. The AI agent proved effective at uncovering hidden structure and generating hypotheses. The symbolic component consists of computer algebra, constraint solvers, and simulated annealing, which provides rigorous verification and exhaustive enumeration. Human steering supplied the critical research pivot that transformed a dead end into a productive inquiry. Our analysis reveals that multi-model deliberation among frontier LLMs proved reliable for criticism and error detection but unreliable for constructive claims. The resulting human-AI mathematical contribution, a tight lower bound of $4n(n{-}1)/9$, is achieved via a novel class of near-perfect permutations. The bound was formally verified in Lean 4. Our experiments show that neurosymbolic systems can indeed produce genuine discoveries in pure mathematics.","abs_url":"https://arxiv.org/abs/2603.08322","pdf_url":"https://arxiv.org/pdf/2603.08322v1","primary_category":"cs.AI","published":"2026-03-09","tags":["formalization","combinatorics"],"lean_note":"Formally verifies a tight lower bound on Latin-square imbalance discovered via neurosymbolic collaboration in Lean 4.","thumb":"thumbs/2603.08322.jpg"},{"id":"2603.08139","title":"Formalizing the stability of the two Higgs doublet model potential into Lean: identifying an error in the literature","authors":["Joseph Tooby-Smith"],"abstract":"In 2006, using the best methods and techniques available at the time, Maniatis, von Manteuffel, Nachtmann and Nagel published a now widely cited paper on the stability of the two Higgs doublet model (2HDM) potential. Twenty years on, it is now easier to apply the process of formalization into an interactive theorem prover to this work thanks to projects like Mathlib and Physlib (the latter formerly PhysLean and Lean-QuantumInfo), and to ask for a higher standard of mathematical correctness. Doing so has revealed an error in the arguments of this 2006 paper, invalidating their main theorem on the stability of the 2HDM potential. This case is noteworthy because to the best of our knowledge it is the first non-trivial error in a physics paper found through formalization. It was one of the first papers where formalization was attempted, which raises the uncomfortable question of how many physics papers would not pass this higher level of scrutiny.","abs_url":"https://arxiv.org/abs/2603.08139","pdf_url":"https://arxiv.org/pdf/2603.08139v2","primary_category":"hep-ph","published":"2026-03-09","tags":["formalization","physics"],"lean_note":"Formalizes the two-Higgs-doublet-model potential stability in Lean with Physlib, uncovering an error in the 2006 literature.","thumb":"thumbs/2603.08139.jpg"},{"id":"2603.06931","title":"LLM2SMT: Building an SMT Solver with Zero Human-Written Code","authors":["Mikoláš Janota","Mirek Olšák"],"abstract":"Whether LLMs can reason or write software is widely debated, but whether they can write software that itself reasons is largely unexplored. We present a case study in which an LLM coding agent builds a complete DPLL(T)-style SMT solver for QF_UF with zero human-written code. The solver implements the Nieuwenhuis-Oliveras congruence closure algorithm, includes preprocessing, and emits Lean proofs for unsatisfiable instances. We describe the development process and key challenges, and show that the resulting solver is competitive on SMT-LIB benchmarks.","abs_url":"https://arxiv.org/abs/2603.06931","pdf_url":"https://arxiv.org/pdf/2603.06931v2","primary_category":"cs.LO","published":"2026-03-06","tags":["verification","tooling","ai-agents"],"lean_note":"An LLM-built DPLL(T)-style SMT solver for QF_UF emits Lean proofs for unsatisfiable instances, certified with Lean's grind tactic.","repo":"https://github.com/MikolasJanota/llm2smt","thumb":"thumbs/2603.06931.jpg"},{"id":"2603.04376","title":"Formalization in Lean of faithfully flat descent of projectivity","authors":["Liran Shaul"],"abstract":"We formalize in Lean the following foundational result in commutative algebra: Let $R \\to S$ be a faithfully flat map of (not necessarily noetherian) commutative rings, and let $P$ be an arbitrary $R$-module. Then $P$ is projective over $R$ if and only if $S\\otimes_R P$ is projective over $S$. This formalizes and verifies Perry's fix of a subtle gap in the classical work of Raynaud and Gruson, a result which is a key ingredient in the study of finitistic dimension of commutative noetherian rings.","abs_url":"https://arxiv.org/abs/2603.04376","pdf_url":"https://arxiv.org/pdf/2603.04376v1","primary_category":"math.AC","published":"2026-03-04","tags":["formalization","algebra"],"lean_note":"Formalizes in Lean that faithfully flat descent preserves projectivity of an arbitrary module, verifying Perry's fix of a gap in Raynaud-Gruson.","thumb":"thumbs/2603.04376.jpg"},{"id":"2603.04235","title":"2-Coloring Cycles in One Round","authors":["Maxime Flin","Alesya Raevskaya","Ronja Stimpert","Jukka Suomela","Qingxin Yang"],"abstract":"We show that there is a one-round randomized distributed algorithm that can 2-color cycles such that the expected fraction of monochromatic edges is less than 0.24118. We also show that a one-round algorithm cannot achieve a fraction less than 0.23879. Before this work, the best upper and lower bounds were 0.25 and 0.2. Our proof was largely discovered and developed by large language models, and both the upper and lower bounds have been formalized in Lean 4.","abs_url":"https://arxiv.org/abs/2603.04235","pdf_url":"https://arxiv.org/pdf/2603.04235v2","primary_category":"cs.DC","published":"2026-03-04","tags":["formalization","combinatorics"],"lean_note":"Formalizes in Lean 4 both upper and lower bounds for one-round randomized 2-coloring of cycles.","repo":"https://github.com/suomela/2-coloring-1-round","thumb":"thumbs/2603.04235.jpg"},{"id":"2603.02668","title":"SorryDB: Can AI Provers Complete Real-World Lean Theorems?","authors":["Austin Letson","Leopoldo Sarra","Auguste Poiroux","Oliver Dressler","Paul Lezeau","Dhyan Aranha","Frederick Pu","Aaron Hill","Miguel Corredera Hidalgo","Julian Berman","George Tsoukalas","Lenny Taelman"],"abstract":"We present SorryDB, a dynamically-updating benchmark of open Lean tasks drawn from 78 real world formalization projects on GitHub. Unlike existing static benchmarks, often composed of competition problems, hillclimbing the SorryDB benchmark will yield tools that are aligned to the community needs, more usable by mathematicians, and more capable of understanding complex dependencies. Moreover, by providing a continuously updated stream of tasks, SorryDB mitigates test-set contamination and offers a robust metric for an agent's ability to contribute to novel formal mathematics projects. We evaluate a collection of approaches, including generalist large language models, agentic approaches, and specialized symbolic provers, over a selected snapshot of 1000 tasks from SorryDB. We show that current approaches are complementary: even though an agentic approach based on Gemini Flash is the most performant, it is not strictly better than other off-the-shelf large-language models, specialized provers, or even a curated list of Lean tactics.","abs_url":"https://arxiv.org/abs/2603.02668","pdf_url":"https://arxiv.org/pdf/2603.02668v1","primary_category":"cs.AI","published":"2026-03-03","tags":["benchmark","dataset","theorem-proving-ml","ai-agents"],"lean_note":"Introduces a dynamically-updating benchmark of open `sorry` tasks scraped from 78 active real-world Lean GitHub projects and evaluates provers on filling them.","repo":"https://github.com/SorryDB/SorryDB","thumb":"thumbs/2603.02668.jpg"},{"id":"2603.01463","title":"Implementing Dependent Type Theory Inhabitation and Unification","authors":["Chase Norman","Jeremy Avigad"],"abstract":"Dependent type theory is the foundation of many modern proof assistants. Inhabitation and unification are undecidable problems that are useful for theorem proving and program synthesis. We introduce Canonical-min, a sound and complete solver for inhabitation and unification in dependent type theory, implemented in 185 lines of Lean code. This paper describes a novel implementation of dependent type theory and a monadic framework to transform the type checker into a performant solver. Finally, we introduce DTTBench, a benchmark for type inhabitation in dependent type theory.","abs_url":"https://arxiv.org/abs/2603.01463","pdf_url":"https://arxiv.org/pdf/2603.01463v1","primary_category":"cs.LO","published":"2026-03-02","tags":["tactic-automation","tooling","benchmark","type-theory"],"lean_note":"Implements Canonical-min, a sound and complete inhabitation/unification solver for dependent type theory, in 185 lines of Lean, exposed as a Lean tactic.","repo":"https://github.com/chasenorman/Canonical-min","thumb":"thumbs/2603.01463.jpg"},{"id":"2603.01056","title":"A Classifying Topos for the Spectrum of Equivalences","authors":["Kenan Oggad"],"abstract":"What makes two computational systems equivalent? Topos theory answers with classifying toposes: a system's semantic content is encoded in the geometric theory it classifies, and two presentations are equivalent when their classifying toposes coincide. Process algebra answers with the linear time-branching time spectrum of van Glabbeek: a hierarchy of behavioral equivalences from trace equivalence to bisimilarity, each determined by which observations can distinguish processes. We show these are aspects of a single structure in which behavioral abstraction is localization. Each labeled transition system receives a geometric theory $\\mathbb{T}_M$ whose classifying topos $\\mathcal{E}[\\mathbb{T}_M]$ determines its provable geometric sequents. Mutual simulation is strictly coarser than bisimulation, strictly coarser than topos equivalence; diamond-only Hennessy-Milner logic characterizes the bisimulation-invariant fragment of geometric logic -- a geometric van Benthem theorem. Grothendieck topologies yield $J_{\\mathrm{bisim}} \\subsetneq J_{\\mathrm{sim}} \\subsetneq J_{\\mathrm{trace}}$, constructive for trace and bisimulation; a counterexample shows the observation-class approach inadequate for simulation, motivating Caramello's duality. Energy-topology extends this to all 13 named equivalences. Lattice closure yields 30 elements including 17 unnamed hybrids absent because the energy-game framework computes but does not close. $L_{30}$ is indecomposable with $S \\to F = \\mathrm{IF}$; a Geometric Closure Theorem computes presheaf Heyting implications at a single free extension. The hierarchy, bi-Heyting structure, and Closure Theorem are proved constructively with no known process-algebraic proof. The spectrum is a finite sub-poset of an infinite coframe whose operations (meets, implications, subtractions) yield structure inaccessible from process algebra. Formalized in Lean 4/Mathlib.","abs_url":"https://arxiv.org/abs/2603.01056","pdf_url":"https://arxiv.org/pdf/2603.01056v1","primary_category":"cs.LO","published":"2026-03-01","tags":["formalization","mathlib","category-theory","logic"],"lean_note":"Constructively formalizes the classifying-topos hierarchy of behavioral equivalences and a Geometric Closure Theorem in Lean 4/Mathlib.","thumb":"thumbs/2603.01056.jpg"},{"id":"2603.00896","title":"Unbiasing symmetric monoidal categories in Lean","authors":["Robin Carlier"],"abstract":"We present a formalization in Lean 4, within the framework of the mathematical library Mathlib, of the unbiasing process for symmetric monoidal categories. This is realized by extending the data of a symmetric monoidal category to a Cat-valued pseudofunctor from the (2,1)-category of spans of finite sets, encoding tensor products of higher arities and their coherences. The construction relies on a formalization of Mac Lane's coherence theorem using Piceghello's presentation of free symmetric monoidal categories as symmetric lists, and uses an encoding of universal formulas via an appropriate Kleisli bicategory.","abs_url":"https://arxiv.org/abs/2603.00896","pdf_url":"https://arxiv.org/pdf/2603.00896v1","primary_category":"math.CT","published":"2026-03-01","tags":["formalization","category-theory","mathlib"],"lean_note":"Formalizes the unbiasing process for symmetric monoidal categories in Lean 4 within Mathlib.","thumb":"thumbs/2603.00896.jpg"},{"id":"2602.23520","title":"Zero-Error Recovery under Deterministic Partial Views: Matroid Bounds and Verifiable Realizability","authors":["Tristan Simas"],"abstract":"Zero-error recovery under deterministic partial views is graph recovery for the induced confusability relation. A finite family of coordinate-subset observations determines a graph on latent states; $T$-ary exact recovery is graph $T$-colorability, block composition is strong powering, and asymptotic recoverability is Shannon capacity. Coordinate structure gives tractable certificates inside the graph semantics. For affine realized state families with explicit linear presentations, restricted coordinate ranks form a representable matroid certificate giving polynomial-time upper bounds on one-shot confusability and asymptotic capacity, with rank additivity matching direct-sum block composition. In the full tuple-space coordinate model, the realizable confusability relations are exactly the upward-closed coordinate-agreement families. Transitive confusability is equivalent to intersection closure of the generated agreement family, yielding a cluster graph whose capacity is determined by connected components. Host-level realizability determines when the latent state family is canonical. Verifiable rate-$1$ realizability for structural facts holds if and only if the host provides zero-delay synchronization and structural side-information; eleven representative host architectures instantiate the criterion. The same clique-size bit-budget bound governs both the graph-level and host-level layers. All cited results are mechanized in Lean 4 against a shared formalization library.","abs_url":"https://arxiv.org/abs/2602.23520","pdf_url":"https://arxiv.org/pdf/2602.23520v9","primary_category":"cs.IT","published":"2026-02-26","tags":["formalization","combinatorics"],"lean_note":"All cited zero-error recovery and matroid-bound results are mechanized in Lean 4 against a shared formalization library.","repo":"https://doi.org/10.5281/zenodo.18141365","thumb":"thumbs/2602.23520.jpg"},{"id":"2602.22631","title":"TorchLean: Formalizing Neural Networks in Lean","authors":["Robert Joseph George","Jennifer Cruden","Will Adkisson","Xiangru Zhong","Huan Zhang","Anima Anandkumar"],"abstract":"Neural networks are increasingly deployed in scientific, safety critical, and mission critical pipelines, yet verification and analysis are often performed outside the programming environment that defines and runs the model. This creates a semantic gap between the executed network and the analyzed artifact: guarantees can depend on implicit conventions about operator semantics, tensor layouts, preprocessing, floating-point behavior, graph transformations, accelerated kernels, and external certificates. We present TorchLean, a unified framework for formalizing, executing, and verifying neural networks in Lean 4. TorchLean treats learned models as executable programs and mathematical objects with a shared semantics for computation, verification, and theorem proving. The framework provides a PyTorch style API for typed tensors, layers, objectives, optimizers, automatic differentiation, and graph programs, with eager and compiled execution paths that lower to a common computation-graph representation. TorchLean supports exact and finite-precision tensor semantics, verified reverse-mode differentiation, interval and affine bound propagation, CROWN/LiRPA style certificate checking, import/export workflows, and CUDA-backed execution through explicit FFI boundaries. It also includes semantic layers for attention and FlashAttention, state-space sequence models, diffusion and sampling processes, probability kernels, reinforcement-learning objectives and Markov decision processes, and self-supervised objectives such as masked autoencoding, JEPA-style predictive views, and variance/correlation-based anti-collapse losses. Together, these components provide a semantic foundation for verified machine learning, where executable neural network artifacts, verification procedures, runtime boundaries, and mathematical claims can be stated and related inside one theorem-proving environment.","abs_url":"https://arxiv.org/abs/2602.22631","pdf_url":"https://arxiv.org/pdf/2602.22631v2","primary_category":"cs.MS","published":"2026-02-26","tags":["tooling","verification","ml-theory"],"lean_note":"Presents TorchLean, a Lean 4 framework for formalizing, executing, and verifying neural networks with verified differentiation and bound propagation.","repo":"https://github.com/lean-dojo/TorchLean","thumb":"thumbs/2602.22631.jpg"},{"id":"2602.20866","title":"DeCo: A Core Calculus for Incremental Functional Programming with Generic Data Types","authors":["Timon Böhler","Tobias Reinhard","David Richter","Mira Mezini"],"abstract":"Incrementalization speeds up computations by avoiding unnecessary recomputations and by efficiently reusing previous results. While domain-specific techniques achieve impressive speedups, e.g., in the context of database queries, they are difficult to generalize. Meanwhile, general approaches offer little support for incrementalizing domain-specific operations. In this work, we present DeCo, a novel core calculus for incremental functional programming with support for a wide range of user-defined data types. Despite its generic nature, our approach statically incrementalizes domain-specific operations on user-defined data types. It is, hence, more fine-grained than other generic techniques which resort to treating domain-specific operations as black boxes. We mechanized our work in Lean and proved it sound, meaning incrementalized execution computes the same result as full reevaluation. We also provide an executable implementation with case studies featuring examples from linear algebra, relational algebra, dictionaries, trees, and conflict-free replicated data types, plus a brief performance evaluation on linear and relational algebra and on trees.","abs_url":"https://arxiv.org/abs/2602.20866","pdf_url":"https://arxiv.org/pdf/2602.20866v1","primary_category":"cs.PL","published":"2026-02-24","tags":["verification","formalization","type-theory"],"lean_note":"Mechanizes the DeCo incremental-computation calculus in Lean and proves its incrementalization transformation value-preserving (sound).","repo":"https://doi.org/10.5281/zenodo.18757667","thumb":"thumbs/2602.20866.jpg"},{"id":"2602.20186","title":"A Symplectic Proof of the Quantum Singleton Bound","authors":["Frederick Dehmel","Shilun Li"],"abstract":"We present a symplectic linear-algebraic proof of the Quantum Singleton Bound for stabiliser quantum error-correcting codes together with a Lean4 formalisation of the linear-algebraic argument. The proof is formulated in the language of finite-dimensional symplectic vector spaces modelling Pauli operators and relies on distance-based erasure correctability and the cleaning lemma. Using a dimension-counting argument within the symplectic stabiliser framework, we derive the bound $k + 2(d-1) \\le n$ for any $[[n, k, d]]$ stabiliser code. This approach isolates the algebraic structure underlying the bound and avoids the heavier analytic machinery that appears in entropy-based proofs, while remaining well-suited to formal verification.","abs_url":"https://arxiv.org/abs/2602.20186","pdf_url":"https://arxiv.org/pdf/2602.20186v2","primary_category":"quant-ph","published":"2026-02-21","tags":["formalization","quantum"],"lean_note":"Gives a symplectic linear-algebraic proof of the Quantum Singleton Bound with a Lean4 formalization of the argument.","repo":"https://github.com/Shilun-Allan-Li/tcslib","thumb":"thumbs/2602.20186.jpg"},{"id":"2602.18767","title":"Nazrin: An Atomic Neural Proof Automation Tactic in Lean 4","authors":["Leni Aniva","Iori Oikawa","David Dill","Clark Barrett"],"abstract":"In Machine-Assisted Theorem Proving, a theorem proving agent searches for a sequence of expressions and tactics that can prove a statement in a proof assistant. In this work, we introduce several novel concepts and capabilities to address obstacles faced by machine-assisted theorem proving. We first present a set of \\textbf{atomic tactics}, a small finite set of tactics capable of proving any provable statement in Lean. We then introduce a \\textbf{transposing atomization} algorithm which turns arbitrary proof expressions into a series of atomic tactics. We next introduce the \\textbf{ExprGraph} data structure, which provides a succinct representation for Lean expressions. Finally, we present the \\textbf{Nazrin Prover}, short for \\textbf{N}eural \\textbf{A}tomi\\textbf{z}e\\textbf{r} for \\textbf{In}habitation Problems, a graph neural network-based theorem proving agent using atomic tactics and ExprGraph. Nazrin circumvents many challenges faced by existing proving agents by exclusively dispatching atomic tactics, and it is robust enough to both train and evaluate on consumer-grade hardware. We demonstrate the potential of tools like Nazrin using theorems from Lean's standard library and from Mathlib.","abs_url":"https://arxiv.org/abs/2602.18767","pdf_url":"https://arxiv.org/pdf/2602.18767v3","primary_category":"cs.LO","published":"2026-02-21","tags":["theorem-proving-ml","tactic-automation","ai-agents","mathlib"],"lean_note":"Introduces a finite set of atomic tactics, an atomization algorithm, and a GNN-based prover (Nazrin) trained on Lean standard library and Mathlib proofs.","thumb":"thumbs/2602.18767.jpg"},{"id":"2602.18657","title":"DSLean: A Framework for Type-Correct Interoperability Between Lean 4 and External DSLs","authors":["Tate Rowney","Riyaz Ahuja","Jeremy Avigad","Sean Welleck"],"abstract":"Domain-specific languages (DSLs) mediate interactions between interactive proof assistants and external automation, but translating between the prover's internal representation and such DSLs is a tedious engineering chore. To simplify this task, we present DSLean, a framework for bidirectional translation between expressions in the Lean proof assistant and external syntax. DSLean requires only a specification of an external language and its Lean equivalents, abstracting away meta-level implementation details. We demonstrate DSLean's capabilities by implementing three new automation tactics, providing access to external solvers for interval arithmetic, ordinary differential equations, and ring ideal membership.","abs_url":"https://arxiv.org/abs/2602.18657","pdf_url":"https://arxiv.org/pdf/2602.18657v2","primary_category":"cs.LO","published":"2026-02-20","tags":["tooling","tactic-automation"],"lean_note":"DSLean is a Lean 4 framework for bidirectional translation between Lean expressions and external DSL syntax for solver tactics.","repo":"https://doi.org/10.5281/zenodo.50621","thumb":"thumbs/2602.18657.jpg"},{"id":"2602.18307","title":"VeriSoftBench: Repository-Scale Formal Verification Benchmarks for Lean","authors":["Yutong Xin","Qiaochu Chen","Greg Durrett","Işil Dillig"],"abstract":"Large language models have achieved striking results in interactive theorem proving, particularly in Lean. However, most benchmarks for LLM-based proof automation are drawn from mathematics in the Mathlib ecosystem, whereas proofs in software verification are developed inside definition-rich codebases with substantial project-specific libraries. We introduce VeriSoftBench, a benchmark of 500 Lean 4 proof obligations drawn from open-source formal-methods developments and packaged to preserve realistic repository context and cross-file dependencies. Our evaluation of frontier LLMs and specialized provers yields three observations. First, provers tuned for Mathlib-style mathematics transfer poorly to this repository-centric setting. Second, success is strongly correlated with transitive repository dependence: tasks whose proofs draw on large, multi-hop dependency closures are less likely to be solved. Third, providing curated context restricted to a proof's dependency closure improves performance relative to exposing the full repository, but nevertheless leaves substantial room for improvement. Our benchmark and evaluation suite are released at https://github.com/utopia-group/VeriSoftBench.","abs_url":"https://arxiv.org/abs/2602.18307","pdf_url":"https://arxiv.org/pdf/2602.18307v1","primary_category":"cs.SE","published":"2026-02-20","tags":["benchmark","verification","theorem-proving-ml"],"lean_note":"Introduces VeriSoftBench, a benchmark of 500 Lean 4 proof obligations from open-source formal-methods software repositories.","repo":"https://github.com/utopia-group/VeriSoftBench","thumb":"thumbs/2602.18307.jpg"},{"id":"2602.17064","title":"Formalization of Two Fixed-Point Algorithms in Hilbert Spaces","authors":["Yifan Bai","Yantao Li","Jian Yu","Jingwei Liang"],"abstract":"Iterative algorithms are fundamental tools for approximating fixed-points of nonexpansive operators in real Hilbert spaces. Among them, Krasnosel'skiĭ--Mann iteration and Halpern iteration are two widely used schemes. In this work, we formalize the convergence of these two fixed-point algorithms in the interactive theorem prover Lean4 based on type dependent theory. To this end, weak convergence and topological properties in the infinite-dimensional real Hilbert space are formalized. Definition and properties of nonexpansive operators are also provided. As a useful tool in convex analysis, we then formalize the Fejér monotone sequence. Building on these foundations, we verify the convergence of both the iteration schemes. Our formalization provides reusable components for machine-checked convergence analysis of fixed-point iterations and theories of convex analysis in real Hilbert spaces. Our code is available at https://github.com/TTony2019/fixed-point-iterations-in-lean.","abs_url":"https://arxiv.org/abs/2602.17064","pdf_url":"https://arxiv.org/pdf/2602.17064v1","primary_category":"math.OC","published":"2026-02-19","tags":["formalization","analysis"],"lean_note":"Formalizes convergence of Krasnoselskii-Mann and Halpern fixed-point iterations for nonexpansive operators in real Hilbert spaces in Lean 4.","repo":"https://github.com/TTony2019/fixed-point-iterations-in-lean","thumb":"thumbs/2602.17064.jpg"},{"id":"2602.17016","title":"M2F: Automated Formalization of Mathematical Literature at Scale","authors":["Zichen Wang","Wanli Ma","Zhenyu Ming","Gong Zhang","Kun Yuan","Zaiwen Wen"],"abstract":"Automated formalization of mathematics enables mechanical verification but remains limited to isolated theorems and short snippets. Scaling to textbooks and research papers is largely unaddressed, as it requires managing cross-file dependencies, resolving imports, and ensuring that entire projects compile end-to-end. We present M2F (Math-to-Formal), the first agentic framework for end-to-end, project-scale autoformalization in Lean. The framework operates in two stages. The statement compilation stage splits the document into atomic blocks, orders them via inferred dependencies, and repairs declaration skeletons until the project compiles, allowing placeholders in proofs. The proof repair stage closes these holes under fixed signatures using goal-conditioned local edits. Throughout both stages, M2F keeps the verifier in the loop, committing edits only when toolchain feedback confirms improvement. In approximately three weeks, M2F converts long-form mathematical sources into a project-scale Lean library of 153,853 lines from 479 pages textbooks on real analysis and convex analysis, fully formalized as Lean declarations with accompanying proofs. This represents textbook-scale formalization at a pace that would typically require months or years of expert effort. On FATE-H, we achieve $96\\%$ proof success (vs.\\ $80\\%$ for a strong baseline). Together, these results demonstrate that practical, large-scale automated formalization of mathematical literature is within reach. The full generated Lean code from our runs is available at https://github.com/optsuite/ReasBook.git.","abs_url":"https://arxiv.org/abs/2602.17016","pdf_url":"https://arxiv.org/pdf/2602.17016v1","primary_category":"cs.AI","published":"2026-02-19","tags":["autoformalization","ai-agents","analysis"],"lean_note":"M2F is an agentic, verifier-in-the-loop framework that autoformalizes textbook-scale mathematics into compiling Lean projects with proofs.","repo":"https://github.com/optsuite/M2F","thumb":"thumbs/2602.17016.jpg"},{"id":"2602.16554","title":"MerLean: An Agentic Framework for Autoformalization in Quantum Computation","authors":["Yuanjie Ren","Jinzheng Li","Yidi Qi"],"abstract":"We introduce MerLean, a fully automated agentic framework for autoformalization in quantum computation. MerLean extracts mathematical statements from \\LaTeX{} source files, formalizes them into verified Lean~4 code built on Mathlib, and translates the result back into human-readable \\LaTeX{} for semantic review. We evaluate MerLean on three theoretical quantum computing papers producing 2,050 Lean declarations from 114 statements in total. MerLean achieves end-to-end formalization on all three papers, reducing the verification burden to only the newly introduced definitions and axioms. Our results demonstrate that agentic autoformalization can scale to frontier research, offering both a practical tool for machine-verified peer review and a scalable engine for mining high-quality synthetic data to train future reasoning models. Our approach can also be generalized to any other rigorous research in mathematics and theoretical physics.","abs_url":"https://arxiv.org/abs/2602.16554","pdf_url":"https://arxiv.org/pdf/2602.16554v1","primary_category":"cs.LO","published":"2026-02-18","tags":["autoformalization","ai-agents","quantum"],"lean_note":"MerLean autoformalizes quantum-computing paper statements into verified Lean 4 code on Mathlib.","repo":"https://zenodo.org/records/11551307","thumb":"thumbs/2602.16554.jpg"},{"id":"2602.15965","title":"FLoPS: Semantics, Operations, and Properties of P3109 Floating-Point Representations in Lean","authors":["Tung-Che Chang","Sehyeok Park","Jay P Lim","Santosh Nagarakatte"],"abstract":"The upcoming IEEE-P3109 standard for low-precision floating-point arithmetic can become the foundation of future machine learning hardware and software. Unlike IEEE-754, P3109 introduces a parametric framework defined by bitwidth, precision, signedness, and domain. This flexibility results in a vast combinatorial space of formats -- some with as little as one bit of precision -- alongside novel features such as stochastic rounding and saturation arithmetic. These deviations create a unique verification gap that this paper intends to address. This paper presents FLoPS, Formalization in Lean of the P3109 Standard, which is a comprehensive formal model of P3109 in Lean. Our work serves as a rigorous, machine-checked specification that facilitates deep analysis of the standard. We demonstrate the model's utility by verifying foundational properties and analyzing key algorithms within the P3109 context. Specifically, we reveal that FastTwoSum exhibits a novel property of computing exact \"overflow error\" under saturation using any rounding mode, whereas previously established properties of the ExtractScalar algorithm fail for formats with one bit of precision. This work provides a verified foundation for reasoning about P3109 and enables formal verification of future numerical software. Our Lean development is open source and publicly available.","abs_url":"https://arxiv.org/abs/2602.15965","pdf_url":"https://arxiv.org/pdf/2602.15965v3","primary_category":"cs.MS","published":"2026-02-17","tags":["verification","formalization","tooling"],"lean_note":"Builds a comprehensive formal model of the IEEE-P3109 low-precision floating-point standard in Lean and verifies properties of algorithms like FastTwoSum.","thumb":"thumbs/2602.15965.jpg"},{"id":"2602.15409","title":"Hennessy-Milner Logic in CSLib, the Lean Computer Science Library","authors":["Fabrizio Montesi","Marco Peressotti","Alexandre Rademaker"],"abstract":"We present a library-level formalisation of Hennessy-Milner Logic (HML) - a foundational logic for labelled transition systems (LTSs) - for the Lean Computer Science Library (CSLib). Our development includes the syntax, satisfaction relation, and denotational semantics of HML, as well as a complete metatheory including the Hennessy-Milner theorem - bisimilarity coincides with theory equivalence for image-finite LTSs. Our development emphasises generality and reusability: it is parametric over arbitrary LTSs, definitions integrate with CSLib's infrastructure (such as the formalisation of bisimilarity), and proofs leverage Lean's automation (notably the grind tactic). All code is publicly available in CSLib and can be readily applied to systems that use its LTS API.","abs_url":"https://arxiv.org/abs/2602.15409","pdf_url":"https://arxiv.org/pdf/2602.15409v1","primary_category":"cs.LO","published":"2026-02-17","tags":["formalization","logic","mathlib"],"lean_note":"Formalizes Hennessy-Milner Logic syntax, semantics, and the Hennessy-Milner theorem in Lean's CSLib, using the grind tactic and CSLib's LTS API.","thumb":"thumbs/2602.15409.jpg"},{"id":"2602.15078","title":"Computer Science as Infrastructure: the Spine of the Lean Computer Science Library (CSLib)","authors":["Christopher Henson","Fabrizio Montesi"],"abstract":"Following in the footsteps of the success of Mathlib - the centralised library of formalised mathematics in Lean - CSLib is a rapidly-growing centralised library of formalised computer science and software. In this paper, we present its founding technical principles, operation, abstractions, and semantic framework. We contribute reusable semantic interfaces (reduction and labelled transition systems), proof automation, CI/testing support for maintaining automation and compatibility with Mathlib, and the first substantial developments of languages and models.","abs_url":"https://arxiv.org/abs/2602.15078","pdf_url":"https://arxiv.org/pdf/2602.15078v1","primary_category":"cs.LO","published":"2026-02-16","tags":["tooling","verification","mathlib"],"lean_note":"Presents CSLib, a centralized Lean library of formalized computer science with Mathlib-compatible automation.","thumb":"thumbs/2602.15078.jpg"},{"id":"2602.12891","title":"Pursuit of Truth and Beauty in Lean 4: Formally Verified Theory of Grammars, Optimization, Matroids","authors":["Martin Dvorak"],"abstract":"This thesis documents a voyage towards truth and beauty via formal verification of theorems. To this end, we develop libraries in Lean 4 that present definitions and results from diverse areas of MathematiCS (i.e., Mathematics and Computer Science). The aim is to create code that is understandable, believable, useful, and elegant. The code should stand for itself as much as possible without a need for documentation; however, this text redundantly documents our code artifacts and provides additional context that isn't present in the code. This thesis is written for readers who know Lean 4 but are not familiar with any of the topics presented. We manifest truth and beauty in three formalized areas of MathematiCS (optimization theory, matroid theory, and the theory of grammars). In the pursuit of truth, we focus on identifying the trusted code in each project and presenting it faithfully. We emphasize the readability and believability of definitions rather than choosing definitions that are easier to work with. In search for beauty, we focus on the philosophical framework of Roger Scruton, who emphasizes that beauty is not a mere decoration but, most importantly, beauty is the means for shaping our place in the world and a source of redemption, where it can be viewed as a substitute for religion.","abs_url":"https://arxiv.org/abs/2602.12891","pdf_url":"https://arxiv.org/pdf/2602.12891v3","primary_category":"cs.LO","published":"2026-02-13","tags":["formalization","combinatorics"],"lean_note":"Thesis develops Lean 4 libraries formalizing optimization theory, matroid theory, and the theory of grammars.","thumb":"thumbs/2602.12891.jpg"},{"id":"2602.12772","title":"Formalizing Gröbner Basis Theory in Lean","authors":["Junyu Guo","Hao Shen","Junqi Liu","Lihong Zhi"],"abstract":"We present a formalization of Gröbner basis theory in Lean 4, built on top of Mathlib's infrastructure for multivariate polynomials and monomial orders. Our development covers the core foundations of Gröbner basis theory, including polynomial division with remainder, Buchberger's criterion, and the existence and uniqueness of reduced Gröbner bases. We develop the theory uniformly for polynomial rings indexed by arbitrary types, enabling the treatment of Gröbner bases in rings with infinitely many variables. Furthermore, we connect the finite and infinite settings by showing that infinite-variable reduced Gröbner bases can be characterized via reduced Gröbner bases on finite-variable subrings through monomial-order embeddings and filter-based limit constructions.","abs_url":"https://arxiv.org/abs/2602.12772","pdf_url":"https://arxiv.org/pdf/2602.12772v1","primary_category":"math.AC","published":"2026-02-13","tags":["formalization","mathlib","algebra"],"lean_note":"Formalizes Gröbner basis theory in Lean 4 atop Mathlib's multivariate polynomial and monomial-order infrastructure.","repo":"https://github.com/Hagb/lean-groebner","thumb":"thumbs/2602.12772.jpg"},{"id":"2602.09464","title":"AlgoVeri: An Aligned Benchmark for Verified Code Generation on Classical Algorithms","authors":["Haoyu Zhao","Ziran Yang","Jiawei Li","Deyuan He","Zenan Li","Chi Jin","Venugopal V. Veeravalli","Aarti Gupta","Sanjeev Arora"],"abstract":"Vericoding refers to the generation of formally verified code from rigorous specifications. Recent AI models show promise in vericoding, but a unified methodology for cross-paradigm evaluation is lacking. Existing benchmarks test only individual languages/tools (e.g., Dafny, Verus, and Lean) and each covers very different tasks, so the performance numbers are not directly comparable. We address this gap with AlgoVeri, a benchmark that evaluates vericoding of $77$ classical algorithms in Dafny, Verus, and Lean. By enforcing identical functional contracts, AlgoVeri reveals critical capability gaps in verification systems. While frontier models achieve tractable success in Dafny ($40.3$% for Gemini-3 Flash), where high-level abstractions and SMT automation simplify the workflow, performance collapses under the systems-level memory constraints of Verus ($24.7$%) and the explicit proof construction required by Lean (7.8%). Beyond aggregate metrics, we uncover a sharp divergence in test-time compute dynamics: Gemini-3 effectively utilizes iterative repair to boost performance (e.g., tripling pass rates in Dafny), whereas GPT-OSS saturates early. Finally, our error analysis shows that language design affects the refinement trajectory: while Dafny allows models to focus on logical correctness, Verus and Lean trap models in persistent syntactic and semantic barriers. All data and evaluation code can be found at https://github.com/haoyuzhao123/algoveri.","abs_url":"https://arxiv.org/abs/2602.09464","pdf_url":"https://arxiv.org/pdf/2602.09464v2","primary_category":"cs.SE","published":"2026-02-10","tags":["benchmark","theorem-proving-ml","verification"],"lean_note":"Includes 77 classical algorithms with identical functional contracts in Lean (alongside Dafny and Verus) and evaluates LLM vericoding on the Lean track.","repo":"https://github.com/haoyuzhao123/algoveri","thumb":"thumbs/2602.09464.jpg"},{"id":"2602.08692","title":"PBLean: Pseudo-Boolean Proof Certificates for Lean 4","authors":["Stefan Szeider"],"abstract":"We present PBLean, a method for importing VeriPB pseudo-Boolean (PB) proof certificates into Lean 4. Key to our approach is reflection: a Boolean checker function whose soundness is fully proved in Lean and executed as compiled native code. Our method scales to proofs with tens of thousands of steps that would exhaust memory under explicit proof-term construction. Our checker supports all VeriPB kernel rules, including cutting-plane derivations, proof-by-contradiction subproofs, and redundance-based reasoning for symmetry breaking. In contrast to external verified checkers that produce verdicts, our integration yields Lean theorems that can serve as composable lemmas in larger formal developments. To derive theorems about the original combinatorial problems rather than about PB constraints alone, we support verified encodings. This closes the trust gap between solver output and problem semantics since the constraint translation and its correctness proof are both formalized in Lean. We demonstrate the approach on various combinatorial problems.","abs_url":"https://arxiv.org/abs/2602.08692","pdf_url":"https://arxiv.org/pdf/2602.08692v2","primary_category":"cs.LO","published":"2026-02-09","tags":["tooling","verification"],"lean_note":"PBLean imports VeriPB pseudo-Boolean proof certificates into Lean 4 via a soundness-proved reflective checker.","repo":"https://github.com/leansolving/pblean","thumb":"thumbs/2602.08692.jpg"},{"id":"2602.11202","title":"interwhen: A Generalizable Framework for Steering Reasoning Models with Test-time Verification","authors":["Vishak K Bhat","Prateek Chanda","Vijval Ekbote","Ashmit Khandelwal","Maitreyi Swaroop","Vineeth N. Balasubramanian","Subbarao Kambhampati","Nagarajan Natarajan","Amit Sharma"],"abstract":"Reasoning models produce long traces of intermediate decisions and tool calls, making test-time verification important for ensuring correctness. Existing approaches either verify only the final answer, which misses early errors, or rely on branch-and-verify strategies that explore multiple trajectories. We introduce interwhen, a single-trajectory verification framework that steers model behavior by providing feedback on intermediate reasoning traces. It addresses two key challenges. First, given a set of verifiers, obtaining verifiable states from the reasoning trace typically requires prompt engineering or external task decomposition into fixed steps. Instead, we propose a monitoring system that periodically polls the reasoning trace and forks inference of the reasoning model to recover intermediate states. Verifiers are run asynchronously alongside generation, adding negligible overhead on correct executions and intervening only when violations occur. Second, beyond math and code, a central challenge for process verification is the scarcity of verifiers. interwhen addresses this through automatic verifier synthesis from natural-language policy documents. Given a policy, it can generate code-based verifiers, including provably correct verifiers in Lean and z3. Together, these contributions yield a plug-and-play test-time verification system that can improve task completion and policy compliance of any reasoning agent. On reasoning benchmarks where policies encode mathematical or logical constraints, interwhen achieves near-perfect accuracy for reasoning models using a fraction of the tokens of baselines. On agentic benchmarks with policy-based verifier generation, it enables improvements in task quality for SLMs without any finetuning, e.g., task completion rate of Qwen3-30B jumps from 32% to 87% on the telecom domain in tau2-bench. Code at https://github.com/microsoft/interwhen.","abs_url":"https://arxiv.org/abs/2602.11202","pdf_url":"https://arxiv.org/pdf/2602.11202v3","primary_category":"cs.LO","published":"2026-02-05","tags":["verification","ai-agents","theorem-proving-ml","logic"],"lean_note":"Synthesizes provably correct verifiers in Lean (and z3) from natural-language policies to check intermediate reasoning states.","thumb":"thumbs/2602.11202.jpg"},{"id":"2602.05095","title":"Dead ends in square-free digit walks","authors":["Evan Chen","Chris Cummins","Ben Eltschig","Dejan Grubisic","Leopold Haller","Letong Hong","Andranik Kurghinyan","Kenny Lau","Hugh Leather","Seewoo Lee","Aram Markosyan","Ken Ono","Manooshree Patel","Gaurang Pendharkar","Vedant Rathi","Alex Schneidman","Volker Seeker","Shubho Sengupta","Ishan Sinha","Jimmy Xin","Jujian Zhang"],"abstract":"We study \"dead ends\" in square-free digit walks: square-free integers $N$ such that, in base $b$, every one-digit extension $bN+d$ is non-square-free. In base $10$, the stochastic independence model of Miller et al. suggests that infinite square-free walks occur with probability near $1$, corresponding to an asymptotic dead-end density of $\\approx 5.218\\times 10^{-5}$. We prove that the true asymptotic dead-end density satisfies \\[ c_{\\mathrm{dead}} \\approx 1.317\\times 10^{-9}, \\] roughly a factor of $\\sim 4\\times 10^4$ smaller than the prediction. For every base $b\\geq 2$, we prove that dead-end densities exist and are given by a closed-form expression (as a finite alternating sum of Euler products). The argument is fully formalized in Lean/Mathlib, and was produced automatically by AxiomProver from a natural-language statement of the problem.","abs_url":"https://arxiv.org/abs/2602.05095","pdf_url":"https://arxiv.org/pdf/2602.05095v2","primary_category":"math.CO","published":"2026-02-04","tags":["autoformalization","combinatorics","number-theory"],"lean_note":"Argument formalized in Lean/Mathlib and produced automatically by AxiomProver from a natural-language statement.","repo":"https://github.com/AxiomMath/dead-ends","thumb":"thumbs/2602.05095.jpg"},{"id":"2602.05090","title":"Almost all primes are partially regular","authors":["Evan Chen","Chris Cummins","Ben Eltschig","Dejan Grubisic","Leopold Haller","Letong Hong","Andranik Kurghinyan","Kenny Lau","Hugh Leather","Seewoo Lee","Aram Markosyan","Ken Ono","Manooshree Patel","Gaurang Pendharkar","Vedant Rathi","Alex Schneidman","Volker Seeker","Shubho Sengupta","Ishan Sinha","Jimmy Xin","Jujian Zhang"],"abstract":"For odd primes $p$, we let $K_p:=\\mathbb{Q}(ζ_p)$ be the $p$th cyclotomic field and let $ω$ denote its Teichmuller character. For $α>1/2$, we say that an odd prime $p$ is partially regular if the eigenspaces of the $p$-Sylow subgroup of $\\operatorname{Cl}(K_p)$ under the Galois action vanish for all characters $ω^{p-2k}$ with \\[ 2\\le 2k \\le \\frac{\\sqrt{p}}{(\\log p)^α}. \\] Equivalently, $p\\nmid \\operatorname{num}(B_{2k})$ throughout this range. We prove that a density-one subset of primes is partially regular in this sense. By Leopoldt reflection, this yields a partial Vandiver Theorem: for a density-one set of primes $p$, the even eigenspaces $A_p(ω^{2k})$ vanish for all even $2k$ satisfying the inequality above. This result has consequences for Kubota-Leopoldt $p$-adic $L$-functions, congruences between cusp forms and Eisenstein series, and $p$-torsion in algebraic $K$-groups. The theorem proving partial regularity for almost all $p$ is fully formalized in Lean/Mathlib and was produced automatically by AxiomProver from a natural-language statement of the conjecture.","abs_url":"https://arxiv.org/abs/2602.05090","pdf_url":"https://arxiv.org/pdf/2602.05090v1","primary_category":"math.NT","published":"2026-02-04","tags":["formalization","autoformalization","number-theory"],"lean_note":"Theorem on partial regularity of almost all primes is fully formalized in Lean/Mathlib, produced automatically by AxiomProver.","repo":"https://github.com/AxiomMath/partial-regularity","thumb":"thumbs/2602.05090.jpg"},{"id":"2602.04846","title":"CSLib: The Lean Computer Science Library","authors":["Clark Barrett","Swarat Chaudhuri","Fabrizio Montesi","Jim Grundy","Pushmeet Kohli","Leonardo de Moura","Alexandre Rademaker","Sorrachai Yingchareonthawornchai"],"abstract":"We introduce CSLib, an open-source framework for proving computer-science-related theorems and writing formally verified code in the Lean proof assistant. CSLib aims to be for computer science what Lean's Mathlib is for mathematics. Mathlib has been tremendously impactful: it is a key reason for Lean's popularity within the mathematics research community, and it has also played a critical role in the training of AI systems for mathematical reasoning. However, the base of computer science knowledge in Lean is currently quite limited. CSLib will vastly enhance this knowledge base and provide infrastructure for using this knowledge in real-world verification projects. By doing so, CSLib will (1) enable the broad use of Lean in computer science education and research, and (2) facilitate the manual and AI-aided engineering of large-scale formally verified systems.","abs_url":"https://arxiv.org/abs/2602.04846","pdf_url":"https://arxiv.org/pdf/2602.04846v1","primary_category":"cs.LO","published":"2026-02-04","tags":["tooling","mathlib","verification"],"lean_note":"Introduces CSLib, an open-source Lean computer-science library aiming to be the CS counterpart of Mathlib.","repo":"https://github.com/verse-lab/loom","thumb":"thumbs/2602.04846.jpg"},{"id":"2602.03722","title":"Parity of $k$-differentials in genus zero and one","authors":["Dawei Chen","Evan Chen","Kenny Lau","Ken Ono","Jujian Zhang"],"abstract":"Here we completely determine the spin parity of $k$-differentials with prescribed zero and pole orders on Riemann surfaces of genus zero and one. This result was previously obtained conditionally by the first author and Quentin Gendron assuming the truth of a number-theoretic hypothesis Conjecture A.10. We prove this hypothesis by reformulating it in terms of Jacobi symbols, reducing the proof to a combinatorial identity and standard facts about Jacobi symbols. The proof was obtained by AxiomProver and the system formalized the proof of the combinatorial identity in Lean/Mathlib (see the Appendix).","abs_url":"https://arxiv.org/abs/2602.03722","pdf_url":"https://arxiv.org/pdf/2602.03722v1","primary_category":"math.NT","published":"2026-02-03","tags":["autoformalization","number-theory"],"lean_note":"AxiomProver formalized the proof of a combinatorial identity in Lean/Mathlib for the spin-parity result.","repo":"https://github.com/AxiomMath/parity-differential","thumb":"thumbs/2602.03722.jpg"},{"id":"2602.03716","title":"Fel's Conjecture on Syzygies of Numerical Semigroups","authors":["Evan Chen","Chris Cummins"," GSM","Dejan Grubisic","Leopold Haller","Letong Hong","Andranik Kurghinyan","Kenny Lau","Hugh Leather","Seewoo Lee","Aram Markosyan","Ken Ono","Manooshree Patel","Gaurang Pendharkar","Vedant Rathi","Alex Schneidman","Volker Seeker","Shubho Sengupta","Ishan Sinha","Jimmy Xin","Jujian Zhang"],"abstract":"Let $S=\\langle d_1,\\dots,d_m\\rangle$ be a numerical semigroup and $k[S]$ its semigroup ring. The Hilbert numerator of $k[S]$ determines normalized alternating syzygy power sums $K_p(S)$ encoding alternating power sums of syzygy degrees. Fel conjectured an explicit formula for $K_p(S)$, for all $p\\ge 0$, in terms of the gap power sums $G_r(S)=\\sum_{g\\notin S} g^r$ and universal symmetric polynomials $T_n$ evaluated at the generator power sums $σ_k=\\sum_i d_i^k$ (and $δ_k=(σ_k-1)/2^k$). We prove Fel's conjecture via exponential generating functions and coefficient extraction, solating the universal identities for $T_n$ needed for the derivation. The argument is fully formalized in Lean/Mathlib, and was produced automatically by AxiomProver from a natural-language statement of the conjecture.","abs_url":"https://arxiv.org/abs/2602.03716","pdf_url":"https://arxiv.org/pdf/2602.03716v1","primary_category":"math.CO","published":"2026-02-03","tags":["autoformalization","combinatorics","algebra"],"lean_note":"Proof of Fel's conjecture on numerical-semigroup syzygies formalized in Lean/Mathlib via AxiomProver.","repo":"https://github.com/AxiomMath/fel-polynomial","thumb":"thumbs/2602.03716.jpg"},{"id":"2602.02990","title":"Learning to Repair Lean Proofs from Compiler Feedback","authors":["Evan Wang","Simon Chess","Daniel Lee","Siyuan Ge","Ajit Mallavarapu","Jarod Alper","Vasily Ilin"],"abstract":"As neural theorem provers become increasingly agentic, the ability to interpret and act on compiler feedback is critical. However, existing Lean datasets consist almost exclusively of correct proofs, offering little supervision for understanding and repairing failures. We study Lean proof repair as a supervised learning problem: given an erroneous proof and compiler feedback, predict both a corrected proof and a natural-language diagnosis grounded in the same feedback. We introduce APRIL (Automated Proof Repair in Lean), a dataset of 260,000 supervised tuples pairing systematically generated proof failures with compiler diagnostics and aligned repair and explanation targets. Training language models on APRIL substantially improves repair accuracy and feedback-conditioned reasoning; in our single-shot repair evaluation setting, a finetuned 4B-parameter model outperforms the strongest open-source baseline. We view diagnostic-conditioned supervision as a complementary training signal for feedback-using provers. Our dataset is available at https://huggingface.co/datasets/uw-math-ai/APRIL.","abs_url":"https://arxiv.org/abs/2602.02990","pdf_url":"https://arxiv.org/pdf/2602.02990v2","primary_category":"cs.LG","published":"2026-02-03","tags":["dataset","theorem-proving-ml","tooling"],"lean_note":"Introduces APRIL, a 260K-tuple dataset pairing erroneous Lean proofs with compiler diagnostics and aligned repair plus explanation targets.","thumb":"thumbs/2602.02990.jpg"},{"id":"2602.13247","title":"Integral Curves and Flows on Banach Manifolds in Lean","authors":["Weichen Winston Yin","Yury Kudryashov"],"abstract":"We present a formalisation of the existence and uniqueness theorems of integral curves of vector fields on Banach manifolds in the Lean theorem prover. First, we formalize properties of differential equations on Banach spaces (the Picard-Lindelöf theorem, the Grönwall inequality, and corollaries), and then transfer results to abstract Banach manifolds. Built upon the differential and integral calculus and Banach manifolds libraries in Mathlib, our work aims to lay the foundation for dynamical systems and differential geometry libraries that are general, robust, and friendly to classical mathematicians.","abs_url":"https://arxiv.org/abs/2602.13247","pdf_url":"https://arxiv.org/pdf/2602.13247v1","primary_category":"math.DG","published":"2026-02-02","tags":["formalization","mathlib","analysis"],"lean_note":"Formalizes existence and uniqueness of integral curves of vector fields on Banach manifolds in Lean atop Mathlib.","thumb":"thumbs/2602.13247.jpg"},{"id":"2602.02285","title":"Statistical Learning Theory in Lean 4: Empirical Processes from Scratch","authors":["Yuanhe Zhang","Jason D. Lee","Fanghui Liu"],"abstract":"We present the first comprehensive Lean 4 formalization of statistical learning theory (SLT) grounded in empirical process theory. Our end-to-end formal infrastructure implement the missing contents in latest Lean 4 Mathlib library, including a complete development of Gaussian Lipschitz concentration, the first formalization of Dudley's entropy integral theorem for sub-Gaussian processes, and an application to least-squares (sparse) regression with a sharp rate. The project was carried out using a human-AI collaborative workflow, in which humans design proof strategies and AI agents execute tactical proof construction, leading to the human-verified Lean 4 toolbox for SLT. Beyond implementation, the formalization process exposes and resolves implicit assumptions and missing details in standard SLT textbooks, enforcing a granular, line-by-line understanding of the theory. This work establishes a reusable formal foundation and opens the door for future developments in machine learning theory. The code is available at https://github.com/YuanheZ/lean-stat-learning-theory","abs_url":"https://arxiv.org/abs/2602.02285","pdf_url":"https://arxiv.org/pdf/2602.02285v1","primary_category":"cs.LG","published":"2026-02-02","tags":["formalization","mathlib","ml-theory"],"lean_note":"First comprehensive Lean 4 formalization of statistical learning theory grounded in empirical process theory.","repo":"https://github.com/YuanheZ/lean-stat-learning-theory","thumb":"thumbs/2602.02285.jpg"},{"id":"2602.01372","title":"Robust Sublinear Convergence Rates for Iterative Bregman Projections","authors":["Gabriel Peyré"],"abstract":"Entropic regularization provides a simple way to approximate linear programs whose constraints split into two or more tractable blocks. The resulting objectives are amenable to cyclic Kullback-Leibler (KL) Bregman projections, with Sinkhorn-type algorithms for optimal transport, matrix scaling, and barycenters as canonical examples. This paper gives a general blueprint for proving $O(1/k)$ dual convergence rate with a constant that scales only linearly in $1/γ$, where $γ$ is the entropic regularization parameter. We call such rates \"robust\", because this mild dependence on $γ$ underpins favorable complexity bounds for approximating the unregularized problem via alternating KL projections. The blueprint reduces the proof to a uniform primal bound and a dual bound for a quotient norm induced by the constraint split. To make these inputs usable, we propose two helper results, which rely on the non-expansiveness of the dual iterations in this quotient dual norm. Instantiating this blueprint for graph-structured transport yields a new flow-Sinkhorn algorithm for the Wasserstein-1 distance on graphs. It achieves $\\varepsilon$-additive accuracy on the transshipment cost in $O(p\\,\\mathrm{diameter}^3/\\varepsilon^{4})$ arithmetic operations (up to logarithmic factors), where $p$ is the number of edges. We also provide a machine-checked Lean formalization of the core blueprint and its graph-$\\mathrm{W}_1$ instantiation.","abs_url":"https://arxiv.org/abs/2602.01372","pdf_url":"https://arxiv.org/pdf/2602.01372v2","primary_category":"math.OC","published":"2026-02-01","tags":["formalization","verification","analysis","probability"],"lean_note":"Provides a machine-checked Lean formalization of the convergence-rate blueprint and its graph Wasserstein-1 instantiation.","repo":"https://github.com/gpeyre/flow-sinkhorn","thumb":"thumbs/2602.01372.jpg"},{"id":"2602.01291","title":"Construction-Verification: A Benchmark for Applied Mathematics in Lean 4","authors":["Bowen Yang","Yi Yuan","Chenyi Li","Ziyu Wang","Liangqi Li","Bo Zhang","Zhe Li","Zaiwen Wen"],"abstract":"Recent advances in large language models have demonstrated impressive capabilities in mathematical formalization. However, existing benchmarks focus on logical verification of declarative propositions, often neglecting the task of explicitly synthesizing solutions. This limitation is particularly acute in applied mathematics domains, where the goal is frequently to derive concrete values or executable algorithms rather than solely proving theorems. To address this, we introduce a Lean 4 framework that enforces a construction-verification workflow, compelling the agent to define explicit solutions before proving their correctness. We curate a comprehensive benchmark AMBER (Applied Mathematics BEnchmark for Reasoning) spanning core domains of applied mathematics, including convex analysis, optimization, numerical algebra, and high-dimensional probability. Aside from theorem proving, our benchmark features complex tasks such as evaluation, algorithm design, and representation transformation. Experiments reveal that current models face significant difficulties with these constructive tasks. Notably, we observe that general-purpose reasoning models consistently outperform specialized theorem provers. We attribute this to a degradation of instruction following capabilities in specialized models. Fine-tuning on proof corpora appears to induce ``tactical overfitting\", compromising the ability to adhere to complex constructive requirements, whereas general models retain the versatility needed for multi-task formal reasoning.","abs_url":"https://arxiv.org/abs/2602.01291","pdf_url":"https://arxiv.org/pdf/2602.01291v1","primary_category":"cs.LO","published":"2026-02-01","tags":["benchmark","theorem-proving-ml"],"lean_note":"Introduces AMBER, a Lean 4 construction-verification benchmark for applied mathematics tasks.","thumb":"thumbs/2602.01291.jpg"},{"id":"2602.02561","title":"MathlibLemma: Folklore Lemma Generation and Benchmark for Formal Mathematics","authors":["Xinyu Liu","Zixuan Xie","Amir Moeini","Claire Chen","Shuze Daniel Liu","Yu Meng","Aidong Zhang","Shangtong Zhang"],"abstract":"While the ecosystem of Lean and Mathlib has enjoyed celebrated success in formal mathematical reasoning with the help of large language models (LLMs), the absence of many folklore lemmas in Mathlib remains a persistent barrier that limits Lean's usability as an everyday tool for mathematicians like \\LaTeX{} or Maple. To address this, we introduce MathlibLemma, a modular LLM-based pipeline for automated folklore-lemma mining: the discovery, formalization, and proving of reusable intermediate facts that mathematicians often take for granted but that are not always present in formal libraries. At its core, MathlibLemma proactively mines the missing connective tissue of mathematics. The pipeline produces a verified library of folklore-style lemmas, including 1,506 Lean-checked proofs that pass a proof-bypass screen; a small curated pilot subset has also been merged into Mathlib, providing external evidence that selected outputs can meet expert library standards. Leveraging this pipeline, we further construct the MathlibLemma benchmark, a suite of 4,028 non-trivial type-checked Lean statements spanning a broad range of mathematical domains. By transforming the role of LLMs from passive consumers to active contributors, this work takes a step toward AI-assisted expansion of formal mathematical libraries.","abs_url":"https://arxiv.org/abs/2602.02561","pdf_url":"https://arxiv.org/pdf/2602.02561v3","primary_category":"cs.LO","published":"2026-01-30","tags":["benchmark","mathlib","theorem-proving-ml"],"lean_note":"Builds MathlibLemma, an LLM pipeline mining and proving folklore lemmas, with a 4028-statement Lean benchmark.","thumb":"thumbs/2602.02561.jpg"},{"id":"2601.22554","title":"LeanArchitect: Automating Blueprint Generation for Humans and AI","authors":["Thomas Zhu","Pietro Monticone","Jeremy Avigad","Sean Welleck"],"abstract":"Large-scale formalization projects in Lean rely on blueprints: structured dependency graphs linking informal mathematical exposition to formal declarations. While blueprints are central to human collaboration, existing tooling treats the informal ($\\LaTeX$) and formal (Lean) components as largely decoupled artifacts, leading to maintenance overhead and limiting integration with AI automation. We present LeanArchitect, a Lean package for extracting, managing, and exporting blueprint data directly from Lean code. LeanArchitect introduces a declarative annotation mechanism that associates formal declarations with blueprint metadata, automatically infers dependency information, and generates $\\LaTeX$ blueprint content synchronized with the Lean development. This design eliminates duplication between formal and informal representations and eases fine-grained progress tracking for both human contributors and AI-based theorem provers. We demonstrate the practicality of LeanArchitect through the automated conversion of several large existing blueprint-driven projects, and through a human--AI collaboration case study formalizing a multivariate Taylor theorem. Our results show that LeanArchitect improves maintainability, exposes latent inconsistencies in existing blueprints, and provides an effective interface for integrating AI tools into real-world formalization workflows.","abs_url":"https://arxiv.org/abs/2601.22554","pdf_url":"https://arxiv.org/pdf/2601.22554v1","primary_category":"cs.LO","published":"2026-01-30","tags":["tooling","formalization","ai-agents"],"lean_note":"A Lean package that extracts and manages blueprint dependency graphs directly from annotated Lean declarations, generating synchronized LaTeX blueprints.","repo":"https://github.com/hanwenzhu/LeanArchitect","thumb":"thumbs/2601.22554.jpg"},{"id":"2601.19327","title":"A generalization of Boppana's entropy inequality","authors":["Boon Suan Ho"],"abstract":"In recent progress on the union-closed sets conjecture, a key lemma has been Boppana's entropy inequality: $h(x^2)\\geφxh(x)$, where $φ=(1+\\sqrt5)/2$ and $h(x)=-x\\log x-(1-x)\\log(1-x)$. In this note, we prove that the generalized inequality $α_kh(x^k)\\ge x^{k-1}h(x)$, first conjectured by Yuster, holds for real $k>1$, where $α_k$ is the unique positive solution to $x(1+x)^{k-1}=1$. This implies an analogue of the union-closed sets conjecture for approximate $k$-union closed set systems. We also formalize our proof in Lean 4.","abs_url":"https://arxiv.org/abs/2601.19327","pdf_url":"https://arxiv.org/pdf/2601.19327v1","primary_category":"math.CO","published":"2026-01-27","tags":["formalization","combinatorics"],"lean_note":"Formalizes in Lean 4 a generalized Boppana entropy inequality relevant to the union-closed sets conjecture.","repo":"https://github.com/boonsuan/entropy-inequality","thumb":"thumbs/2601.19327.jpg"},{"id":"2601.18944","title":"Neural Theorem Proving for Verification Conditions: A Real-World Benchmark","authors":["Qiyuan Xu","Xiaokun Luan","Renxi Wang","Joshua Ong Jun Leang","Peixin Wang","Haonan Li","Wenda Li","Conrad Watt"],"abstract":"Theorem proving is fundamental to program verification, where the automated proof of Verification Conditions (VCs) remains a primary bottleneck. Real-world program verification frequently encounters hard VCs that existing Automated Theorem Provers (ATPs) cannot prove, leading to a critical need for extensive manual proofs that burden practical application. While Neural Theorem Proving (NTP) has achieved significant success in mathematical competitions, demonstrating the potential of machine learning approaches to formal reasoning, its application to program verification--particularly VC proving--remains largely unexplored. Despite existing work on annotation synthesis and verification-related theorem proving, no benchmark has specifically targeted this fundamental bottleneck: automated VC proving. This work introduces Neural Theorem Proving for Verification Conditions (NTP4VC), presenting the first real-world multi-language benchmark for this task. From real-world projects such as Linux and Contiki-OS kernel, our benchmark leverages industrial pipelines (Why3 and Frama-C) to generate semantically equivalent test cases across formal languages of Isabelle, Lean, and Rocq. We evaluate large language models (LLMs), both general-purpose and those fine-tuned for theorem proving, on NTP4VC. Results indicate that although LLMs show promise in VC proving, significant challenges remain for program verification, highlighting a large gap and opportunity for future research.","abs_url":"https://arxiv.org/abs/2601.18944","pdf_url":"https://arxiv.org/pdf/2601.18944v2","primary_category":"cs.AI","published":"2026-01-26","tags":["benchmark","theorem-proving-ml","verification"],"lean_note":"NTP4VC benchmark generates verification-condition proving tasks across Isabelle, Lean, and Rocq, evaluating LLMs and hammers on Lean among the three.","repo":"https://github.com/xqyww123/NTP4VC","thumb":"thumbs/2601.18944.jpg"},{"id":"2602.00101","title":"A Formal Approach to AMM Fee Mechanisms with Lean 4","authors":["Marco Dessalvi","Massimo Bartoletti","Alberto Lluch-Lafuente"],"abstract":"Decentralized Finance (DeFi) has revolutionized financial markets by enabling complex asset-exchange protocols without trusted intermediaries. Automated Market Makers (AMMs) are a central component of DeFi, providing the core functionality of swapping assets of different types at algorithmically computed exchange rates. Several mainstream AMM implementations are based on the constant-product model, which ensures that swaps preserve the product of the token reserves in the AMM -- up to a \\emph{trading fee} used to incentivize liquidity provision. Trading fees substantially complicate the economic properties of AMMs, and for this reason some AMM models abstract them away in order to simplify the analysis. However, trading fees have a non-trivial impact on users' trading strategies, making it crucial to develop refined AMM models that precisely account for their effects. We extend a foundational model of AMMs by introducing a new parameter, the trading fee $φ\\in(0,1]$, into the swap rate function. Fee amounts increase inversely proportional to $φ$. When $φ= 1$, no fee is applied and the original model is recovered. We analyze the resulting fee-adjusted model from an economic perspective. We show that several key properties of the swap rate function, including output-boundedness and monotonicity, are preserved. At the same time, other properties - most notably additivity - no longer hold. We precisely characterize this deviation by deriving a generalized form of additivity that captures the effect of swaps in the presence of trading fees. We prove that when $φ< 1$, executing a single large swap yields strictly greater profit than splitting the trade into smaller ones. Finally, we derive a closed-form solution to the arbitrage problem in the presence of trading fees and prove its uniqueness. All results are formalized and machine-checked in the Lean 4 proof assistant.","abs_url":"https://arxiv.org/abs/2602.00101","pdf_url":"https://arxiv.org/pdf/2602.00101v1","primary_category":"q-fin.MF","published":"2026-01-24","tags":["formalization","verification","economics"],"lean_note":"Formalizes and machine-checks in Lean 4 an extended AMM model with trading fees and its economic properties.","repo":"https://github.com/Uniswap/v2-periphery","thumb":"thumbs/2602.00101.jpg"},{"id":"2601.17473","title":"LeanTutor: Towards a Verified AI Mathematical Proof Tutor","authors":["Manooshree Patel","Rayna Bhattacharyya","Thomas Lu","Arnav Mehta","Niels Voss","Narges Norouzi","Gireeja Ranade"],"abstract":"This paper considers the development of an AI-based provably-correct mathematical proof tutor. While Large Language Models (LLMs) allow seamless communication in natural language, they are error prone. Theorem provers such as Lean allow for provable-correctness, but these are hard for students to learn. We present a proof-of-concept system (LeanTutor) by combining the complementary strengths of LLMs and theorem provers. LeanTutor is composed of three modules: (i) an autoformalizer/proof-checker, (ii) a next-step generator, and (iii) a natural language feedback generator. To evaluate the system, we introduce PeanoBench, a dataset of 371 Peano Arithmetic proofs in human-written natural language and formal language, derived from the Natural Numbers Game.","abs_url":"https://arxiv.org/abs/2601.17473","pdf_url":"https://arxiv.org/pdf/2601.17473v2","primary_category":"cs.LG","published":"2026-01-24","tags":["autoformalization","ai-agents","dataset","logic"],"lean_note":"An autoformalizer/proof-checker module uses Lean to verify student proof steps, paired with the PeanoBench Lean/NL proof dataset."},{"id":"2601.16172","title":"Inference-Time Diversity in RL-Trained Lean Theorem Provers: A Diagnostic Study","authors":["Zachary Burton"],"abstract":"RL-trained Lean theorem provers mode-collapse at inference time: on miniF2F-test with DeepSeek-Prover-V1.5-RL, doubling the i.i.d.\\ sampling budget from $k{=}32$ to $k{=}64$ produces zero additional solved theorems (42/244 in both cases). A fixed schedule of 15 tactic skeletons breaks this plateau and recovers a $+45%$ relative improvement at $k{=}16$ (mean $Δ= +12.3 \\pm 4.2$ theorems across $n{=}3$ seeds, sign preserved in every seed). A controlled diversity ablation rules out the prompt-diversity confound: tactic skeletons help, paraphrases match the baseline, and irrelevant Lean comments actively degrade. A leave-one-out formalization-difficulty stratification reveals a structural-content gradient across the three perturbations. The phenomenon is RL-specific: V1.5-Base proves zero theorems regardless of intervention, identifying RL as the stage that creates the proof capability which subsequently collapses; extending to two additional 7B Lean provers, RL-trained DeepSeek-Prover-V2-7B contributes $+3$ frontier solves no i.i.d.\\ baseline can reach despite a flat aggregate, while SFT-trained Goedel-Prover does not ($-10.0 \\pm 4.4$ theorems, $n{=}3$, sign preserved every seed). Inference-time structural diversity is a cheap, complementary axis for RL-trained provers, orthogonal to scaling model size or training compute.","abs_url":"https://arxiv.org/abs/2601.16172","pdf_url":"https://arxiv.org/pdf/2601.16172v2","primary_category":"cs.AI","published":"2026-01-22","tags":["theorem-proving-ml","meta-study"],"lean_note":"Diagnoses inference-time mode collapse of RL-trained Lean theorem provers on miniF2F.","thumb":"thumbs/2601.16172.jpg"},{"id":"2601.14495","title":"Hint-Based SMT Proof Reconstruction","authors":["Joshua Clune","Haniel Barbosa","Jeremy Avigad"],"abstract":"There are several paradigms for integrating interactive and automated theorem provers, combining the convenience of powerful automation with strong soundness guarantees. We introduce a new approach for reconstructing proofs found by SMT solvers which we intend to be complementary with existing techniques. Rather than verifying or replaying a full proof produced by the SMT solver, or at the other extreme, rediscovering the solver's proof from just the set of premises it uses, we explore an approach which helps guide an interactive theorem prover's internal automation by leveraging derived facts during solving, which we call hints. This makes it possible to extract more information from the SMT solver's proof without the cost of retaining a dependency on the SMT solver itself. We implement a tactic in the Lean proof assistant, called QuerySMT, which leverages hints from the cvc5 SMT solver to improve existing Lean automation. We evaluate QuerySMT's performance on relevant Lean benchmarks, compare it to other tools available in Lean relating to SMT solving, and show that the hints generated by cvc5 produce a clear improvement in existing automation's performance.","abs_url":"https://arxiv.org/abs/2601.14495","pdf_url":"https://arxiv.org/pdf/2601.14495v1","primary_category":"cs.LO","published":"2026-01-20","tags":["tactic-automation","tooling"],"lean_note":"Implements QuerySMT, a Lean tactic leveraging cvc5 SMT hints to improve Lean's internal automation.","repo":"https://github.com/JOSHCLUNE/QuerySMT","thumb":"thumbs/2601.14495.jpg"},{"id":"2601.14252","title":"Semantic Identity Compression: Zero-Error Laws, Rate-Distortion, and Neurosymbolic Necessity","authors":["Tristan Simas"],"abstract":"Symbolic systems operate over precise identities: variables denote specific objects, pointers target precise memory locations, and database keys refer to singular records. Neural embeddings generalize by compressing away semantic detail, but this compression creates collision ambiguity: multiple distinct entities can share the same representation value. Exact identity recovery requires additional information precisely when representation fibers have size greater than one. The residual cost is controlled by a single combinatorial object: the collision-fiber geometry of the representation map $π$. Let $A_π=\\max_u |π^{-1}(u)|$ be the largest collision fiber. The finite laws include a tight fixed-length converse $L \\ge \\log_2 A_π$, an exact finite-block scaling law, a pointwise adaptive budget $\\lceil \\log_2 |π^{-1}(u)|\\rceil$, and an exact fiberwise rate-distortion law for arbitrary finite sources via recoverable-mass decomposition across representation fibers. The uniform single-block formula $D^\\star(L)=\\max(0,1-2^L/a)$ appears as a closed-form special case when all mass lies on one collision block, where $a = A_π$ is the collision block size. The same fiber geometry determines query complexity and canonical structure for distinguishing families. Because this residual ambiguity is structural rather than representation-specific, symbolic identity mechanisms (handles, keys, pointers, nominal tags) are the necessary system-level complement to any non-injective semantic representation. All main results are machine-checked in Lean 4.","abs_url":"https://arxiv.org/abs/2601.14252","pdf_url":"https://arxiv.org/pdf/2601.14252v6","primary_category":"cs.IT","published":"2026-01-20","tags":["formalization","logic"],"lean_note":"Machine-checks all main semantic-identity-compression laws in Lean 4.","thumb":"thumbs/2601.14252.jpg"},{"id":"2601.14027","title":"Numina-Lean-Agent: An Open and General Agentic Reasoning System for Formal Mathematics","authors":["Junqi Liu","Zihao Zhou","Zekai Zhu","Marco Dos Santos","Weikun He","Jiawei Liu","Ran Wang","Yunzhou Xie","Junqiao Zhao","Qiufeng Wang","Lihong Zhi","Jia Li","Wenda Li"],"abstract":"Agentic systems have recently become the dominant paradigm for formal theorem proving, achieving strong performance by coordinating multiple models and tools. However, existing approaches often rely on task-specific pipelines and trained formal provers, limiting their flexibility and reproducibility. In this paper, we propose the paradigm that directly uses a general coding agent as a formal math reasoner. This paradigm is motivated by (1) A general coding agent provides a natural interface for diverse reasoning tasks beyond proving, (2) Performance can be improved by simply replacing the underlying base model, without training, and (3) MCP enables flexible extension and autonomous calling of specialized tools, avoiding complex design. Based on this paradigm, we introduce Numina-Lean-Agent, which combines Claude Code with Numina-Lean-MCP to enable autonomous interaction with Lean, retrieval of relevant theorems, informal proving and auxiliary reasoning tools. Using Claude Opus 4.5 as the base model, Numina-Lean-Agent solves all problems in Putnam 2025 (12 / 12), matching the best closed-source system. Beyond benchmark evaluation, we further demonstrate its generality by interacting with mathematicians to successfully formalize the Brascamp-Lieb theorem. We release Numina-Lean-Agent and all solutions at https://github.com/project-numina/numina-lean-agent.","abs_url":"https://arxiv.org/abs/2601.14027","pdf_url":"https://arxiv.org/pdf/2601.14027v1","primary_category":"cs.AI","published":"2026-01-20","tags":["ai-agents","theorem-proving-ml","autoformalization"],"lean_note":"Combines Claude Code with a Numina-Lean-MCP to drive autonomous interaction with Lean for formal theorem proving and formalization.","repo":"https://github.com/project-numina/numina-lean-agent","thumb":"thumbs/2601.14027.jpg"},{"id":"2601.12032","title":"Speaking to Silicon: Neural Communication with Bitcoin Mining ASICs","authors":["Francisco Angulo de Lafuente","Vladimir Veselov","Richard Goodman"],"abstract":"This definitive research memoria presents a comprehensive, mathematically verified paradigm for neural communication with Bitcoin mining Application-Specific Integrated Circuits (ASICs), integrating five complementary frameworks: thermodynamic reservoir computing, hierarchical number system theory, algorithmic analysis, network latency optimization, and machine-checked mathematical formalization. We establish that obsolete cryptocurrency mining hardware exhibits emergent computational properties enabling bidirectional information exchange between AI systems and silicon substrates. The research program demonstrates: (1) reservoir computing with NARMA-10 Normalized Root Mean Square Error (NRMSE) of 0.8661; (2) the Thermodynamic Probability Filter (TPF) achieving 92.19% theoretical energy reduction; (3) the Virtual Block Manager achieving +25% effective hashrate; and (4) hardware universality across multiple ASIC families including Antminer S9, Lucky Miner LV06, and Goldshell LB-Box. A significant contribution is the machine-checked mathematical formalization using Lean 4 and Mathlib, providing unambiguous definitions, machine-verified theorems, and reviewer-proof claims. Key theorems proven include: independence implies zero leakage, predictor beats baseline implies non-independence (the logical core of TPF), energy savings theoretical maximum, and Physical Unclonable Function (PUF) distinguishability witnesses. Vladimir Veselov's hierarchical number system theory explains why early-round information contains predictive power. This work establishes a new paradigm: treating ASICs not as passive computational substrates but as active conversational partners whose thermodynamic state encodes exploitable computational information.","abs_url":"https://arxiv.org/abs/2601.12032","pdf_url":"https://arxiv.org/pdf/2601.12032v1","primary_category":"cs.NE","published":"2026-01-17","tags":["verification","probability"],"lean_note":"Provides a Lean 4 and Mathlib formalization with machine-verified theorems about ASIC information leakage.","repo":"https://github.com/Abraxas1010/speaking-to-silicon","thumb":"thumbs/2601.12032.jpg"},{"id":"2601.11757","title":"Sequencelib: A Computational Platform for Formalizing the OEIS in Lean","authors":["Walter Moreira","Joe Stubbs"],"abstract":"The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-accessible database cataloging interesting integer sequences and associated theorems. With more than 12,000 citations, the OEIS is one of the most highly cited resources in all of theoretical mathematics. In this paper, we present Sequencelib, a project to formalize the mathematics contained within the OEIS using the Lean programming language. Sequencelib includes a library of Lean formalizations of OEIS sequences as well as metaprogramming tools for programmatically attaching OEIS metadata to Lean definitions and deriving theorems about their values. Further, we describe OEIS-LT, a highly scalable Lean server that exposes these tools via a low-latency API. Finally, using OEIS-LT and prior work of Gauthier, et al., we describe a computational pipeline that formalized more than 25,000 sequences from the OEIS and proved more than 1.6 million theorems about their values. Our method makes use of a transpiler, available in OEIS-LT, that is capable of translating a subset of Standard ML to Lean, together with a set of performance improvement transformations and proofs of correctness.","abs_url":"https://arxiv.org/abs/2601.11757","pdf_url":"https://arxiv.org/pdf/2601.11757v1","primary_category":"cs.LO","published":"2026-01-16","tags":["formalization","tooling","number-theory","dataset"],"lean_note":"Formalizes OEIS integer sequences in Lean with metaprogramming tools and a Lean tool server, formalizing 25,000+ sequences and proving 1.6M+ value theorems.","thumb":"thumbs/2601.11757.jpg"},{"id":"2601.07444","title":"Formalization of Amicable Numbers Theory","authors":["Zhipeng Chen","Haolun Tang","Jingyi Zhan"],"abstract":"This paper presents a formalization of the theory of amicable numbers in the Lean~4 proof assistant. Two positive integers $m$ and $n$ are called an amicable pair if the sum of proper divisors of $m$ equals $n$ and the sum of proper divisors of $n$ equals $m$. Our formalization introduces the proper divisor sum function $\\propersum(n) = σ(n) - n$, defines the concepts of amicable pairs and amicable numbers, and computationally verifies historically famous amicable pairs. Furthermore, we formalize basic structural theorems, including symmetry, non-triviality, and connections to abundant/deficient numbers. A key contribution is the complete formal proof of the classical Thābit formula (9th century), using index-shifting and the \\texttt{zify} tactic. Additionally, we provide complete formal proofs of both Thābit's rule and Euler's generalized rule (1747), two fundamental theorems for generating amicable pairs. A major achievement is the first complete formalization of the Borho-Hoffmann breeding method (1986), comprising 540 lines with 33 theorems and leveraging automated algebra tactics (\\texttt{zify} and \\texttt{ring}) to verify complex polynomial identities. We also formalize extensions including sociable numbers (aliquot cycles), betrothed numbers (quasi-amicable pairs), parity constraint theorems, and computational search bounds for coprime pairs ($>10^{65}$). We verify the smallest sociable cycle of length 5 (Poulet's cycle) and computationally verify specific instances. The formalization comprises 2076 lines of Lean code organized into Mathlib-candidate and paper-specific modules, with 139 theorems and all necessary infrastructure for divisor sum multiplicativity and coprimality reasoning.","abs_url":"https://arxiv.org/abs/2601.07444","pdf_url":"https://arxiv.org/pdf/2601.07444v1","primary_category":"cs.LO","published":"2026-01-12","tags":["formalization","number-theory"],"lean_note":"Formalizes the theory of amicable numbers in Lean 4, including Thabit's rule, Euler's rule, and the Borho-Hoffmann method.","repo":"https://github.com/chainstart/amicable-numbers","thumb":"thumbs/2601.07444.jpg"},{"id":"2601.07421","title":"Resolution of Erdős Problem #728: a writeup of Aristotle's Lean proof","authors":["Nat Sothanaphan"],"abstract":"We provide a writeup of a resolution of Erdős Problem #728; this is the first Erdős problem (a problem proposed by Paul Erdős which has been collected in the Erdős Problems website) regarded as fully resolved autonomously by an AI system. The system in question is a combination of GPT-5.2 Pro by OpenAI and Aristotle by Harmonic, operated by Kevin Barreto. The final result of the system is a formal proof written in Lean, which we translate to informal mathematics in the present writeup for wider accessibility. The proved result is as follows. We show a logarithmic-gap phenomenon regarding factorial divisibility: For any constants $0<C_1<C_2$ and $0 < \\varepsilon < 1/2$ there exist infinitely many triples $(a,b,n)\\in\\mathbb N^3$ with $\\varepsilon n \\le a,b \\le (1-\\varepsilon)n$ such that \\[ a!\\,b!\\mid n!\\,(a+b-n)!\\qquad\\text{and}\\qquad C_1\\log n < a+b-n < C_2\\log n. \\] The argument reduces this to a binomial divisibility $\\binom{m+k}{k}\\mid\\binom{2m}{m}$ and studies it prime-by-prime. By Kummer's theorem, $ν_p\\binom{2m}{m}$ translates into a carry count for doubling $m$ in base $p$. We then employ a counting argument to find, in each scale $[M,2M]$, an integer $m$ whose base-$p$ expansions simultaneously force many carries when doubling $m$, for every prime $p\\le 2k$, while avoiding the rare event that one of $m+1,\\dots,m+k$ is divisible by an unusually high power of $p$. These \"carry-rich but spike-free\" choices of $m$ force the needed $p$-adic inequalities and the divisibility. The overall strategy is similar to results regarding divisors of $\\binom{2n}{n}$ studied earlier by Erdős and by Pomerance.","abs_url":"https://arxiv.org/abs/2601.07421","pdf_url":"https://arxiv.org/pdf/2601.07421v5","primary_category":"math.NT","published":"2026-01-12","tags":["theorem-proving-ml","ai-agents","number-theory"],"lean_note":"Writes up, in informal mathematics, a Lean proof produced autonomously by an AI system that resolved Erdős Problem #728.","repo":"https://github.com/plby/lean-proofs","thumb":"thumbs/2601.07421.jpg"},{"id":"2601.07175","title":"Optimal Equivariant Matchings on the 6-Cube with an Application to the King Wen Sequence","authors":["Alejandro Radisic"],"abstract":"We study equivariant perfect matchings on the Boolean hypercube $\\B^6$ under the Klein four-group $K_4 = \\langle \\comp, \\rev \\rangle$ generated by bitwise complement and reversal. Among matchings using only $\\comp$ or $\\rev$ pairings, there is a unique Hamming-cost minimizer, given by a simple ``reverse-priority rule'': pair each element with its reversal unless it is a palindrome, in which case pair it with its complement. This matching has total Hamming cost 120, compared to 192 for the complement-only matching. The historically significant King Wen sequence of the I Ching realizes precisely this matching. Pure Hamming minimization over the full $K_4$ action is different: allowing $\\comp \\circ \\rev$ lowers the cost to 96. The King Wen rule is recovered, however, as the unique Hamming-weight-preserving optimum: it minimizes failures of Hamming-weight preservation before Hamming distance, and it is stable for the weighted energy $α|Δw|+βd_H$ throughout the open region $α>β$. The finite orbit counts and case distinctions are checked in Lean~4.","abs_url":"https://arxiv.org/abs/2601.07175","pdf_url":"https://arxiv.org/pdf/2601.07175v3","primary_category":"math.GM","published":"2026-01-12","tags":["formalization","combinatorics"],"lean_note":"Checks finite orbit counts and case distinctions for equivariant matchings on the 6-cube in Lean 4.","thumb":"thumbs/2601.07175.jpg"},{"id":"2601.06795","title":"GDEPO: Group Dual-dynamic and Equal-right Advantage Policy Optimization with Enhanced Training Data Utilization for Sample-Constrained Reinforcement Learning","authors":["Zhengqing Yan","Xinyang Liu","Yi Zhang","Fan Guo","ChengXun Jia","Junchen Wan","Yao Liu","Qi Liu","Jihao Huang","Kang Song"],"abstract":"Automated Theorem Proving (ATP) represents a fundamental challenge in Artificial Intelligence (AI), requiring the construction of machine-verifiable proofs in formal languages such as Lean to evaluate AI reasoning capabilities. Reinforcement learning (RL), particularly the high-performance Group Relative Policy Optimization (GRPO) algorithm, has emerged as a mainstream approach for this task. However, in ATP scenarios, GRPO faces two critical issues: when composite rewards are used, its relative advantage estimation may conflict with the binary feedback from the formal verifier; meanwhile, its static sampling strategy may discard entire batches of data if no valid proof is found, resulting in zero contribution to model updates and significant data waste. To address these limitations, we propose Group Dual-dynamic and Equal-right-advantage Policy Optimization (GDEPO), a method incorporating three core mechanisms: 1) dynamic additional sampling, which resamples invalid batches until a valid proof is discovered; 2) equal-right advantage, decoupling the sign of the advantage function (based on correctness) from its magnitude (modulated by auxiliary rewards) to ensure stable and correct policy updates; and 3) dynamic additional iterations, applying extra gradient steps to initially failed but eventually successful samples to accelerate learning on challenging cases. Experiments conducted on three datasets of varying difficulty (MinF2F-test, MathOlympiadBench, PutnamBench) confirm the effectiveness of GDEPO, while ablation studies validate the necessity of its synergistic components. The proposed method enhances data utilization and optimization efficiency, offering a novel training paradigm for ATP.","abs_url":"https://arxiv.org/abs/2601.06795","pdf_url":"https://arxiv.org/pdf/2601.06795v3","primary_category":"cs.AI","published":"2026-01-11","tags":["theorem-proving-ml","benchmark"],"lean_note":"Trains an LLM theorem prover with RL using the Lean verifier's binary correctness signal to gate sampling and decouple advantage sign from auxiliary rewards.","thumb":"thumbs/2601.06795.jpg"},{"id":"2601.06692","title":"The Axiom of Consent: Friction Dynamics in Multi-Agent Coordination","authors":["Murad Farzulla"],"abstract":"Multi-agent systems must coordinate despite heterogeneous preferences, asymmetric stakes, and imperfect information. When coordination fails, friction emerges: measurable resistance such as deadlock, thrashing, or conflict. We derive a formal framework for coordination friction from a single axiom: actions affecting agents require their authorization in proportion to stakes. From this axiom of consent we establish the kernel triple $(α, σ, \\varepsilon)$ -- alignment, stake, and entropy -- as sufficient statistics for any resource-allocation configuration, and propose a friction functional whose simplest candidate form $F = σ(1+\\varepsilon)/(1+α)$ predicts that friction rises with stakes and entropy and falls with alignment. We stress that this form is a phenomenological ansatz, not a theorem -- the simplest expression satisfying our desiderata -- whose empirical adequacy, in particular whether the alignment dependence is monotone, remains open. A companion study tests it in a multi-agent reinforcement-learning environment, finds the linear alignment dependence falsified by a U-shaped relationship, and motivates a quadratic form $F = σ(1+\\varepsilon)/(1+α^2)$ that we characterize axiomatically as a refinement for future confirmation. The Replicator-Optimization Mechanism governs selection over coordination strategies: lower-friction configurations persist longer, making consent-respecting arrangements dynamical attractors rather than normative ideals. We give formal definitions for resource consent, coordination legitimacy, and friction-aware allocation, a measurement apparatus, and machine-checked Lean 4 proofs of the core comparative-statics. Illustrative applications to cryptocurrency governance and political legitimacy show one architecture spanning domains, offered as candidate unification, not established identity.","abs_url":"https://arxiv.org/abs/2601.06692","pdf_url":"https://arxiv.org/pdf/2601.06692v2","primary_category":"cs.MA","published":"2026-01-10","tags":["formalization","economics"],"lean_note":"Provides machine-checked Lean 4 proofs of the core comparative-statics of a consent-based coordination-friction framework.","repo":"https://github.com/studiofarzulla/lean-formalizations","thumb":"thumbs/2601.06692.jpg"},{"id":"2601.06363","title":"The Replicator-Optimization Mechanism: A Scale-Relative Formalism for Persistence-Conditioned Dynamics with Application to Consent-Based Metaethics","authors":["Murad Farzulla"],"abstract":"This paper formalizes a widely used dynamical class--replicator-mutator dynamics and Price-style selection-and-transmission--and makes explicit the modeling choices (scale, atomic unit, interaction topology, transmission kernel) that determine how this class instantiates across domains. The backbone is known; we do not claim to have discovered selection. The novel contributions are threefold: (i) a scale-relative kernel parameterization where atomic units are themselves parameters, enabling systematic instantiation across physics, biology, economics, cognition, and social organization; (ii) a consent-friction instantiation for political philosophy, where friction is the primitive, legitimacy functions as survival probability, and belief-transfer functions as mutation kernel; and (iii) a derivation path from social contract theory rather than from biology or physics, arriving at the same formal structure via an independent route. We provide a bridge principle connecting descriptive dynamics to instrumental normativity: if agents prefer lower expected friction, then \"ought\" claims are shorthand for policies that reduce expected friction under the specified dynamics. This conditional structure avoids the is-ought fallacy while grounding normative discourse in empirically tractable dynamics. We address pathological cases (authoritarian stability, suppressed friction) through explicit modeling of latent versus observed friction. The framework generates testable predictions through operationalization of friction, legitimacy, and belief-transfer dynamics, and is falsifiable at the level of measurement apparatus rather than formal structure.","abs_url":"https://arxiv.org/abs/2601.06363","pdf_url":"https://arxiv.org/pdf/2601.06363v2","primary_category":"econ.TH","published":"2026-01-10","tags":["formalization","economics"],"lean_note":"Includes machine-checked Lean proofs of core algebraic results for a scale-relative replicator-optimization formalism.","repo":"https://github.com/studiofarzulla/consent-rom-empirical","thumb":"thumbs/2601.06363.jpg"},{"id":"2601.03768","title":"Agentic Proof Automation: A Case Study","authors":["Yichen Xu","Martin Odersky"],"abstract":"Proof engineering is notoriously labor-intensive: proofs that are straightforward on paper often require lengthy scripts in theorem provers. Recent advances in large language models (LLMs) create new opportunities for proof automation: modern LLMs not only generate proof scripts, but also support agentic behavior, exploring codebases and iteratively refining their outputs against prover feedback. These advances enable an emerging scheme where LLM-based agents undertake most proof engineering under human guidance. Humans provide mathematical insight (definitions, theorems, proof strategies); agents handle the mechanical work of proof development. We call this scheme agentic proof automation. We present this scheme through a case study: mechanizing the semantic type soundness of a sophisticated formal system, System Capless, in Lean 4, comprising over 14,000 lines of code. Using off-the-shelf LLM agents with a single lightweight proof-checking tool, the agents completed 189 proof engineering tasks with an 87% success rate, only 16% requiring human intervention. The case study demonstrates that agents are capable proof engineers that substantially boost productivity, though they fall short in creative reasoning and still require human guidance in certain cases. We release an interactive explorer where readers can examine all agent interactions; the mechanization is open-sourced for experiments and extensions.","abs_url":"https://arxiv.org/abs/2601.03768","pdf_url":"https://arxiv.org/pdf/2601.03768v1","primary_category":"cs.PL","published":"2026-01-07","tags":["verification","ai-agents","type-theory"],"lean_note":"Case study mechanizing System Capless type soundness in Lean 4 via agentic LLM proof automation.","repo":"https://github.com/linyxus/apa-explorer","thumb":"thumbs/2601.03768.jpg"},{"id":"2601.01255","title":"A Blueprint for the Formalization of Seymour's Matroid Decomposition Theorem","authors":["Ivan Sergeev","Martin Dvorak","Cameron Rampell","Mark Sandey","Pietro Monticone"],"abstract":"This document is a blueprint for the formalization in Lean of the structural theory of regular matroids underlying Seymour's decomposition theorem. We present a modular account of regularity via totally unimodular representations, show that regularity is preserved under $1$-, $2$-, and $3$-sums, and establish regularity for several special classes of matroids, including graphic, cographic, and the matroid $R_{10}$. The blueprint records the logical structure of the proof, the precise dependencies between results, and their correspondence with Lean declarations. It is intended both as a guide for the ongoing formalization effort and as a human-readable reference for the organization of the proof.","abs_url":"https://arxiv.org/abs/2601.01255","pdf_url":"https://arxiv.org/pdf/2601.01255v1","primary_category":"math.CO","published":"2026-01-03","tags":["formalization","combinatorics","mathlib"],"lean_note":"A blueprint mapping the structural theory of regular matroids and Seymour's decomposition theorem to Lean declarations for an ongoing formalization.","thumb":"thumbs/2601.01255.jpg"},{"id":"2601.00995","title":"Grain Theory: Type-Level Granularity Correctness in Data Pipelines","authors":["Nikos Karayannidis"],"abstract":"Data transformation correctness is a fundamental challenge in data engineering: how can we verify that pipelines produce correct results before executing on production data? Existing practice relies on iterative testing over materialized data. A common cause of errors is the absence of formal reasoning about grain -- the level of detail of data -- so transformations inadvertently change granularity, yielding pathologies like fan traps (metric duplication) and chasm traps (data loss). We introduce grain theory, a type-theoretic framework that elevates grain to a composable property of any algebraic data type. It has two phases. First, a denotation of data: grain itself, defined by irreducibility and isomorphism, with no reference to functional dependencies; three grain relations forming a bounded lattice whose axioms recover Armstrong's on product types; the entity key as a derived grain; and grain-determined behavioral classes -- together the type-level triple (G[R], EK[R], BC[R]). Second, a denotation of transformations: every transformation $h$ has a grain lift $\\varphi(h)$. For collections of product types under the relational algebra we prove an equi-join grain inference theorem and present CalcG, a decidable algorithm that composes grain lifts across a pipeline DAG. The central theorem -- the grain homomorphism -- ties the phases together: grain projection commutes with transformation, and grain lifts compose ($\\varphi(h_2 \\circ h_1) = \\varphi(h_2) \\circ \\varphi(h_1)$). Grain-correctness is therefore verifiable at design time, before any code or query runs. As corollaries, fan traps emerge as schema-detectable grain-relation violations; chasm traps localize to a specific ordering-chain pattern; and behavioral-class violations, such as point-in-time queries on the wrong collection type, become compile-time type errors. All theorems are mechanically verified in Lean 4.","abs_url":"https://arxiv.org/abs/2601.00995","pdf_url":"https://arxiv.org/pdf/2601.00995v2","primary_category":"cs.DB","published":"2026-01-02","tags":["verification","type-theory"],"lean_note":"Mechanically verifies grain theory's homomorphism and grain-lift composition theorems in Lean 4.","repo":"https://github.com/nkarag/grain-theory-artifacts","thumb":"thumbs/2601.00995.jpg"},{"id":"2512.24796","title":"LeanCat: A Benchmark Suite for Formal Category Theory in Lean (Part I: 1-Categories)","authors":["Rongge Xu","Hui Dai","Yiming Fu","Jiedong Jiang","Tianjiao Nie","Junkai Wang","Holiverse Yang","Zhi-Hao Zhang"],"abstract":"While large language models (LLMs) have demonstrated impressive capabilities in formal theorem proving, current benchmarks fail to adequately measure library-grounded abstraction -- the ability to reason with high-level interfaces and reusable structures central to modern mathematics and software engineering. We introduce LeanCat, a challenging benchmark comprising 100 fully formalized category-theory tasks in Lean. Unlike algebra or arithmetic, category theory serves as a rigorous stress test for structural, interface-level reasoning. Our evaluation reveals a severe abstraction gap: the best state-of-the-art model solves only 12.0% of tasks at pass@4, with performance collapsing from 55.0% on Easy tasks to 0.0% on High-difficulty tasks, highlighting a failure in compositional generalization. To overcome this, we evaluate LeanBridge, a retrieval-augmented agent that employs a retrieve-generate-verify loop. LeanBridge achieves a peak success rate of 24.0% -- doubling the performance of the best static baseline. These results empirically demonstrate that iterative refinement and dynamic library retrieval are not merely optimizations but strict necessities for neuro-symbolic reasoning in abstract domains. LeanCat offers a compact, reusable testbed for tracking progress toward reliable, research-level formalization.","abs_url":"https://arxiv.org/abs/2512.24796","pdf_url":"https://arxiv.org/pdf/2512.24796v2","primary_category":"cs.LO","published":"2025-12-31","tags":["benchmark","theorem-proving-ml","category-theory","ai-agents"],"lean_note":"Introduces LeanCat, 100 fully formalized category-theory theorem tasks in Lean 4 used to benchmark LLM provers.","repo":"https://github.com/sciencraft/LeanCat","thumb":"thumbs/2512.24796.jpg"},{"id":"2512.21137","title":"Declarative distributed algorithms as axiomatic theories in three-valued modal logic over semitopologies","authors":["Murdoch J. Gabbay"],"abstract":"We illustrate how to formally specify distributed algorithms as declarative axiomatic theories in a modal logic, using as illustrative examples a simple voting protocol, a simple broadcast protocol (Bracha Broadcast), and a simple agreement protocol (Crusader Agreement). The methods scale well and have been used to find errors in a proposed industrial protocol. The key novelty is to use modal logic to capture a declarative, high-level representation of essential system properties -- the logical essence of the algorithm -- while abstracting away from explicit state transitions of an abstract machine that implements it. It is like the difference between specifying code in a functional or logic programming language, versus specifying code in an imperative one. Thus we present axiomatisations of Declarative Bracha Broacast and Declarative Crusader Agreement. A logical axiomatisation in the style we propose provides a precise, compact, human-readable specification that abstractly captures essential system properties, while eliding low-level implementation details; it is more precise than a natural language description, yet more abstract than source code or a logical specification thereof. This creates new opportunities for reasoning about correctness, resilience, and failure, and could serve as a foundation for human- and machine verification efforts, design improvements, and even alternative protocol implementations. The proofs in this paper have been formalised in Lean 4.","abs_url":"https://arxiv.org/abs/2512.21137","pdf_url":"https://arxiv.org/pdf/2512.21137v3","primary_category":"cs.LO","published":"2025-12-24","tags":["verification","logic","formalization"],"lean_note":"Correctness proofs for declarative axiomatic specifications of distributed protocols are formalised in Lean 4.","repo":"https://github.com/XRPLF/rippled","thumb":"thumbs/2512.21137.jpg"},{"id":"2512.17260","title":"Seed-Prover 1.5: Mastering Undergraduate-Level Theorem Proving via Learning from Experience","authors":["Jiangjie Chen","Wenxiang Chen","Jiacheng Du","Jinyi Hu","Zhicheng Jiang","Allan Jie","Xiaoran Jin","Xing Jin","Chenggang Li","Wenlei Shi","Zhihong Wang","Mingxuan Wang","Chenrui Wei","Shufa Wei","Huajian Xin","Fan Yang","Weihao Gao","Zheng Yuan","Tianyang Zhan","Zeyu Zheng","Tianxi Zhou","Thomas Hanwen Zhu"],"abstract":"Large language models have recently made significant progress to generate rigorous mathematical proofs. In contrast, utilizing LLMs for theorem proving in formal languages (such as Lean) remains challenging and computationally expensive, particularly when addressing problems at the undergraduate level and beyond. In this work, we present \\textbf{Seed-Prover 1.5}, a formal theorem-proving model trained via large-scale agentic reinforcement learning, alongside an efficient test-time scaling (TTS) workflow. Through extensive interactions with Lean and other tools, the model continuously accumulates experience during the RL process, substantially enhancing the capability and efficiency of formal theorem proving. Furthermore, leveraging recent advancements in natural language proving, our TTS workflow efficiently bridges the gap between natural and formal languages. Compared to state-of-the-art methods, Seed-Prover 1.5 achieves superior performance with a smaller compute budget. It solves \\textbf{88\\% of PutnamBench} (undergraduate-level), \\textbf{80\\% of Fate-H} (graduate-level), and \\textbf{33\\% of Fate-X} (PhD-level) problems. Notably, using our system, we solved \\textbf{11 out of 12 problems} from Putnam 2025 within 9 hours. Our findings suggest that scaling learning from experience, driven by high-quality formal feedback, holds immense potential for the future of formal mathematical reasoning.","abs_url":"https://arxiv.org/abs/2512.17260","pdf_url":"https://arxiv.org/pdf/2512.17260v1","primary_category":"cs.CL","published":"2025-12-19","tags":["theorem-proving-ml","benchmark","ai-agents"],"lean_note":"Trains a formal theorem-proving model via agentic reinforcement learning that interacts with Lean to prove undergraduate-and-beyond problems.","repo":"https://github.com/ByteDance-Seed/Seed-Prover","thumb":"thumbs/2512.17260.jpg"},{"id":"2601.15294","title":"KnowTeX: Visualizing Mathematical Dependencies","authors":["Elif Uskuplu","Lawrence S. Moss","Valeria de Paiva"],"abstract":"Mathematical knowledge exists in many forms, ranging from informal textbooks and lecture notes to large formal proof libraries, yet moving between these representations remains difficult. Informal texts hide dependencies, while formal systems expose every detail in ways that are not always human-readable. Dependency graphs offer a middle ground by making visible the structure of results, definitions, and proofs. We present KnowTeX, a standalone, user-friendly tool that extends the ideas of Lean's Blueprints, enabling the visualization of conceptual dependencies directly from LaTeX sources. Using a simple \"uses\" command, KnowTeX extracts relationships among statements and generates previewable graphs in DOT and TikZ formats. Applied to mathematical texts, such graphs clarify core results, support education and formalization, and provide a resource for aligning informal and formal mathematical representations. We argue that dependency graphs should become a standard feature of mathematical writing, benefiting both human readers and automated systems.","abs_url":"https://arxiv.org/abs/2601.15294","pdf_url":"https://arxiv.org/pdf/2601.15294v1","primary_category":"cs.HC","published":"2025-12-16","tags":["tooling","formalization"],"lean_note":"Presents KnowTeX, a tool extending Lean's Blueprint system to generate mathematical dependency graphs from LaTeX sources.","repo":"https://github.com/ElifUskuplu/KnowTex","thumb":"thumbs/2601.15294.jpg"},{"id":"2512.14575","title":"Extremal descendant integrals on moduli spaces of curves: An inequality discovered and proved in collaboration with AI","authors":["Johannes Schmitt"],"abstract":"For the pure $ψ$-class intersection numbers $D(\\textbf{e})=\\langle τ_{e_1} \\cdots τ_{e_n} \\rangle_g$ on the moduli space $\\overline{\\mathcal{M}}_{g,n}$ of stable curves, we determine for which choices of $\\textbf{e}=(e_1, \\ldots, e_n)$ the value of $D(\\textbf{e})$ becomes extremal. The intersection number is minimal for powers of a single $ψ$-class (i.e. all $e_i$ but one vanish), whereas maximal values are obtained for balanced vectors ($|e_i - e_j| \\leq 1$ for all $i,j$). The proof uses the nefness of the $ψ$-classes combined with Khovanskii--Teissier log-concavity. Apart from the mathematical content, this paper is also meant as an experiment in collaborations between human mathematicians and AI models: the proof of the above result was found and formulated by the AI models GPT-5 and Gemini 3 Pro. Large parts of the paper were drafted by Claude Opus 4.5, and a part of the argument was formalized in Lean with the help of Claude Code and GPT-5.2. The paper aims for maximal transparency on the authorship of different sections and the employed AI tools (including prompts and conversation logs).","abs_url":"https://arxiv.org/abs/2512.14575","pdf_url":"https://arxiv.org/pdf/2512.14575v1","primary_category":"math.AG","published":"2025-12-16","tags":["formalization","algebra","combinatorics","geometry"],"lean_note":"A combinatorial optimization theorem underlying the extremal-descendant result was formalized and machine-checked in Lean 4.","repo":"https://github.com/seewoo5/awesome-ai-for-math","thumb":"thumbs/2512.14575.jpg"},{"id":"2512.14252","title":"Gödel's Poetry","authors":["Kelly J. Davis"],"abstract":"Formal, automated theorem proving has long been viewed as a challenge to artificial intelligence. We introduce here a new approach to computer theorem proving, one that employs specialized language models for Lean4 proof generation combined with recursive decomposition of difficult theorems into simpler entailing propositions. These models are coordinated through a multi-agent architecture that orchestrates autoformalization (if required), proof generation, decomposition of difficult theorems into simpler entailing propositions, and recursive proof (and/or decomposition) of these propositions. Without decomposition, we achieve a 90.4% pass rate on miniF2F. With decomposition, this is significantly improved. A key technical contribution lies in our extension of the Kimina Lean Server with abstract syntax tree (AST) parsing capabilities to facilitate automated, recursive proof decomposition. The system is made available on PyPI as goedels-poetry (at https://pypi.org/project/goedels-poetry ), and the open-source implementation KellyJDavis/goedels-poetry (at https://github.com/KellyJDavis/goedels-poetry ) facilitates both adaptation to alternative language models and extension with custom functionality.","abs_url":"https://arxiv.org/abs/2512.14252","pdf_url":"https://arxiv.org/pdf/2512.14252v1","primary_category":"cs.AI","published":"2025-12-16","tags":["theorem-proving-ml","autoformalization","ai-agents","tooling"],"lean_note":"Uses specialized language models for Lean4 proof generation with recursive theorem decomposition via an AST-extended Kimina Lean Server.","repo":"https://github.com/KellyJDavis/goedels-poetry","thumb":"thumbs/2512.14252.jpg"},{"id":"2512.13999","title":"A verified implementation of the Misra and Gries edge coloring algorithm","authors":["Arohee Bhoja"],"abstract":"Vizing's theorem states that every simple undirected graph can be edge-colored using fewer than $Δ+ 1$ colors, where $Δ$ is the graph's maximum degree. The original proof was given through a polynomial-time algorithmic procedure that iteratively extends a partial coloring until it becomes complete. In this work, I used the Lean theorem prover to produce a verified implementation of the Misra and Gries edge-coloring algorithm, a modified version of Vizing's original method. The focus is on building libraries for relevant mathematical objects and rigorously maintaining required invariants.","abs_url":"https://arxiv.org/abs/2512.13999","pdf_url":"https://arxiv.org/pdf/2512.13999v1","primary_category":"cs.DM","published":"2025-12-16","tags":["verification","formalization","combinatorics"],"lean_note":"Uses the Lean theorem prover to build a verified implementation of the Misra-Gries edge-coloring algorithm with maintained invariants.","thumb":"thumbs/2512.13999.jpg"},{"id":"2512.10317","title":"Translating Informal Proofs into Formal Proofs Using a Chain of States","authors":["Ziyu Wang","Bowen Yang","Chenyi Li","Yuan Zhang","Shihao Zhou","Bin Dong","Zaiwen Wen"],"abstract":"We address the problem of translating informal mathematical proofs expressed in natural language into formal proofs in Lean4 under a constrained computational budget. Our approach is grounded in two key insights. First, informal proofs tend to proceed via a sequence of logical transitions - often implications or equivalences - without explicitly specifying intermediate results or auxiliary lemmas. In contrast, formal systems like Lean require an explicit representation of each proof state and the tactics that connect them. Second, each informal reasoning step can be viewed as an abstract transformation between proof states, but identifying the corresponding formal tactics often requires nontrivial domain knowledge and precise control over proof context. To bridge this gap, we propose a two stage framework. Rather than generating formal tactics directly, we first extract a Chain of States (CoS), a sequence of intermediate formal proof states aligned with the logical structure of the informal argument. We then generate tactics to transition between adjacent states in the CoS, thereby constructing the full formal proof. This intermediate representation significantly reduces the complexity of tactic generation and improves alignment with informal reasoning patterns. We build dedicated datasets and benchmarks for training and evaluation, and introduce an interactive framework to support tactic generation from formal states. Empirical results show that our method substantially outperforms existing baselines, achieving higher proof success rates.","abs_url":"https://arxiv.org/abs/2512.10317","pdf_url":"https://arxiv.org/pdf/2512.10317v2","primary_category":"cs.LO","published":"2025-12-11","tags":["autoformalization","theorem-proving-ml","tactic-automation","benchmark"],"lean_note":"Translates informal proofs into Lean 4 via an intermediate Chain of States representation, with custom datasets and benchmarks.","thumb":"thumbs/2512.10317.jpg"},{"id":"2512.09280","title":"A Modular Lean 4 Framework for Confluence and Strong Normalization of Lambda Calculi with Products and Sums","authors":["Arthur Ramos","Anjolina Oliveira","Ruy de Queiroz","Tiago de Veras"],"abstract":"We present Metatheory, a comprehensive library for programming language foundations in Lean 4. The library features a modular framework for proving confluence of abstract rewriting systems using three classical proof techniques: the diamond property, Newmans lemma, and the Hindley-Rosen lemma. These are instantiated across six case studies including untyped lambda calculus, combinatory logic, term rewriting, simply typed lambda calculus, and STLC with products and sums. All theorems are fully mechanized with zero axioms or sorry statements. We provide complete proofs of de Bruijn substitution infrastructure and demonstrate strong normalization via logical relations. To our knowledge, this is the first comprehensive confluence and normalization framework for Lean 4.","abs_url":"https://arxiv.org/abs/2512.09280","pdf_url":"https://arxiv.org/pdf/2512.09280v1","primary_category":"cs.LO","published":"2025-12-10","tags":["verification","type-theory","tooling","logic"],"lean_note":"Presents Metatheory, a Lean 4 library mechanizing confluence and strong normalization for lambda calculi with products and sums.","repo":"https://github.com/arthuraa/metatheory","thumb":"thumbs/2512.09280.jpg"},{"id":"2512.07766","title":"Formalized Hopfield Networks and Boltzmann Machines","authors":["Matteo Cipollina","Michail Karatarakis","Freek Wiedijk"],"abstract":"Neural networks are widely used, yet their analysis and verification remain challenging. In this work, we present a Lean 4 formalization of neural networks, covering both deterministic and stochastic models. We first formalize Hopfield networks, recurrent networks that store patterns as stable states. We prove convergence and the correctness of Hebbian learning, a training rule that updates network parameters to encode patterns, here limited to the case of pairwise-orthogonal patterns. We then consider stochastic networks, where updates are probabilistic and convergence is to a stationary distribution. As a canonical example, we formalize the dynamics of Boltzmann machines and prove their ergodicity, showing convergence to a unique stationary distribution using a new formalization of the Perron-Frobenius theorem.","abs_url":"https://arxiv.org/abs/2512.07766","pdf_url":"https://arxiv.org/pdf/2512.07766v1","primary_category":"cs.LG","published":"2025-12-08","tags":["formalization","verification","ml-theory","probability"],"lean_note":"Formalizes Hopfield networks and Boltzmann machines in Lean 4, proving convergence, Hebbian-learning correctness, and ergodicity.","thumb":"thumbs/2512.07766.jpg"},{"id":"2512.07087","title":"The Equational Theories Project: Advancing Collaborative Mathematical Research at Scale","authors":["Matthew Bolan","Joachim Breitner","Jose Brox","Nicholas Carlini","Mario Carneiro","Floris van Doorn","Martin Dvorak","Andrés Goens","Aaron Hill","Harald Husum","Hernán Ibarra Mejia","Zoltan A. Kocsis","Bruno Le Floch","Amir Livne Bar-on","Lorenzo Luccioli","Douglas McNeil","Alex Meiburg","Pietro Monticone","Pace P. Nielsen","Emmanuel Osalotioman Osazuwa","Giovanni Paolini","Marco Petracci","Bernhard Reinke","David Renshaw","Marcus Rossel","Cody Roux","Jérémy Scanvic","Shreyas Srinivas","Anand Rao Tadipatri","Terence Tao","Vlad Tsyrklevich","Fernando Vaquerizo-Villar","Daniel Weber","Fan Zheng"],"abstract":"We report on the Equational Theories Project (ETP), an online collaborative pilot project to explore new ways to collaborate in mathematics with machine assistance. The project successfully determined all 22 028 942 edges of the implication graph between the 4694 simplest equational laws on magmas, by a combination of human-generated and automated proofs, all validated by the formal proof assistant language Lean. As a result of this project, several new constructions of magmas satisfying specific laws were discovered, and several auxiliary questions were also addressed, such as the effect of restricting attention to finite magmas.","abs_url":"https://arxiv.org/abs/2512.07087","pdf_url":"https://arxiv.org/pdf/2512.07087v2","primary_category":"math.RA","published":"2025-12-08","tags":["formalization","theorem-proving-ml","tactic-automation","algebra"],"lean_note":"All 22,028,942 implications between 4694 equational laws on magmas were formalized and validated in Lean with Mathlib.","repo":"https://github.com/teorth/equational_theories","thumb":"thumbs/2512.07087.jpg"},{"id":"2512.06952","title":"Resource-Bounded Type Theory: Compositional Cost Analysis via Graded Modalities","authors":["Mirco A. Mannucci","Corey Thuro"],"abstract":"We present a compositional framework for certifying resource bounds in typed programs. Terms are typed with synthesized bounds drawn from an abstract resource lattice, enabling uniform treatment of time, memory, gas, and domain-specific costs. We introduce a graded feasibility modality with co-unit and monotonicity laws. Our main result is a syntactic cost soundness theorem for the recursion-free simply-typed fragment: if a closed term has synthesized bound b under a given budget, its operational cost is bounded by b. We provide a syntactic term model in the topos of presheaves over the lattice -- where resource bounds index a cost-stratified family of definable values -- with cost extraction as a natural transformation. We prove canonical forms via reification and establish initiality of the syntactic model: it embeds uniquely into all resource-bounded models. A case study demonstrates compositional reasoning for binary search using Lean's native recursion with separate bound proofs.","abs_url":"https://arxiv.org/abs/2512.06952","pdf_url":"https://arxiv.org/pdf/2512.06952v1","primary_category":"cs.LO","published":"2025-12-07","tags":["type-theory","verification"],"lean_note":"Implements a resource-bounded binary search using Lean 4's well-founded recursion with separately proven logarithmic cost bounds.","repo":"https://github.com/CoreyThuro/RB-TT","thumb":"thumbs/2512.06952.jpg"},{"id":"2512.06393","title":"Conflict-Aware Fusion: Mitigating Logic Inertia in Large Language Models via Structured Cognitive Priors","authors":["Qiming Bao","Xiaoxuan Fu","Michael Witbrock"],"abstract":"Large language models (LLMs) achieve high accuracy on many reasoning benchmarks but remain brittle under structural perturbations of rule-based systems. We introduce a diagnostic framework with four stress tests -- redundant vs. essential rule deletion, contradictory-rule injection, logic-preserving rewrites, and multi-law stacking -- and use it to expose Logic Inertia: the tendency of generative LLMs (Qwen2/3, TinyLlama, GPT-4o, Gemma-3-4B-IT) and the encoder-only BERT baseline to persist along learned deductive trajectories under inconsistent premises. The collapse is sharp: untreated baselines fall from accuracy 1.00 on the base task to 0.00 on contradiction injection (instance-level exact match), and GPT-4o resolves only 56.0% of contradiction cases. We propose Conflict-Aware Fusion, a four-stage training pipeline that enforces verification-before-deduction as a learned structural prior: (i) SFT establishes the verification preamble; (ii) DPO sharpens the halt-on-contradiction decision boundary; (iii) Logical Invariance REgularisation (LIRE) penalises divergence between logically equivalent rule formulations via symmetric KL; (iv) Reinforcement Learning from Verification Feedback (RLVF) uses a symbolic forward-chaining engine as a deterministic oracle reward, jointly optimising invariance and sensitivity. The pipeline saturates all four primary stress tests for both 1.5B and 8B backbones. We further validate a Phase 2 extension that replaces the propositional oracle with a Lean 4 kernel, attaining 99.0% kernel agreement on the 105 classically-derivable (T) questions within a stratified 187-question Lean-translated sample (overall 71.7% across both polarities), providing a sound upgrade path to formally verified RL training. Code and benchmark: https://github.com/14H034160212/lemo","abs_url":"https://arxiv.org/abs/2512.06393","pdf_url":"https://arxiv.org/pdf/2512.06393v7","primary_category":"cs.AI","published":"2025-12-06","tags":["theorem-proving-ml","verification","logic"],"lean_note":"Uses the Lean 4 kernel as a deterministic verification oracle for reinforcement-learning reward in an LLM reasoning pipeline.","repo":"https://github.com/14H034160212/lemo","thumb":"thumbs/2512.06393.jpg"},{"id":"2512.05962","title":"Whatever Remains Must Be True: Filtering Drives Reasoning in LLMs, Shaping Diversity","authors":["Germán Kruszewski","Pierre Erbacher","Jos Rozen","Marc Dymetman"],"abstract":"Reinforcement Learning (RL) has become the de facto standard for tuning LLMs to solve tasks involving reasoning. However, growing evidence shows that models trained in such way often suffer from a significant loss in diversity. We argue that this arises because RL implicitly optimizes the \"mode-seeking\" or \"zero-forcing\" Reverse KL to a target distribution causing the model to concentrate mass on certain high-probability regions of the target while neglecting others. In this work, we instead begin from an explicit target distribution, obtained by filtering out incorrect answers while preserving the relative probabilities of correct ones. Starting from a pre-trained LLM, we approximate this target distribution using the $α$-divergence family, which unifies prior approaches and enables direct control of the precision-diversity trade-off by interpolating between mode-seeking and mass-covering divergences. On a Lean theorem-proving benchmark, our method achieves state-of-the-art performance along the coverage-precision Pareto frontier, outperforming all prior methods on the coverage axis.","abs_url":"https://arxiv.org/abs/2512.05962","pdf_url":"https://arxiv.org/pdf/2512.05962v2","primary_category":"cs.LG","published":"2025-12-05","tags":["theorem-proving-ml","benchmark"],"lean_note":"Trains theorem provers on a Lean benchmark using DeepSeek-Prover and the open Lean Workbook dataset to improve proof diversity.","repo":"https://github.com/naver/alpha-dpg","thumb":"thumbs/2512.05962.jpg"},{"id":"2512.05750","title":"Formalizing Polynomial Laws and the Universal Divided Power Algebra","authors":["Antoine Chambert-Loir","María Inés de Frutos-Fernández"],"abstract":"The goal of this paper is to present an ongoing formalization, in the framework provided by the Lean/Mathlib mathematical library, of the construction by Roby (1965) of the universal divided power algebra. This is an analogue, in the theory of divided powers, of the classical algebra of polynomials. It is a crucial tool in the development of crystalline cohomology; it is also used in $p$-adic Hodge theory to define the crystalline period ring. As an algebra, this universal divided power algebra has a fairly simple definition that shows that it is a graded algebra. The main difficulty in Roby's theorem lies in constructing a divided power structure on its augmentation ideal. To that aim, Roby identified the graded pieces with another universal structure: homogeneous polynomial laws.We formalize the first steps of the theory of polynomial laws and show how future work will allow to complete the formalization of the above-mentioned divided power structure. We report on various difficulties that appeared in this formalization: taking care of universes, extending to semirings some aspects of the Mathlib library, and coping with several instances of \"invisible mathematics\".","abs_url":"https://arxiv.org/abs/2512.05750","pdf_url":"https://arxiv.org/pdf/2512.05750v1","primary_category":"cs.LO","published":"2025-12-05","tags":["formalization","mathlib","algebra"],"lean_note":"Formalizes polynomial laws and the universal divided power algebra and its graded structure in Lean/Mathlib.","repo":"https://github.com/mariainesdff/divided_power_algebra_journal","thumb":"thumbs/2512.05750.jpg"},{"id":"2512.03175","title":"The Seifert-van Kampen Theorem via Computational Paths: A Formalized Approach to Computing Fundamental Groups","authors":["Arthur F. Ramos","Tiago M. L. de Veras","Ruy J. G. B. de Queiroz","Anjolina G. de Oliveira"],"abstract":"The Seifert-van Kampen theorem computes the fundamental group of a space from the fundamental groups of its constituents. We develop a modular SVK framework within the setting of computational paths - an approach to equality where witnesses are explicit sequences of rewrites governed by the LNDEQ-TRS. Our contributions are: (i) pushouts as higher-inductive types with modular typeclass assumptions for computation rules; (ii) free products and amalgamated free products as quotients of word representations; (iii) an SVK equivalence schema parametric in user-supplied encode/decode structure; and (iv) instantiations for classical spaces - figure-eight (pi_1(S^1 v S^1) = Z * Z), 2-sphere (pi_1(S^2) = 1), and 3-sphere (pi_1(S^3) = 1) with Hopf fibration context. Recent extensions include higher homotopy groups pi_n via weak infinity-groupoid structure (with pi_2 abelian via Eckmann-Hilton), and pi_1 >= 1 in the 1-groupoid truncated setting; truncation levels connecting the framework to HoTT; automated path simplification tactics; basic covering space theory with pi_1-actions on fibers; fibration theory with long exact sequences; and Eilenberg-MacLane space characterization (S^1 = K(Z,1)). The development is formalized in Lean 4 with 41,130 lines across 107 modules, using 36 kernel axioms for HIT type-constructor declarations.","abs_url":"https://arxiv.org/abs/2512.03175","pdf_url":"https://arxiv.org/pdf/2512.03175v3","primary_category":"cs.LO","published":"2025-12-02","tags":["formalization","topology","type-theory","logic"],"lean_note":"Formalizes a Seifert-van Kampen framework via computational paths in Lean 4, with 41,130 lines across 107 modules.","repo":"https://github.com/Arthur742Ramos/ComputationalPathsLean","thumb":"thumbs/2512.03175.jpg"},{"id":"2512.00997","title":"IndiMathBench: Autoformalizing Mathematical Reasoning Problems with a Human Touch","authors":["Param Biyani","Shashank Kirtania","Yasharth Bajpai","Sumit Gulwani","Ashish Tiwari"],"abstract":"Reliable autoformalization remains challenging even in the era of large language models (LLMs). The scarcity of high-quality training data is a major bottleneck. Expert annotation requires substantial time and deep expertise in both mathematics and theorem proving. We introduce IndiMathBench, a human-verified benchmark designed to evaluate mathematical theorem proving, curated using an AI-powered human-assisted pipeline for formalizing natural language problems in Lean. IndiMathBench is composed of 312 formal Lean 4 theorems paired with their corresponding informal problem statements, sourced from Indian Mathematics Olympiads. Through category-based retrieval, iterative compiler feedback, and multi-model ensembles, our pipeline generates candidate formalizations that experts efficiently validate via an interactive dashboard with automated quality summaries. Evaluation across multiple frontier models demonstrates that autoformalization remains challenging, with substantial gaps between syntactic validity and semantic correctness, while theorem proving success rates remain low even with iterative refinement, demonstrating that \\benchmark~presents a challenging testbed for mathematical reasoning. IndiMathBench is available at https://github.com/prmbiy/IndiMathBench.","abs_url":"https://arxiv.org/abs/2512.00997","pdf_url":"https://arxiv.org/pdf/2512.00997v2","primary_category":"cs.AI","published":"2025-11-30","tags":["benchmark","autoformalization","dataset","theorem-proving-ml"],"lean_note":"Introduces IndiMathBench, 312 human-verified Lean 4 theorems from Indian Math Olympiads, built via compiler-feedback autoformalization.","repo":"https://github.com/prmbiy/IndiMathBench","thumb":"thumbs/2512.00997.jpg"},{"id":"2512.00910","title":"Effros' theorem on transitive group actions with a glimpse into descriptive set theory","authors":["Jochen Wengenroth"],"abstract":"The main aim of this note is to prove a version of a celebrated theorem of Effros about transitive group actions in a non-metrizable setting, these parts have been formalized and verified with Lean by Lara Toledano. We do not claim any originality since the given proof is in fact very close to one of van Mill. Our presentation is however completely self-contained and may serve as an appetizer to descriptive set theory. It also contains a few results about Suslin spaces (continuous images of separable completely metrizable spaces, which are often called analytic) which are extremely useful in measure theory.","abs_url":"https://arxiv.org/abs/2512.00910","pdf_url":"https://arxiv.org/pdf/2512.00910v1","primary_category":"math.FA","published":"2025-11-30","tags":["formalization","topology","logic"],"lean_note":"A non-metrizable version of Effros' theorem on transitive group actions was formalized and verified in Lean by Lara Toledano.","thumb":"thumbs/2512.00910.jpg"},{"id":"2512.00657","title":"Computational Paths Form a Weak ω-Groupoid","authors":["Arthur F. Ramos","Tiago M. L. de Veras","Ruy J. G. B. de Queiroz","Anjolina G. de Oliveira"],"abstract":"Lumsdaine (2010) and van den Berg-Garner (2011) proved that types in Martin-Löf type theory carry the structure of weak ω-groupoids. Their proofs, while foundational, rely on abstract properties of the identity type without providing explicit computational content for coherence witnesses. We establish an analogous result for computational paths -- an alternative formulation of equality where witnesses are explicit sequences of rewrites from the LNDEQ-TRS term rewriting system. Our main result is that computational paths on any type form a weak ω-groupoid with fully explicit coherence data. The groupoid operations -- identity, composition, and inverse -- are defined at every dimension, and the coherence laws (associativity, unit laws, inverse laws) are witnessed by concrete rewrite derivations rather than abstract existence proofs. The construction provides: (i) a proper tower of n-cells for all dimensions, with 2-cells as derivations between paths and higher cells mediating between lower-dimensional witnesses; (ii) explicit pentagon and triangle coherences built from the rewrite rules; and (iii) contractibility at dimensions $\\geq 3$, ensuring all parallel higher cells are connected. The contractibility property is derived from the normalization algorithm of the rewrite system, grounding the higher-dimensional structure in concrete computational content. The entire construction has been formalized in Lean 4, providing machine-checked verification of the weak ω-groupoid structure.","abs_url":"https://arxiv.org/abs/2512.00657","pdf_url":"https://arxiv.org/pdf/2512.00657v1","primary_category":"cs.LO","published":"2025-11-29","tags":["formalization","type-theory","category-theory","logic"],"lean_note":"Formalizes the weak ω-groupoid structure of computational paths in Lean 4 with machine-checked coherence witnesses.","thumb":"thumbs/2512.00657.jpg"},{"id":"2511.23152","title":"A Differentiable Measure of Algebraic Complexity: Provably Exact Discovery of Group Structures","authors":["Dongsung Huh","Lior Horesh","Halyun Jeong"],"abstract":"Discovering discrete algebraic rules from data is a fundamental challenge in machine learning. We formalize this problem through Cayley-table completion -- an algebraic counterpart to classical matrix completion -- where the degree of associativity violation replaces linear rank as the intrinsic measure of complexity. We provide a rigorous landscape analysis of HyperCube, an operator-valued tensor factorization, on the fully observed target table $δ$, proving that its global infimum $H_{\\inf}(δ) := \\inf_{Θ\\in F_δ} H(Θ)$ implicitly defines an exact differentiable measure for this complexity. We show that HyperCube's native objective $H(Θ)$ decomposes into two components: geometric alignment (collinearity) and an inverse $\\ell_2$ penalty. We establish that these continuous variational pressures induce core discrete properties: collinearity enforces associativity (Collinearity--Associativity Equivalence), and the inverse $\\ell_2$ penalty reduces to an exact inverse rank penalty within the collinear manifold, driving the parameters toward full-rank unitarity. Consequently, we derive an absolute lower bound $H(Θ) \\ge H_{\\inf}(δ) \\ge 3 \\, |δ|$, where $|δ|$ is the target table size. We prove this absolute floor is attained if and only if the target is isotopic to a group, and characterize the global minimizer as the regular representation of the underlying group (up to unitary gauge), resolving the central open conjecture of Huh (2025). This work serves as an existence proof that certain discrete algebraic structures can be exactly characterized by differentiable measures, enabling gradient-based discovery without the need for combinatorial search. All theoretical results are mechanically verified in Lean 4 and confirmed via small-scale experiments.","abs_url":"https://arxiv.org/abs/2511.23152","pdf_url":"https://arxiv.org/pdf/2511.23152v4","primary_category":"cs.LG","published":"2025-11-28","tags":["formalization","algebra","ml-theory"],"lean_note":"All theoretical results on Cayley-table completion and the associativity gap are mechanically verified in Lean 4.","thumb":"thumbs/2511.23152.jpg"},{"id":"2512.00081","title":"The Orientation Boundary for Step-Duplicating Recursors: Mechanized Impossibility, Escape, and Certification","authors":["Moses Rahnama"],"abstract":"We formalize the orientation boundary for first-order step-duplicating recursors, centered on the Right-Duplicating Recursor Schema (RDRS), $\\mathrm{recur}(b,s,\\mathrm{succ}(n))\\to\\mathrm{wrap}(s,\\mathrm{recur}(b,s,n))$. In Lean 4, the no-go side excludes twelve base direct-measure classes (two unconditional, six scalar growth, four tracked vector / pair), with arctic / tropical matrix continuations, a WPO-facing polynomial-branch corollary, and a KBO obstruction. Four meta-theorems organize the stack: projected-primary dominance, scalar-projection lift, mixed-matrix scalarization, and the symbolic comparator barrier. The surface spans 72 schema-level dup-step impossibilities and 80 concrete-system global-step theorems, with a 76-row RDRS method-universe closeout and a semantic capstone proving every payload-erasing semantic direct measure is counter-dominated. The successful side carries a transparency-essentiality witness, a dependency-pair projection escape, a generalized polynomial barrier under frozen-base failure, computable witness extractors, a coefficient-table decision procedure, and mutual-recursion / synchronized-SCC barriers. The witness calculus KO7 has a two-layer chain. Its guarded fragment is strongly normalizing, root-confluent, and normalizable, with single-exponential contextual derivation bounds and an exact $ω^ω$ ordinal calibration below $ω^ω\\cdot 2$. The full unguarded system is root-terminating via a nonlinear polynomial witness and a specialized MPO, with context-closed strong normalization lifted through every constructor position. A checked TPDB export and a Lean-side replay of the FAST certificate connect the development to TTT2 / CeTA. To our knowledge, this is the first mechanized object-level barrier theorem on a fixed terminating system, proved without reductions or undecidability arguments.","abs_url":"https://arxiv.org/abs/2512.00081","pdf_url":"https://arxiv.org/pdf/2512.00081v10","primary_category":"cs.LO","published":"2025-11-26","tags":["formalization","logic","verification"],"lean_note":"Mechanizes in Lean 4 an orientation-boundary impossibility theorem for first-order step-duplicating recursors, with TPDB/CeTA certificate replay.","repo":"https://github.com/MosesRahnama/The-Orientation-Boundary","thumb":"thumbs/2512.00081.jpg"},{"id":"2511.21104","title":"BRIDGE: Building Representations In Domain Guided Program Synthesis","authors":["Robert Joseph George","Carson Eisenach","Udaya Ghai","Dominique Perrault-Joncas","Anima Anandkumar","Dean Foster"],"abstract":"Large language models can generate plausible code, but remain brittle for formal verification in proof assistants such as Lean. A central scalability challenge is that verified synthesis requires consistent artifacts across several coupled domains: executable code, formal specifications, theorem statements, and proof attempts. Existing approaches often treat these artifacts separately. We present BRIDGE, a structured prompting framework for multi-artifact program synthesis. BRIDGE decomposes generation into three interconnected domains: Code, Specification, and Theorem/Proof, and uses domain-specific intermediate reasoning to connect them. In Lean, BRIDGE often follows a code-first workflow, using the generated implementation as a semantic anchor for downstream specification, theorem statement, and proof-attempt generation. Across 178 algorithmic problems and five LLMs, BRIDGE improves Lean executable correctness by up to nearly 1.5x over direct prompting and can be roughly 2x more sample efficient at comparable generation lengths. We further find that specification-oriented prompting improves Python pass rates by up to 17.5 percentage points. Beyond inference-time prompting, supervised fine-tuning on BRIDGE-style reasoning traces yields nearly 1.5x higher Lean pass success than code-only fine-tuning, suggesting that these intermediate representations provide a learnable inductive bias. BRIDGE provides a practical framework for scaling verified synthesis while highlighting the remaining gap between executable correctness and full formal proof generation.","abs_url":"https://arxiv.org/abs/2511.21104","pdf_url":"https://arxiv.org/pdf/2511.21104v4","primary_category":"cs.LG","published":"2025-11-26","tags":["theorem-proving-ml","verification","benchmark"],"lean_note":"BRIDGE structures LLM program synthesis into code, specification, and theorem/proof artifacts, measuring Lean executable correctness on VERINA and CLEVER.","repo":"https://github.com/lean-dojo/BRIDGE","thumb":"thumbs/2511.21104.jpg"},{"id":"2511.20118","title":"Formalization of Brownian motion in Lean","authors":["Rémy Degenne","David Ledvinka","Etienne Marion","Peter Pfaffelhuber"],"abstract":"Brownian motion is a building block in modern probability theory. In this paper, we describe a formalization of Brownian motion using the Lean theorem prover. We build on the existing measure-theoretic foundations in Lean's mathematical library, Mathlib, and we develop several key components needed for the construction of Brownian motion, including the Carathéodory and Kolmogorov extension theorems, Gaussian measures in Banach spaces, and the Kolmogorov-Chentsov theorem for path continuity.","abs_url":"https://arxiv.org/abs/2511.20118","pdf_url":"https://arxiv.org/pdf/2511.20118v2","primary_category":"math.PR","published":"2025-11-25","tags":["formalization","mathlib","probability","analysis"],"lean_note":"Formalizes the construction of Brownian motion in Lean, building on Mathlib's measure-theoretic foundations.","repo":"https://github.com/RemyDegenne/brownian-motion","thumb":"thumbs/2511.20118.jpg"},{"id":"2511.19833","title":"Average-Rare Order Ideals in Functional Preorders","authors":["Masahiro Hachimori","Kenji Kashiwabara"],"abstract":"We prove that for the preorder induced by a function f: V -> V, the family of all order ideals is average-rare, that is, its normalized degree sum (nds) is nonpositive. As a base case in our reduction, we establish the same result for functional partial orders (or rooted forests). We also propose a conjecture related to Frankl's Conjecture. All proofs have been formally verified in the proof assistant Lean 4.","abs_url":"https://arxiv.org/abs/2511.19833","pdf_url":"https://arxiv.org/pdf/2511.19833v1","primary_category":"math.CO","published":"2025-11-25","tags":["formalization","combinatorics","mathlib"],"lean_note":"All theorems on average-rarity of order-ideal families of functional preorders are machine-checked in Lean 4 with definitions added to mathlib4.","repo":"https://github.com/kashiwabarakenji/avg-rare","thumb":"thumbs/2511.19833.jpg"},{"id":"2511.19142","title":"Formalizing Computational Paths and Fundamental Groups in Lean","authors":["Arthur F. Ramos","Anjolina G. de Oliveira","Ruy J. G. B. de Queiroz","Tiago M. L. de Veras"],"abstract":"Computational paths treat propositional equality as explicit paths built from labelled deduction steps and rewrite rules. This view originates in work by de Queiroz and collaborators [1] and yields a weak groupoid structure for equality, together with a computational account of homotopy inspired by homotopy type theory. In this paper we present a complete mechanization of this framework in Lean 4 and show how it supports concrete homotopy theoretic computations. Our contributions are threefold. First, we formalize the theory of computational paths in Lean, including path formation, composition, inverses, and a rewrite system that identifies redundant or trivial paths. We prove that equality types with computational paths carry a weak groupoid structure in the sense of the original theory. Second, we organize this material into a reusable Lean library, ComputationalPathsLean, which exposes an interface for paths, rewrites, and loop spaces. This library allows later developments to treat computational paths as a drop-in replacement for propositional equality when reasoning about homotopical structure. Third, we apply the library to six canonical examples in algebraic topology. We give Lean proofs that the fundamental group of the circle is isomorphic to the integers, the cylinder and Mobius band also have fundamental group isomorphic to the integers (via retraction to the circle), the fundamental group of the torus is isomorphic to the product of two copies of the integers, the fundamental group of the Klein bottle is isomorphic to the semidirect product Z cross Z, and the fundamental group of the real projective plane is isomorphic to Z_2. These case studies demonstrate that the computational paths approach scales to nontrivial homotopical computations in a modern proof assistant. All the definitions and proofs described here are available in an open-source Lean 4 repository.","abs_url":"https://arxiv.org/abs/2511.19142","pdf_url":"https://arxiv.org/pdf/2511.19142v2","primary_category":"cs.LO","published":"2025-11-24","tags":["formalization","topology","type-theory"],"lean_note":"Mechanizes computational paths and proves fundamental groups of canonical spaces in Lean 4 via the ComputationalPathsLean library.","repo":"https://github.com/Arthur742Ramos/ComputationalPathsLean","thumb":"thumbs/2511.19142.jpg"},{"id":"2511.18760","title":"HERMES: Towards Efficient and Verifiable Mathematical Reasoning in LLMs","authors":["Azim Ospanov","Zijin Feng","Jiacheng Sun","Haoli Bai","Xin Shen","Farzan Farnia"],"abstract":"Informal mathematics has been central to modern large language model (LLM) reasoning, offering flexibility and efficient construction of arguments. However, purely informal reasoning is prone to logical gaps and subtle errors that are difficult to detect and correct. In contrast, formal theorem proving provides rigorous, verifiable mathematical reasoning, where each inference step is checked by a trusted compiler, but lacks the exploratory freedom of informal problem-solving. This mismatch leaves current LLM-based math agents without a principled way to combine the strengths of both paradigms. In this work, we introduce Hermes, the first tool-assisted agent that explicitly interleaves informal reasoning with formally verified proofs in Lean. The framework performs intermediate formal checking to prevent reasoning drift and a memory module for proof continuity across multi-step reasoning chains, enabling both exploration and verification. We evaluate Hermes on four challenging mathematical reasoning benchmarks using LLMs of varying parameter scales, from small models to state-of-the-art systems. Across all settings, Hermes reliably improves the reasoning accuracy of base models while substantially reducing reasoning token usage and computational cost compared to reward-based approaches. On difficult datasets such as AIME and HARDMath2, Hermes@1 achieves up to a 40% accuracy improvement while using 80% fewer total inference FLOPs. When scaled at test time, Hermes@5 boosts accuracy further by 20%. The implementation and codebase are publicly available at https://github.com/aziksh-ospanov/HERMES.","abs_url":"https://arxiv.org/abs/2511.18760","pdf_url":"https://arxiv.org/pdf/2511.18760v2","primary_category":"cs.AI","published":"2025-11-24","tags":["theorem-proving-ml","verification","ai-agents","autoformalization"],"lean_note":"A tool-assisted agent interleaves informal LLM reasoning with proof steps verified by a Lean 4 REPL backend.","repo":"https://github.com/aziksh-ospanov/HERMES","thumb":"thumbs/2511.18760.jpg"},{"id":"2511.17645","title":"BlockCert: Certified Blockwise Extraction of Transformer Mechanisms","authors":["Sandro Andric"],"abstract":"Mechanistic interpretability aspires to reverse-engineer neural networks into explicit algorithms, while model editing seeks to modify specific behaviours without retraining. Both areas are typically evaluated with informal evidence and ad-hoc experiments, with few explicit guarantees about how far an extracted or edited model can drift from the original on relevant inputs. We introduce BlockCert, a framework for certified blockwise extraction of transformer mechanisms, and outline how a lightweight extension can support certified local edits. Given a pre-trained transformer and a prompt distribution, BlockCert extracts structured surrogate implementations for residual blocks together with machine-checkable certificates that bound approximation error, record coverage metrics, and hash the underlying artifacts. We formalize a simple Lipschitz-based composition theorem in Lean 4 that lifts these local guarantees to a global deviation bound. Empirically, we apply the framework to GPT-2 small, TinyLlama-1.1B-Chat, and Llama-3.2-3B. Across these models we obtain high per-block coverage and small residual errors on the evaluated prompts, and in the TinyLlama setting we show that a fully stitched model matches the baseline perplexity within approximately 6e-5 on stress prompts. Our results suggest that blockwise extraction with explicit certificates is feasible for real transformer language models and offers a practical bridge between mechanistic interpretability and formal reasoning about model behaviour.","abs_url":"https://arxiv.org/abs/2511.17645","pdf_url":"https://arxiv.org/pdf/2511.17645v1","primary_category":"cs.LG","published":"2025-11-20","tags":["verification","ml-theory"],"lean_note":"Formalizes a Lipschitz-based composition theorem in Lean 4 that lifts per-block extraction error bounds to a global model deviation bound.","thumb":"thumbs/2511.17645.jpg"},{"id":"2511.13304","title":"Capturing properties of planar diagrams in Lean proof assistant software","authors":["Alastair Litterick","Alexei Vernitski","Billy Woods"],"abstract":"Automated proof assistants are a technology pre-empting mistakes in mathematics. In our practice we have seen that reasoning about planar diagrams is difficult to both humans and computers. One example that has led to wrong statements in publications is that an orientation-preserving mapping is not always defined by how it acts on triples of elements. In this paper we formalise orientation-preserving mappings in proof assistant software Lean and report on our take-aways.","abs_url":"https://arxiv.org/abs/2511.13304","pdf_url":"https://arxiv.org/pdf/2511.13304v2","primary_category":"math.CO","published":"2025-11-17","tags":["formalization","combinatorics","mathlib"],"lean_note":"Formalizes orientation-preserving mappings of planar diagrams in Lean 4 with mathlib to verify subtle combinatorial claims.","thumb":"thumbs/2511.13304.jpg"},{"id":"2511.12784","title":"Evaluating Autoformalization Robustness via Semantically Similar Paraphrasing","authors":["Hayden Moore","Asfahan Shah"],"abstract":"Large Language Models (LLMs) have recently emerged as powerful tools for autoformalization. Despite their impressive performance, these models can still struggle to produce grounded and verifiable formalizations. Recent work in text-to-SQL, has revealed that LLMs can be sensitive to paraphrased natural language (NL) inputs, even when high degrees of semantic fidelity are preserved. In this paper, we investigate this claim in the autoformalization domain. Specifically, we evaluate the robustness of LLMs generating formal proofs with semantically similar paraphrased NL statements by measuring semantic and compilation validity. Using the formal benchmarks MiniF2F and Lean 4 version of ProofNet, and two modern LLMs, we generate paraphrased natural language statements and cross-evaluate these statements across both models. The results of this paper reveal performance variability across paraphrased inputs, demonstrating that minor shifts in NL statements can significantly impact model outputs.","abs_url":"https://arxiv.org/abs/2511.12784","pdf_url":"https://arxiv.org/pdf/2511.12784v3","primary_category":"cs.CL","published":"2025-11-16","tags":["autoformalization","theorem-proving-ml","benchmark"],"lean_note":"Tests LLM autoformalization robustness on MiniF2F and Lean 4 ProofNet using paraphrased statements, measuring Lean compilation validity.","thumb":"thumbs/2511.12784.jpg"},{"id":"2511.12253","title":"The Search for Constrained Random Generators","authors":["Harrison Goldstein","Hila Peleg","Cassia Torczon","Daniel Sainati","Leonidas Lampropoulos","Benjamin C. Pierce"],"abstract":"Among the biggest challenges in property-based testing (PBT) is the constrained random generation problem: given a predicate on program values, randomly sample from the set of all values satisfying that predicate, and only those values. Efficient solutions to this problem are critical, since the executable specifications used by PBT often have preconditions that input values must satisfy in order to be valid test cases, and satisfying values are often sparsely distributed. We propose a novel approach to this problem using ideas from deductive program synthesis. We present a set of synthesis rules, based on a denotational semantics of generators, that give rise to an automatic procedure for synthesizing correct generators. Our system handles recursive predicates by rewriting them as catamorphisms and then matching with appropriate anamorphisms; this is theoretically simpler than other approaches to synthesis for recursive functions, yet still extremely expressive. Our implementation, Palamedes, is an extensible library for the Lean theorem prover. The synthesis algorithm itself is built on standard proof-search tactics, reducing implementation burden and allowing the algorithm to benefit from further advances in Lean proof automation.","abs_url":"https://arxiv.org/abs/2511.12253","pdf_url":"https://arxiv.org/pdf/2511.12253v2","primary_category":"cs.PL","published":"2025-11-15","tags":["tooling","tactic-automation","verification"],"lean_note":"Implements Palamedes, an extensible Lean library that synthesizes constrained random test generators using Lean proof-search tactics.","repo":"https://doi.org/10.5281/zenodo.19073205","thumb":"thumbs/2511.12253.jpg"},{"id":"2511.10356","title":"SITA: A Framework for Structure-to-Instance Theorem Autoformalization","authors":["Chenyi Li","Wanli Ma","Zichen Wang","Zaiwen Wen"],"abstract":"While large language models (LLMs) have shown progress in mathematical reasoning, they still face challenges in formalizing theorems that arise from instantiating abstract structures in concrete settings. With the goal of auto-formalizing mathematical results at the research level, we develop a framework for structure-to-instance theorem autoformalization (SITA), which systematically bridges the gap between abstract mathematical theories and their concrete applications in Lean proof assistant. Formalized abstract structures are treated as modular templates that contain definitions, assumptions, operations, and theorems. These templates serve as reusable guides for the formalization of concrete instances. Given a specific instantiation, we generate corresponding Lean definitions and instance declarations, integrate them using Lean's typeclass mechanism, and construct verified theorems by checking structural assumptions. We incorporate LLM-based generation with feedback-guided refinement to ensure both automation and formal correctness. Experiments on a dataset of optimization problems demonstrate that SITA effectively formalizes diverse instances grounded in abstract structures.","abs_url":"https://arxiv.org/abs/2511.10356","pdf_url":"https://arxiv.org/pdf/2511.10356v1","primary_category":"cs.AI","published":"2025-11-13","tags":["autoformalization","formalization","theorem-proving-ml"],"lean_note":"SITA autoformalizes instantiations of abstract math structures into Lean definitions and theorems using Lean's typeclass mechanism.","repo":"https://github.com/chenyili0818/SITA","thumb":"thumbs/2511.10356.jpg"},{"id":"2511.06701","title":"Structural Enforcement of Statistical Rigor in AI-Driven Discovery: A Functional Architecture","authors":["Karen Sargsyan"],"abstract":"AI-Scientist systems that use large language models to automate research risk generating spurious discoveries through uncontrolled multiple testing. We present a functional architecture that enforces statistical rigor at two levels: a Haskell embedded DSL (the Research monad) that makes it impossible to test a hypothesis without updating the error budget, and a declarative scaffolding technique that structurally prevents data leakage across the boundary into LLM-generated code. We ground these guarantees in a machine-checked Lean 4 formalization of the LORD++ online FDR control theorem (855 lines, zero sorry), which identifies four sufficient conditions for FDR control. Three are structural conditions -- about information flow, data separation, and test validity -- enforced by the architecture's type system and scaffolding. The fourth is an arithmetic condition: a budget invariant requiring that a wealth process remain non-negative under floating-point computation. We verify this condition over IEEE 754 doubles using SPARK/Ada, whose GNATprove toolchain statically confirms that no rounding sequence can violate the invariant and whose flow analysis independently confirms the predictability condition. The resulting verification chain -- from real-analysis proof to floating-point implementation -- is, to our knowledge, the first for any online FDR control procedure. Monte Carlo simulation (N=2000 hypotheses) and an end-to-end case study confirm that the monadic implementation controls FDR at 1.1% against a 5% target, while a naive approach inflates to 41%.","abs_url":"https://arxiv.org/abs/2511.06701","pdf_url":"https://arxiv.org/pdf/2511.06701v2","primary_category":"cs.SE","published":"2025-11-10","tags":["formalization","verification","probability"],"lean_note":"Provides a machine-checked Lean 4 formalization (855 lines, zero sorry) of the LORD++ online FDR-control theorem.","repo":"https://github.com/karsar/ai-scientist-guards","thumb":"thumbs/2511.06701.jpg"},{"id":"2511.03618","title":"Towards Formalizing Reinforcement Learning Theory","authors":["Shangtong Zhang"],"abstract":"In this paper, we formalize the almost sure convergence of $Q$-learning and linear temporal difference (TD) learning with Markovian samples using the Lean 4 theorem prover based on the Mathlib library. $Q$-learning and linear TD are among the earliest and most influential reinforcement learning (RL) algorithms. The investigation of their convergence properties is not only a major research topic during the early development of the RL field but also receives increasing attention nowadays. This paper formally verifies their almost sure convergence in a unified framework based on the Robbins-Siegmund theorem. The framework developed in this work can be easily extended to convergence rates and other modes of convergence. This work thus makes an important step towards fully formalizing convergent RL results. The code is available at https://github.com/ShangtongZhang/rl-theory-in-lean.","abs_url":"https://arxiv.org/abs/2511.03618","pdf_url":"https://arxiv.org/pdf/2511.03618v1","primary_category":"cs.LG","published":"2025-11-05","tags":["formalization","verification","probability","ml-theory","mathlib"],"lean_note":"Formalizes almost-sure convergence of Q-learning and linear TD learning in the Lean 4 theorem prover atop the Mathlib library.","repo":"https://github.com/ShangtongZhang/rl-theory-in-lean","thumb":"thumbs/2511.03618.jpg"},{"id":"2511.03108","title":"miniF2F-Lean Revisited: Reviewing Limitations and Charting a Path Forward","authors":["Azim Ospanov","Farzan Farnia","Roozbeh Yousefzadeh"],"abstract":"We perform a thorough analysis of the formal and informal statements in the miniF2F benchmark from the perspective of an AI system that is tasked to participate in a math Olympiad consisting of the problems in miniF2F. In such setting, the model has to read and comprehend the problems in natural language, formalize them in Lean language, then proceed with proving the problems, and it will get credit for each problem if the formal proof corresponds to the original informal statement presented to the model. Our evaluation results reveal that the best accuracy of such pipeline can be about 36% using the SoTA models in the literature, considerably lower than the individual SoTA accuracies, 97% and 69% reported in the autoformalization and theorem proving literature. Analyzing the failure modes, we trace back a considerable portion of this drop to discrepancies between the formal and informal statements for more than half of the problems in miniF2F. We proceed with correcting all the errors, discrepancies and simplifications in formal and informal statements, and present the miniF2F-v2 with fully verified formal and informal statements and proofs. Evaluating the full theorem proving pipeline on miniF2F-v2 leads to the best accuracy of 70%, a significant improvement from the 40% on the original miniF2F, yet indicating considerable misalignment between the autoformalization models and theorem provers. Our deep analysis suggests that a higher quality benchmark can help the community better evaluate progress in the field of formal reasoning and also better diagnose the failure and success modes of autoformalization and theorem proving models. Our dataset is available at https://github.com/roozbeh-yz/miniF2F_v2.","abs_url":"https://arxiv.org/abs/2511.03108","pdf_url":"https://arxiv.org/pdf/2511.03108v1","primary_category":"cs.AI","published":"2025-11-05","tags":["benchmark","autoformalization","theorem-proving-ml","dataset"],"lean_note":"Audits and corrects the Lean miniF2F benchmark, releasing miniF2F-v2 with verified formal and informal statements and proofs.","repo":"https://github.com/roozbeh-yz/miniF2F_v2","thumb":"thumbs/2511.03108.jpg"},{"id":"2511.02872","title":"FATE: A Formal Benchmark Series for Frontier Algebra of Multiple Difficulty Levels","authors":["Jiedong Jiang","Wanyi He","Yuefeng Wang","Guoxiong Gao","Yongle Hu","Jingting Wang","Nailin Guan","Peihao Wu","Chunbo Dai","Liang Xiao","Bin Dong"],"abstract":"Recent advances in large language models (LLMs) have demonstrated impressive capabilities in formal theorem proving, particularly on contest-based mathematical benchmarks like the IMO. However, these contests do not reflect the depth, breadth, and abstraction of modern mathematical research. To bridge this gap, we introduce FATE (Formal Algebra Theorem Evaluation), a new benchmark series in formal algebra designed to chart a course toward advanced mathematical reasoning. We present two new components, FATE-H and FATE-X, each with 100 problems in abstract and commutative algebra. The FATE series spans a difficulty spectrum from undergraduate exercises to problems exceeding PhD qualifying exams. Notably, FATE-X is the first formal benchmark to surpass both PhD-level exam difficulty and the coverage of the Mathlib library. Our evaluations of state-of-the-art LLM provers on this new benchmark reveal a stark performance gap compared to contest math: the best model achieves only 3% (pass@64) accuracy on FATE-H and 0% on FATE-X. Our two-stage evaluation reveals that models' natural-language reasoning is notably more accurate than their ability to formalize this reasoning. We systematically classify the common errors that arise during this formalization process. Furthermore, a comparative study shows that a specialized prover can exhibit less effective reflection than general-purpose models, reducing its accuracy at the natural-language stage. We believe FATE provides a robust and challenging benchmark that establishes essential checkpoints on the path toward research-level formal mathematical reasoning.","abs_url":"https://arxiv.org/abs/2511.02872","pdf_url":"https://arxiv.org/pdf/2511.02872v4","primary_category":"cs.LG","published":"2025-11-04","tags":["benchmark","theorem-proving-ml","algebra","mathlib"],"lean_note":"Introduces FATE-H and FATE-X, Lean/Mathlib formal-algebra benchmarks of 100 problems each, evaluating LLM provers on advanced algebra.","repo":"https://github.com/frenzymath/FATE","thumb":"thumbs/2511.02872.jpg"},{"id":"2511.02164","title":"ScenicProver: A Framework for Compositional Probabilistic Verification of Learning-Enabled Systems","authors":["Eric Vin","Kyle A. Miller","Inigo Incer","Sanjit A. Seshia","Daniel J. Fremont"],"abstract":"Full verification of learning-enabled cyber-physical systems (CPS) has long been intractable due to challenges including black-box components and complex real-world environments. Existing tools either provide formal guarantees for limited types of systems or test the system as a monolith, but no general framework exists for compositional analysis of learning-enabled CPS using varied verification techniques over complex real-world environments. This paper introduces ScenicProver, a verification framework that aims to fill this gap. Built upon the Scenic probabilistic programming language, the framework supports: (1) compositional system description with clear component interfaces, ranging from interpretable code to black boxes; (2) assume-guarantee contracts over those components using an extension of Linear Temporal Logic containing arbitrary Scenic expressions; (3) evidence generation through testing, formal proofs via Lean 4 integration, and importing external assumptions; (4) systematic combination of generated evidence using contract operators; and (5) automatic generation of assurance cases tracking the provenance of system-level guarantees. We demonstrate the framework's effectiveness through a case study on an autonomous vehicle's automatic emergency braking system with sensor fusion. By leveraging manufacturer guarantees for radar and laser sensors and focusing testing efforts on uncertain conditions, our approach enables stronger probabilistic guarantees than monolithic testing with the same computational budget.","abs_url":"https://arxiv.org/abs/2511.02164","pdf_url":"https://arxiv.org/pdf/2511.02164v1","primary_category":"cs.LO","published":"2025-11-04","tags":["verification","tooling","ai-agents","probability"],"lean_note":"Generates formal proofs of component contracts via Lean 4 integration within a compositional probabilistic verification framework.","thumb":"thumbs/2511.02164.jpg"},{"id":"2510.26094","title":"Lean4Physics: Comprehensive Reasoning Framework for College-level Physics in Lean4","authors":["Yuxin Li","Minghao Liu","Ruida Wang","Wenzhao Ji","Zhitao He","Rui Pan","Junming Huang","Tong Zhang","Yi R. Fung"],"abstract":"We present **Lean4PHYS**, a comprehensive reasoning framework for college-level physics problems in Lean4. **Lean4PHYS** includes *LeanPhysBench*, a college-level benchmark for formal physics reasoning in Lean4, which contains 200 hand-crafted and peer-reviewed statements derived from university textbooks and physics competition problems. To establish a solid foundation for formal reasoning in physics, we also introduce *PhysLib*, a community-driven repository containing fundamental unit systems and theorems essential for formal physics reasoning. Based on the benchmark and Lean4 repository we composed in **Lean4PHYS**, we report baseline results using major expert Math Lean4 provers and state-of-the-art closed-source models, with the best performance of DeepSeek-Prover-V2-7B achieving only 16% and Claude-Sonnet-4 achieving 35%. We also conduct a detailed analysis showing that our *PhysLib* can achieve an average improvement of 11.75% in model performance. This demonstrates the challenging nature of our *LeanPhysBench* and the effectiveness of *PhysLib*. To the best of our knowledge, this is the first study to provide a physics benchmark in Lean4.","abs_url":"https://arxiv.org/abs/2510.26094","pdf_url":"https://arxiv.org/pdf/2510.26094v1","primary_category":"cs.AI","published":"2025-10-30","tags":["benchmark","physics"],"lean_note":"Introduces Lean4PHYS with PhysLib and the LeanPhysBench benchmark of 200 Lean4 college-physics statements.","repo":"https://github.com/ShirleyLIYuxin/Lean4PHYS","thumb":"thumbs/2510.26094.jpg"},{"id":"2510.25427","title":"RLMEval: Evaluating Research-Level Neural Theorem Proving","authors":["Auguste Poiroux","Antoine Bosselut","Viktor Kunčak"],"abstract":"Despite impressive results on curated benchmarks, the practical impact of large language models (LLMs) on research-level neural theorem proving and proof autoformalization is still limited. We introduce RLMEval, an evaluation suite for these tasks, focusing on research-level mathematics from real-world Lean formalization projects. RLMEval targets the evaluation of neural theorem proving and proof autoformalization on challenging research-level theorems by leveraging real Lean Blueprint formalization projects. Our evaluation of state-of-the-art models on RLMEval, comprising 613 theorems from 6 Lean projects, reveals a significant gap: progress on existing benchmarks does not readily translate to these more realistic settings, with the best model achieving only a 10.3 % pass rate. RLMEval provides a new, challenging benchmark designed to guide and accelerate progress in automated reasoning for formal mathematics.","abs_url":"https://arxiv.org/abs/2510.25427","pdf_url":"https://arxiv.org/pdf/2510.25427v1","primary_category":"cs.CL","published":"2025-10-29","tags":["benchmark","theorem-proving-ml","autoformalization","formalization"],"lean_note":"Builds an evaluation suite of 613 theorems drawn from six real Lean Blueprint formalization projects.","thumb":"thumbs/2510.25427.jpg"},{"id":"2510.24818","title":"Formalization of Auslander--Buchsbaum--Serre criterion in Lean4","authors":["Naillin Guan","Yongle Hu"],"abstract":"We present a comprehensive formalization in the Lean4 theorem prover of the Auslander--Buchsbaum--Serre criterion, which characterizes regular local rings as those Noetherian local rings with finite global dimension. Rather than following the well-known proof that computes the projective dimension of the residue field via quotient by regular sequences and uses the Koszul complex to bound the cotangent space dimension by the global dimension, our approach is built systematically on the formalization of depth defined via the vanishing of Ext functors. We establish key homological results including Rees' theorem, the Auslander--Buchsbaum formula, and Ischebeck's theorem, and further develop the theories of Cohen--Macaulay modules and rings, including a complete formalization of the unmixedness theorem for Cohen--Macaulay rings. To prove the Auslander--Buchsbaum--Serre criterion, we show that maximal Cohen--Macaulay modules over regular local rings are free and establish a weakened form of the Ferrand--Vasconcelos theorem specific for the unique maximal ideal. As corollaries, we deduce that regularity can be checked at maximal ideals and formalize Hilbert's Syzygy Theorem. This work demonstrates how homological algebra can be effectively employed in the formalization of commutative algebra, providing extensive infrastructure for future developments in the field.","abs_url":"https://arxiv.org/abs/2510.24818","pdf_url":"https://arxiv.org/pdf/2510.24818v3","primary_category":"math.AC","published":"2025-10-28","tags":["formalization","algebra","mathlib"],"lean_note":"Formalizes the Auslander-Buchsbaum-Serre criterion characterizing regular local rings in Lean4, with supporting homological algebra.","repo":"https://github.com/Thmoas-Guan/mathlib4_fork","thumb":"thumbs/2510.24818.jpg"},{"id":"2510.24060","title":"Formalizing Schwartz functions and tempered distributions","authors":["Moritz Doll"],"abstract":"Distribution theory is a cornerstone of the theory of partial differential equations. We report on the progress of formalizing the theory of tempered distributions in the interactive proof assistant Lean, which is the first formalization in any proof assistant. We give an overview of the mathematical theory and highlight key aspects of the formalization that differ from the classical presentation. As an application, we prove that the Fourier transform extends to a linear isometry on $L^2$ and we define Sobolev spaces via the Fourier transform on tempered distributions.","abs_url":"https://arxiv.org/abs/2510.24060","pdf_url":"https://arxiv.org/pdf/2510.24060v2","primary_category":"cs.LO","published":"2025-10-28","tags":["formalization","analysis","mathlib"],"lean_note":"Formalizes Schwartz functions and tempered distributions in Lean, the first such formalization in any proof assistant.","repo":"https://github.com/thefundamentaltheor3m/Sphere-Packing-Lean","thumb":"thumbs/2510.24060.jpg"},{"id":"2510.23637","title":"Combining Textual and Structural Information for Premise Selection in Lean","authors":["Job Petrovčič","David Eliecer Narvaez Denis","Ljupčo Todorovski"],"abstract":"Premise selection is a key bottleneck for scaling theorem proving in large formal libraries. Yet existing language-based methods often treat premises in isolation, ignoring the web of dependencies that connects them. We present a graph-augmented approach that combines dense text embeddings of Lean formalizations with graph neural networks over a heterogeneous dependency graph capturing both state-premise and premise-premise relations. On the LeanDojo Benchmark, our method outperforms the ReProver language-based baseline by over 25\\% across standard retrieval metrics. These results suggest that relational information is beneficial for premise selection.","abs_url":"https://arxiv.org/abs/2510.23637","pdf_url":"https://arxiv.org/pdf/2510.23637v2","primary_category":"cs.LG","published":"2025-10-24","tags":["theorem-proving-ml","premise-selection","benchmark"],"lean_note":"Combines text embeddings of Lean formalizations with a GNN over a dependency graph for premise selection on the LeanDojo benchmark.","repo":"https://github.com/JobPetrovcic/GNNReProver","thumb":"thumbs/2510.23637.jpg"},{"id":"2510.21523","title":"Interpretable epistemic uncertainty decomposition in sequential generative models via polynomial chaos surrogates","authors":["Ramón Nartallo-Kaluarachchi","Shashanka Ubaru","Małgorzata J Zimoń","Dongsung Huh","Robert Manson-Sawko","Lior Horesh","Yoshua Bengio"],"abstract":"Sequential generative models conditioned on uncertain rewards are central to AI-driven scientific discovery, yet the epistemic uncertainty they inherit from imperfect reward estimates remains unquantified. We propagate this uncertainty through generative flow networks (GFlowNets) by fitting polynomial chaos expansions (PCEs) to small ensembles of trained models. The PCE coefficients yield analytical Sobol sensitivity indices, providing the first interpretable decomposition of which reward components drive which generative decisions, a capability unavailable from deep ensembles, Bayesian neural networks, or Monte Carlo dropout. Convergence guarantees are established theoretically and four of five are formally verified in the Lean 4 proof assistant. Across three real-world tasks the framework reveals actionable structure invisible to ensembles alone. On the Doyle-Dreher Buchwald-Hartwig dataset catalyst selection is robust ($D_{\\mathrm{catalyst}}\\approx 71$) while additive selection is fragile ($D_{\\mathrm{additive}}\\approx 179$, $2.5\\times$ higher). In fragment-based molecular design the linker position is the most sensitive ($D_{\\mathrm{linker}}\\approx 28$) while decoration positions are the most robust ($D\\approx 14$-$18$), reversing the conventional scaffold-robust / decoration-fragile assumption. On the Sachs protein signalling network, MAPK-cascade edges and PKA/PKC hub edges separate into distinct sensitivity regimes, providing a targeted map for perturbation experiments. Calibration coverage at the 95% level reaches 0.97-1.00 across the dominant steps, and the surrogate evaluates 10{,}000 policy samples in milliseconds - $10^{3}$-$10^{4}\\times$ faster than exhaustive retraining.","abs_url":"https://arxiv.org/abs/2510.21523","pdf_url":"https://arxiv.org/pdf/2510.21523v2","primary_category":"cs.LG","published":"2025-10-24","tags":["verification","ml-theory","probability"],"lean_note":"Convergence guarantees for the polynomial-chaos surrogate are formally verified in the Lean 4 proof assistant (four of five theorems).","repo":"https://github.com/supermanG/uq-gflow-net","thumb":"thumbs/2510.21523.jpg"},{"id":"2510.20728","title":"Co-Designing Quantum Codes with Transversal Diagonal Gates via Multi-Agent Systems","authors":["Xi He","Sirui Lu","Bei Zeng"],"abstract":"Exact scientific discovery requires more than heuristic search: candidate constructions must be turned into exact objects and checked independently. We address this gap by extending TeXRA with an independent Lean 4 verification layer, turning it into a human-guided multi-agent platform for exact scientific discovery. The platform couples symbolic synthesis, combinatorial and linear-programming search, exact reconstruction of numerical candidates, and formal verification in Lean. We apply this platform to nonadditive quantum error-correcting codes with prescribed transversal diagonal gates within the subset-sum linear-programming (SSLP) framework. In the distance-2 regime where logical states occupy distinct residue classes, the platform yields a Lean-certified catalogue of 14,116 codes for $K\\in\\{2,3,4\\}$ and up to six physical qubits, realizing cyclic logical orders 2 through 18, from which we extract closed-form infinite families. We also construct a residue-degenerate $((6,4,2))$ code implementing the logical controlled-phase gate $\\mathrm{diag}(1,1,1,i)$. At distance 3, we resolve the transversal-$T$ problem for $((7,2,3))$ codes within the complementary binary-dihedral $\\mathrm{BD}_{16}$ setting: among the 12 candidates surviving the SSLP filters, 10 admit exact realizations and 2 are excluded by no-go proofs. All accepted constructions, families, and no-go results are formalized and checked in Lean, illustrating how AI-assisted workflows can bridge search, exact reconstruction, and formal proof in the physical sciences.","abs_url":"https://arxiv.org/abs/2510.20728","pdf_url":"https://arxiv.org/pdf/2510.20728v3","primary_category":"quant-ph","published":"2025-10-23","tags":["verification","ai-agents","quantum"],"lean_note":"Quantum error-correcting code constructions and no-go results are formally verified in a Lean 4 layer integrated with the TeXRA multi-agent platform.","repo":"https://github.com/LionSR/lean-qec","thumb":"thumbs/2510.20728.jpg"},{"id":"2510.19804","title":"Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proof","authors":["Boris Alexeev","Dustin G. Mixon"],"abstract":"We resolve a $1000 Erdős prize problem, complete with formal verification generated by a large language model. In over a dozen papers, beginning in 1976 and spanning two decades, Paul Erdős repeatedly posed one of his \"favourite\" conjectures: every finite Sidon set can be extended to a finite perfect difference set. We establish that {1, 2, 4, 8, 13} is a counterexample to this conjecture. During the preparation of this paper, we discovered that although this problem was presumed to be open for half a century, Marshall Hall, Jr. published a different counterexample three decades before Erdős first posed the problem. With a healthy skepticism of this apparent oversight, and out of an abundance of caution, we used ChatGPT to vibe code a Lean proof of both Hall's and our counterexamples.","abs_url":"https://arxiv.org/abs/2510.19804","pdf_url":"https://arxiv.org/pdf/2510.19804v2","primary_category":"math.CO","published":"2025-10-22","tags":["formalization","combinatorics","number-theory"],"lean_note":"Uses ChatGPT to generate a Lean proof verifying counterexamples to an Erds Sidon-set conjecture.","thumb":"thumbs/2510.19804.jpg"},{"id":"2510.15700","title":"ProofOptimizer: Training Language Models to Simplify Proofs without Human Demonstrations","authors":["Alex Gu","Bartosz Piotrowski","Fabian Gloeckle","Kaiyu Yang","Aram H. Markosyan"],"abstract":"Neural theorem proving has advanced rapidly in the past year, reaching IMO gold-medalist capabilities and producing formal proofs that span thousands of lines. Although such proofs are mechanically verified by formal systems like Lean, their excessive length renders them difficult for humans to comprehend and limits their usefulness for mathematical insight. Proof simplification is therefore a critical bottleneck. Yet, training data for this task is scarce, and existing methods -- mainly agentic scaffolding with off-the-shelf LLMs -- struggle with the extremely long proofs generated by RL-trained provers. We introduce ProofOptimizer, the first language model trained to simplify Lean proofs without requiring additional human supervision. ProofOptimizer is trained via expert iteration and reinforcement learning, using Lean to verify simplifications and provide training signal. At inference time, it operates within an iterative proof-shortening workflow, progressively reducing proof length. Experiments show that ProofOptimizer substantially compresses proofs generated by state-of-the-art RL-trained provers on standard benchmarks, reducing proof length by 87% on miniF2F, 57% on PutnamBench, and 49% on Seed-Prover's IMO 2025 proofs. Beyond conciseness, the simplified proofs check faster in Lean and further improve downstream prover performance when reused as training data for supervised finetuning.","abs_url":"https://arxiv.org/abs/2510.15700","pdf_url":"https://arxiv.org/pdf/2510.15700v1","primary_category":"cs.LG","published":"2025-10-17","tags":["theorem-proving-ml","tooling","benchmark"],"lean_note":"Trains a language model to shorten Lean proofs, using Lean to verify each candidate simplification and supply training signal.","repo":"https://github.com/ByteDance-Seed/Seed-Prover","thumb":"thumbs/2510.15700.jpg"},{"id":"2510.15681","title":"ProofBridge: Auto-Formalization of Natural Language Proofs in Lean via Joint Embeddings","authors":["Prithwish Jana","Kaan Kale","Ahmet Ege Tanriverdi","Cruise Song","Sriram Vishwanath","Vijay Ganesh"],"abstract":"Translating human-written mathematical theorems and proofs from natural language (NL) into formal languages (FLs) like Lean 4 has long been a significant challenge for AI. Most state-of-the-art methods either focus on theorem-only NL-to-FL auto-formalization or on FL proof synthesis from FL theorems. In practice, auto-formalization of both theorem and proof still requires human intervention, as seen in AlphaProof's silver-medal performance at the 2024 IMO, where problem statements were manually translated before automated proof synthesis. We present ProofBridge, a unified framework for automatically translating entire NL theorems and proofs into Lean 4. At its core is a joint embedding model that aligns NL and FL (NL-FL) theorem+proof pairs in a shared semantic space, enabling cross-modal retrieval of semantically relevant FL examples to guide translation. ProofBridge integrates retrieval-augmented fine-tuning with iterative proof repair, leveraging Lean's type checker and semantic equivalence feedback to ensure both syntactic correctness and semantic fidelity. Experiments show substantial improvements in proof auto-formalization over strong baselines (including GPT-5, Gemini-2.5, Kimina-Prover, DeepSeek-Prover), with our retrieval-augmented approach yielding significant gains in semantic correctness (SC, via proving bi-directional equivalence) and type correctness (TC, via type-checking theorem+proof) across pass@k metrics on miniF2F-Test-PF, a dataset we curated. In particular, ProofBridge improves cross-modal retrieval quality by up to 3.28x Recall@1 over all-MiniLM-L6-v2, and achieves +31.14% SC and +1.64% TC (pass@32) compared to the baseline Kimina-Prover-RL-1.7B.","abs_url":"https://arxiv.org/abs/2510.15681","pdf_url":"https://arxiv.org/pdf/2510.15681v3","primary_category":"cs.LO","published":"2025-10-17","tags":["autoformalization","theorem-proving-ml","ai-agents"],"lean_note":"ProofBridge auto-formalizes entire natural-language theorems and proofs into Lean 4 using joint NL-FL embeddings and Lean type-checker feedback.","repo":"https://github.com/PrithwishJana/ProofBridge","thumb":"thumbs/2510.15681.jpg"},{"id":"2510.14558","title":"HITrees: Higher-Order Interaction Trees","authors":["Amir Mohammad Fadaei Ayyam","Michael Sammler"],"abstract":"Recent years have witnessed the rise of compositional semantics as a foundation for formal verification of complex systems. In particular, interaction trees have emerged as a popular denotational semantics. Interaction trees achieve compositionality by providing a reusable library of effects. However, their notion of effects does not support higher-order effects, i.e., effects that take or return monadic computations. Such effects are essential to model complex semantic features like parallel composition and call/cc. We introduce Higher-Order Interaction Trees (HITrees), the first variant of interaction trees to support higher-order effects in a non-guarded type theory. HITrees accomplish this through two key techniques: first, by designing the notion of effects such that the fixpoints of effects with higher-order input can be expressed as inductive types inside the type theory; and second, using defunctionalization to encode higher-order outputs into a first-order representation. We implement HITrees in the Lean proof assistant, accompanied by a comprehensive library of effects including concurrency, recursion, and call/cc. Furthermore, we provide two interpretations of HITrees, as state transition systems and as monadic programs. To demonstrate the expressiveness of HITrees, we apply them to define the semantics of a language with parallel composition and call/cc.","abs_url":"https://arxiv.org/abs/2510.14558","pdf_url":"https://arxiv.org/pdf/2510.14558v1","primary_category":"cs.PL","published":"2025-10-16","tags":["formalization","verification","type-theory"],"lean_note":"Higher-Order Interaction Trees and a library of effects (concurrency, recursion, call/cc) are implemented in the Lean proof assistant.","thumb":"thumbs/2510.14558.jpg"},{"id":"2510.14480","title":"Certifying optimal MEV strategies with Lean","authors":["Massimo Bartoletti","Riccardo Marchesin","Roberto Zunino"],"abstract":"Maximal Extractable Value (MEV) refers to a class of attacks to decentralized applications where the adversary profits by manipulating the ordering, inclusion, or exclusion of transactions in a blockchain. Decentralized Finance (DeFi) protocols are a primary target of these attacks, as their logic depends critically on transaction sequencing. To date, MEV attacks have already extracted billions of dollars in value, underscoring their systemic impact on blockchain security. Verifying the absence of MEV attacks requires determining suitable upper bounds, i.e. proving that no adversarial strategy can extract more value (if any) than expected by protocol designers. This problem is notoriously difficult: the space of adversarial strategies is extremely vast, making empirical studies and pen-and-paper reasoning insufficiently rigorous. In this paper, we present the first mechanized formalization of MEV in the Lean theorem prover. We introduce a methodology to construct machine-checked proofs of MEV bounds, providing correctness guarantees beyond what is possible with existing techniques. To demonstrate the generality of our approach, we model and analyse the MEV of two paradigmatic DeFi protocols. Notably, we develop the first machine-checked proof of the optimality of sandwich attacks in Automated Market Makers, a fundamental DeFi primitive.","abs_url":"https://arxiv.org/abs/2510.14480","pdf_url":"https://arxiv.org/pdf/2510.14480v1","primary_category":"cs.CR","published":"2025-10-16","tags":["verification","cryptography","formalization"],"lean_note":"Provides the first mechanized formalization of Maximal Extractable Value in Lean, with machine-checked proofs of MEV upper bounds for DeFi protocols.","thumb":"thumbs/2510.14480.jpg"},{"id":"2510.12787","title":"Ax-Prover: A Deep Reasoning Agentic Framework for Theorem Proving in Mathematics and Quantum Physics","authors":["Benjamin Breen","Marco Del Tredici","Jacob McCarran","Javier Aspuru Mijares","Weichen Winston Yin","Kfir Sulimany","Jacob M. Taylor","Frank H. L. Koppens","Dirk Englund"],"abstract":"We present Ax-Prover, a multi-agent system for automated theorem proving in Lean that can solve problems across diverse scientific domains and operate either autonomously or collaboratively with human experts. To achieve this, Ax-Prover approaches scientific problem solving through formal proof generation, a process that demands both creative reasoning and strict syntactic rigor. Ax-Prover meets this challenge by equipping Large Language Models (LLMs), which provide knowledge and reasoning, with Lean tools via the Model Context Protocol (MCP), which ensure formal correctness. To evaluate its performance as an autonomous prover, we benchmark our approach against frontier LLMs and specialized prover models on two public math benchmarks and on two Lean benchmarks we introduce in the fields of abstract algebra and quantum theory. On public datasets, Ax-Prover is competitive with state-of-the-art provers, while it largely outperforms them on the new benchmarks. This shows that, unlike specialized systems that struggle to generalize, our tool-based agentic theorem prover approach offers a generalizable methodology for formal verification across diverse scientific domains. Furthermore, we demonstrate Ax-Prover's assistant capabilities in a practical use case, showing how it enabled an expert mathematician to formalize the proof of a complex cryptography theorem.","abs_url":"https://arxiv.org/abs/2510.12787","pdf_url":"https://arxiv.org/pdf/2510.12787v4","primary_category":"cs.AI","published":"2025-10-14","tags":["theorem-proving-ml","ai-agents","benchmark","verification"],"lean_note":"Ax-Prover is a multi-agent system proving theorems in Lean via LLMs equipped with Lean tools over MCP.","repo":"https://github.com/project-numina/aimo-progress-prize","thumb":"thumbs/2510.12787.jpg"},{"id":"2510.11986","title":"Conjecturing: An Overlooked Step in Formal Mathematical Reasoning","authors":["Jasivan Alex Sivakumar","Philipp Borchert","Ronald Cardenas","Gerasimos Lampouras"],"abstract":"Autoformalisation, the task of expressing informal mathematical statements in formal language, is often viewed as a direct translation process. This, however, disregards a critical preceding step: conjecturing. Many mathematical problems cannot be formalised directly without first conjecturing a conclusion such as an explicit answer, or a specific bound. Since Large Language Models (LLMs) already struggle with autoformalisation, and the evaluation of their conjecturing ability is limited and often entangled within autoformalisation or proof, it is particularly challenging to understand its effect. To address this gap, we augment existing datasets to create ConjectureBench, and redesign the evaluation framework and metric specifically to measure the conjecturing capabilities of LLMs both as a distinct task and within the autoformalisation pipeline. Our evaluation of foundational models, including GPT-4.1 and DeepSeek-V3.1, reveals that their autoformalisation performance is substantially overestimated when the conjecture is accounted for during evaluation. However, the conjecture should not be assumed to be provided. We design an inference-time method, Lean-FIRe to improve conjecturing and autoformalisation, which, to the best of our knowledge, achieves the first successful end-to-end autoformalisation of 13 PutnamBench problems with GPT-4.1 and 7 with DeepSeek-V3.1. We demonstrate that while LLMs possess the requisite knowledge to generate accurate conjectures, improving autoformalisation performance requires treating conjecturing as an independent task, and investigating further how to correctly integrate it within autoformalisation. Finally, we provide forward-looking guidance to steer future research toward improving conjecturing, an overlooked step of formal mathematical reasoning.","abs_url":"https://arxiv.org/abs/2510.11986","pdf_url":"https://arxiv.org/pdf/2510.11986v1","primary_category":"cs.CL","published":"2025-10-13","tags":["autoformalization","theorem-proving-ml","benchmark","ai-agents"],"lean_note":"Introduces ConjectureBench and the Lean-FIRe method, achieving end-to-end autoformalization of PutnamBench problems in Lean.","thumb":"thumbs/2510.11986.jpg"},{"id":"2510.11944","title":"TopoAlign: A Framework for Aligning Code to Math via Topological Decomposition","authors":["Yupei Li","Philipp Borchert","Gerasimos Lampouras"],"abstract":"Large Language Models (LLMs) excel at both informal and formal (e.g. Lean 4) mathematical reasoning but still struggle with autoformalisation, the task of transforming informal into formal mathematical statements. Autoformalisation helps pair the informal reasoning of LLMs with formal proof assistants which enable machine-verifiable generation and mitigate hallucinations. Yet, the performance of current Math LLMs is constrained by the scarcity of large-scale corpora, particularly those containing pairs of informal and formal statements. Although current models are trained to generate code from natural language instructions, structural and syntactic differences between these and formal mathematics limit effective transfer learning. We propose TopoAlign, a framework that unlocks widely available code repositories as training resources for Math LLMs. TopoAlign decomposes code into docstrings, main functions, and dependency functions, and reassembles these components into analogues that structurally mirror formal statements. This produces structurally aligned code data that can be used for training Math LLMs without requiring additional human annotation. We train two state-of-the-art models, DeepSeek-Math and Herald, and evaluate them on the minif2f, Putnam, and ProofNet benchmarks. TopoAlign provides substantial gains for DeepSeek-Math, improving performance by 17.77% on BEq@10 and 68.82% on typecheck@10. Despite introducing no new mathematical knowledge, our framework achieves gains of 0.12% and 1.09% for Herald on BEq@10 and typecheck@10, respectively, demonstrating that training on aligned code data is beneficial even for specialized models.","abs_url":"https://arxiv.org/abs/2510.11944","pdf_url":"https://arxiv.org/pdf/2510.11944v1","primary_category":"cs.CL","published":"2025-10-13","tags":["autoformalization","theorem-proving-ml","dataset","mathlib"],"lean_note":"Aligns code repositories to Lean 4 formal statements to train Math LLMs for autoformalization, evaluated against Mathlib-derived data.","repo":"https://github.com/huawei-noah/TopoAlign","thumb":"thumbs/2510.11944.jpg"},{"id":"2510.11769","title":"GAR: Generative Adversarial Reinforcement Learning for Formal Theorem Proving","authors":["Ruida Wang","Jiarui Yao","Rui Pan","Shizhe Diao","Tong Zhang"],"abstract":"Solving math problems through verifiable languages such as Lean has significantly impacted both the mathematics and computer science communities. Current state-of-the-art models are often trained with expensive online Reinforcement Learning (RL) or expert iteration. However, these approaches rely on fixed problem sets, which causes inefficient training and limits the model to tackle complex problems. To overcome these limitations, we propose **GAR**: *Generative Adversarial Reinforcement learning*, a comprehensive RL training framework that jointly trains the problem composer and solver in an adversarial loop. **GAR** introduces an implicit curriculum learning mechanism, which aligns task difficulty with the prover's evolving capability. It thereby improves the training efficiency and enables stronger performance of proving advanced theorems. Experiments show that with **GAR** training, Goedel-Prover-V2-8B and DeepSeek-Prover-V2-7B achieve an average relative improvement in pass@32 of **4.20%** on MiniF2F-Test benchmark, while DeepSeek-Prover-V2's pass@32 on ProofNet-Test increases from 22.58% to **25.81%**. Beyond formal proving, **GAR** establishes a general RL paradigm for co-evolution of problem generation and solving under verifiable environments. The training code for this paper is open-sourced in https://github.com/RickySkywalker/GAR-Official","abs_url":"https://arxiv.org/abs/2510.11769","pdf_url":"https://arxiv.org/pdf/2510.11769v2","primary_category":"cs.LG","published":"2025-10-13","tags":["theorem-proving-ml","autoformalization","benchmark"],"lean_note":"GAR trains a problem composer and Lean prover adversarially, evaluated on MiniF2F and ProofNet Lean benchmarks.","repo":"https://github.com/RickySkywalker/GAR-Official","thumb":"thumbs/2510.11769.jpg"},{"id":"2510.10815","title":"DRIFT: Decompose, Retrieve, Illustrate, then Formalize Theorems","authors":["Meiru Zhang","Philipp Borchert","Milan Gritta","Gerasimos Lampouras"],"abstract":"Automating the formalization of mathematical statements for theorem proving remains a major challenge for Large Language Models (LLMs). LLMs struggle to identify and utilize the prerequisite mathematical knowledge and its corresponding formal representation in languages like Lean. Current retrieval-augmented autoformalization methods query external libraries using the informal statement directly, but overlook a fundamental limitation: informal statements lack direct mappings to mathematical theorems and lemmata, nor do those theorems translate trivially into the formal primitives of languages like Lean. To address this, we introduce DRIFT, a novel framework that enables LLMs to decompose informal mathematical statements into smaller, more tractable \"sub-components\". This facilitates targeted retrieval of premises from mathematical libraries such as Mathlib. Additionally, DRIFT retrieves illustrative theorems to help models use premises more effectively in formalization tasks. We evaluate DRIFT across diverse benchmarks (ProofNet, ConNF, and MiniF2F-test) and find that it consistently improves premise retrieval, nearly doubling the F1 score compared to the DPR baseline on ProofNet. Notably, DRIFT demonstrates strong performance on the out-of-distribution ConNF benchmark, with BEq+@10 improvements of 42.25% and 37.14% using GPT-4.1 and DeepSeek-V3.1, respectively. Our analysis shows that retrieval effectiveness in mathematical autoformalization depends heavily on model-specific knowledge boundaries, highlighting the need for adaptive retrieval strategies aligned with each model's capabilities.","abs_url":"https://arxiv.org/abs/2510.10815","pdf_url":"https://arxiv.org/pdf/2510.10815v4","primary_category":"cs.AI","published":"2025-10-12","tags":["autoformalization","mathlib","theorem-proving-ml","benchmark"],"lean_note":"DRIFT decomposes informal statements to retrieve Mathlib premises for Lean autoformalization, evaluated on ProofNet, MiniF2F, and ConNF.","repo":"https://github.com/Formal-Math-Reasoning/DRIFT","thumb":"thumbs/2510.10815.jpg"},{"id":"2510.12829","title":"Mathematics with large language models as provers and verifiers","authors":["Hieu Le Duc","Leo Liberti"],"abstract":"During 2024 and 2025 the discussion about the theorem-proving capabilities of large language models started reporting interesting success stories, mostly to do with difficult exercises (such as problems from the International Mathematical Olympiad), but also with conjectures [Feldman & Karbasi, arXiv:2509.18383v1] formulated for the purpose of verifying whether the artificial intelligence could prove it. In this paper we report a theorem proving feat achieved by ChatGPT by using a protocol involving different prover and verifier instances of the gpt-5 model working collaboratively. To make sure that the produced proofs do not suffer from hallucinations, the final proof is formally verified by the lean proof assistant, and the conformance of premises and conclusion of the lean code is verified by a human. Our methodology is by no means complete or exact. It was nonetheless able to solve five out of six 2025 IMO problems, and close about a third of the sixty-six number theory conjectures in [Cohen, Journal of Integer Sequences, 2025].","abs_url":"https://arxiv.org/abs/2510.12829","pdf_url":"https://arxiv.org/pdf/2510.12829v3","primary_category":"cs.CL","published":"2025-10-11","tags":["theorem-proving-ml","ai-agents","formalization","number-theory"],"lean_note":"Final LLM-generated proofs are formally verified by the Lean proof assistant to guard against hallucinated reasoning.","thumb":"thumbs/2510.12829.jpg"},{"id":"2510.08672","title":"A Formalization of the Generalized Quantum Stein's Lemma in Lean","authors":["Alex Meiburg","Leonardo A. Lessa","Rodolfo R. Soldati"],"abstract":"The Generalized Quantum Stein's Lemma is a theorem in quantum hypothesis testing that provides an operational meaning to the relative entropy within the context of quantum resource theories. Its original proof was found to have a gap, which led to a search for a corrected proof. We formalize the proof presented in [Hayashi and Yamasaki (2024)] in the Lean interactive theorem prover. This is the most technically demanding theorem in physics with a computer-verified proof to date, building with a variety of intermediate results from topology, analysis, and operator algebra. In the process, we rectified minor imprecisions in [HY24]'s proof that formalization forces us to confront, and refine a more precise definition of quantum resource theory. Formalizing this theorem has ensured that our Lean-QuantumInfo library, which otherwise has begun to encompass a variety of topics from quantum information, includes a robust foundation suitable for a larger collaborative program of formalizing quantum theory more broadly.","abs_url":"https://arxiv.org/abs/2510.08672","pdf_url":"https://arxiv.org/pdf/2510.08672v1","primary_category":"quant-ph","published":"2025-10-09","tags":["formalization","verification","quantum","analysis"],"lean_note":"Formalizes the Generalized Quantum Stein's Lemma in Lean, extending a Lean-QuantumInfo library across topology, analysis, and operator algebra.","repo":"https://github.com/Timeroot/Lean-QuantumInfo","thumb":"thumbs/2510.08672.jpg"},{"id":"2510.15940","title":"Lean Finder: Semantic Search for Mathlib That Understands User Intents","authors":["Jialin Lu","Kye Emond","Kaiyu Yang","Swarat Chaudhuri","Weiran Sun","Wuyang Chen"],"abstract":"We present Lean Finder, a semantic search engine for Lean and mathlib that understands and aligns with the intents of mathematicians. Progress in formal theorem proving is often hindered by the difficulty of locating relevant theorems and the steep learning curve of the Lean 4 language, making advancement slow and labor-intensive. Existing Lean search engines, though helpful, rely primarily on informalizations (natural language translation of the formal statements), while largely overlooking the mismatch with real-world user queries. In contrast, we propose a user-centered semantic search tailored to the needs of mathematicians. Our approach begins by analyzing and clustering the semantics of public Lean discussions, then fine-tuning text embeddings on synthesized queries that emulate user intents. We further align Lean Finder with mathematicians' preferences using diverse feedback signals, encoding it with a rich awareness of their goals from multiple perspectives. Evaluations on real-world queries, informalized statements, and proof states demonstrate that our Lean Finder achieves over $30\\%$ relative improvement compared to previous search engines and GPT-4o. In addition, Lean Finder is compatible with LLM-based theorem provers, bridging retrieval with formal reasoning. Lean Finder is available at: https://leanfinder.github.io","abs_url":"https://arxiv.org/abs/2510.15940","pdf_url":"https://arxiv.org/pdf/2510.15940v2","primary_category":"cs.LG","published":"2025-10-08","tags":["tooling","mathlib","theorem-proving-ml"],"lean_note":"Builds Lean Finder, a semantic search engine over Lean and Mathlib fine-tuned to match mathematicians' query intents.","repo":"https://github.com/justincasher/lean-explore","thumb":"thumbs/2510.15940.jpg"},{"id":"2510.06857","title":"Autoformalizer with Tool Feedback","authors":["Qi Guo","Jianing Wang","Jianfei Zhang","Deyang Kong","Xiangzhou Huang","Xiangyu Xi","Wei Wang","Jingang Wang","Xunliang Cai","Shikun Zhang","Wei Ye"],"abstract":"Autoformalization addresses the scarcity of data for Automated Theorem Proving (ATP) by translating mathematical problems from natural language into formal statements. Efforts in recent work shift from directly prompting large language models to training an end-to-end formalizer model from scratch, achieving remarkable advancements. However, existing formalizer still struggles to consistently generate valid statements that meet syntactic validity and semantic consistency. To address this issue, we propose the Autoformalizer with Tool Feedback (ATF), a novel approach that incorporates syntactic and consistency information as tools into the formalization process. By integrating Lean 4 compilers for syntax corrections and employing a multi-LLMs-as-judge approach for consistency validation, the model is able to adaptively refine generated statements according to the tool feedback, enhancing both syntactic validity and semantic consistency. The training of ATF involves a cold-start phase on synthetic tool-calling data, an expert iteration phase to improve formalization capabilities, and Direct Preference Optimization to alleviate ineffective revisions. Experimental results show that ATF markedly outperforms a range of baseline formalizer models, with its superior performance further validated by human evaluations. Subsequent analysis reveals that ATF demonstrates excellent inference scaling properties. Moreover, we open-source Numina-ATF, a dataset containing 750K synthetic formal statements to facilitate advancements in autoformalization and ATP research.","abs_url":"https://arxiv.org/abs/2510.06857","pdf_url":"https://arxiv.org/pdf/2510.06857v1","primary_category":"cs.AI","published":"2025-10-08","tags":["autoformalization","theorem-proving-ml","dataset","tooling"],"lean_note":"Trains an autoformalizer using Lean 4 compiler syntax feedback and releases Numina-ATF, 750K synthetic Lean formal statements.","repo":"https://github.com/qguo-create/Autoformalizer-with-Tool-Feedback","thumb":"thumbs/2510.06857.jpg"},{"id":"2510.21741","title":"Virasoro algebra and Sugawara constructions formally in Lean","authors":["Kalle Kytölä"],"abstract":"We formalize in Lean certain calculational proofs about infinite-dimensional Lie algebras. Specifically, we construct the Virasoro algebra as a central extension of the Witt algebra associated with a nontrivial 2-cocycle, and we construct representations of the Virasoro algebra by Sugawara constructions.","abs_url":"https://arxiv.org/abs/2510.21741","pdf_url":"https://arxiv.org/pdf/2510.21741v1","primary_category":"math.QA","published":"2025-10-06","tags":["formalization","algebra"],"lean_note":"Formalizes in Lean the Virasoro algebra as a central extension of the Witt algebra and its Sugawara representations.","thumb":"thumbs/2510.21741.jpg"},{"id":"2510.04520","title":"Aria: An Agent For Retrieval and Iterative Auto-Formalization via Dependency Graph","authors":["Hanyu Wang","Ruohan Xie","Yutong Wang","Guoxiong Gao","Xintao Yu","Bin Dong"],"abstract":"Accurate auto-formalization of theorem statements is essential for advancing automated discovery and verification of research-level mathematics, yet remains a major bottleneck for LLMs due to hallucinations, semantic mismatches, and their inability to synthesize new definitions. To tackle these issues, we present Aria (Agent for Retrieval and Iterative Autoformalization), a system for conjecture-level formalization in Lean that emulates human expert reasoning via a two-phase Graph-of-Thought process: recursively decomposing statements into a dependency graph and then constructing formalizations from grounded concepts. To ensure semantic correctness, we introduce AriaScorer, a checker that retrieves definitions from Mathlib for term-level grounding, enabling rigorous and reliable verification. We evaluate Aria on diverse benchmarks. On ProofNet, it achieves 91.6% compilation success rate and 68.5% final accuracy, surpassing previous methods. On FATE-X, a suite of challenging algebra problems from research literature, it outperforms the best baseline with 44.0% vs. 24.0% final accuracy. On a dataset of homological conjectures, Aria reaches 42.9% final accuracy while all other models score 0%.","abs_url":"https://arxiv.org/abs/2510.04520","pdf_url":"https://arxiv.org/pdf/2510.04520v1","primary_category":"cs.AI","published":"2025-10-06","tags":["autoformalization","ai-agents","mathlib","theorem-proving-ml"],"lean_note":"Aria is an LLM agent for conjecture-level autoformalization in Lean, grounding terms against Mathlib definitions via the AriaScorer checker.","thumb":"thumbs/2510.04520.jpg"},{"id":"2510.04070","title":"Markov kernels in Mathlib's probability library","authors":["Rémy Degenne"],"abstract":"The probability folder of Mathlib, Lean's mathematical library, makes a heavy use of Markov kernels. We present their definition and properties and describe the formalization of the disintegration theorem for Markov kernels. That theorem is used to define conditional probability distributions of random variables as well as posterior distributions. We then explain how Markov kernels are used in a more unusual way to get a common definition of independence and conditional independence and, following the same principles, to define sub-Gaussian random variables. Finally, we also discuss the role of kernels in our formalization of entropy and Kullback-Leibler divergence.","abs_url":"https://arxiv.org/abs/2510.04070","pdf_url":"https://arxiv.org/pdf/2510.04070v2","primary_category":"cs.DL","published":"2025-10-05","tags":["formalization","mathlib","probability"],"lean_note":"Describes the formalization of Markov kernels and the disintegration theorem in Mathlib's probability library in Lean.","thumb":"thumbs/2510.04070.jpg"},{"id":"2510.01346","title":"Aristotle: IMO-level Automated Theorem Proving","authors":["Tudor Achim","Alex Best","Alberto Bietti","Kevin Der","Mathïs Fédérico","Sergei Gukov","Daniel Halpern-Leistner","Kirsten Henningsgard","Yury Kudryashov","Alexander Meiburg","Martin Michelsen","Riley Patterson","Eric Rodriguez","Laura Scharff","Vikram Shanker","Vladmir Sicca","Hari Sowrirajan","Aidan Swope","Matyas Tamas","Vlad Tenev","Jonathan Thomm","Harold Williams","Lawrence Wu"],"abstract":"We introduce Aristotle, an AI system that combines formal verification with informal reasoning, achieving gold-medal-equivalent performance on the 2025 International Mathematical Olympiad problems. Aristotle integrates three main components: a Lean proof search system, an informal reasoning system that generates and formalizes lemmas, and a dedicated geometry solver. Our system demonstrates state-of-the-art performance with favorable scaling properties for automated theorem proving.","abs_url":"https://arxiv.org/abs/2510.01346","pdf_url":"https://arxiv.org/pdf/2510.01346v2","primary_category":"cs.AI","published":"2025-10-01","tags":["theorem-proving-ml","ai-agents","geometry"],"lean_note":"Aristotle combines a Lean proof search system with informal reasoning and a geometry solver for IMO-level theorem proving.","thumb":"thumbs/2510.01346.jpg"},{"id":"2510.02335","title":"FormalML: A Benchmark for Evaluating Formal Subgoal Completion in Machine Learning Theory","authors":["Xiao-Wen Yang","Zihao Zhang","Jianuo Cao","Zhi Zhou","Zenan Li","Lan-Zhe Guo","Yuan Yao","Taolue Chen","Yu-Feng Li","Xiaoxing Ma"],"abstract":"Large language models (LLMs) have recently demonstrated remarkable progress in formal theorem proving. Yet their ability to serve as practical assistants for mathematicians, filling in missing steps within complex proofs, remains underexplored. We identify this challenge as the task of subgoal completion, where an LLM must discharge short but nontrivial proof obligations left unresolved in a human-provided sketch. To study this problem, we introduce FormalML, a Lean 4 benchmark built from foundational theories of machine learning. Using a translation tactic that converts procedural proofs into declarative form, we extract 4937 problems spanning optimization and probability inequalities, with varying levels of difficulty. FormalML is the first subgoal completion benchmark to combine premise retrieval and complex research-level contexts. Evaluation of state-of-the-art provers highlights persistent limitations in accuracy and efficiency, underscoring the need for more capable LLM-based theorem provers for effective subgoal completion,","abs_url":"https://arxiv.org/abs/2510.02335","pdf_url":"https://arxiv.org/pdf/2510.02335v1","primary_category":"cs.CL","published":"2025-09-26","tags":["benchmark","dataset","theorem-proving-ml","formalization"],"lean_note":"Introduces FormalML, a Lean 4 benchmark of 4937 machine-learning-theory subgoal-completion problems and evaluates LLM provers on it.","repo":"https://github.com/njuyxw/FormalML","thumb":"thumbs/2510.02335.jpg"},{"id":"2509.22908","title":"A benchmark for vericoding: formally verified program synthesis","authors":["Sergiu Bursuc","Theodore Ehrenborg","Shaowei Lin","Lacramioara Astefanoaei","Ionel Emilian Chiosa","Jure Kukovec","Alok Singh","Oliver Butterley","Adem Bizid","Quinn Dougherty","Miranda Zhao","Max Tan","Max Tegmark"],"abstract":"We present and test the largest benchmark for vericoding, LLM-generation of formally verified code from formal specifications - in contrast to vibe coding, which generates potentially buggy code from a natural language description. Our benchmark contains 12,504 formal specifications, with 3,029 in Dafny, 2,334 in Verus/Rust and 7,141 in Lean. Of these, 6,174 are new unseen problems. We find vericoding success rates of 27% in Lean, 44% in Verus/Rust and 82% in Dafny using off-the-shelf LLMs. Adding natural-language descriptions does not significantly improve performance. We also find that LLM progress has improved progress on pure Dafny verification from 68% to 96% over the past year. The benchmark and vericoding results are shared at https://github.com/Beneficial-AI-Foundation/vericoding-benchmark","abs_url":"https://arxiv.org/abs/2509.22908","pdf_url":"https://arxiv.org/pdf/2509.22908v1","primary_category":"cs.SE","published":"2025-09-26","tags":["benchmark","dataset","verification","theorem-proving-ml"],"lean_note":"Includes 7,141 Lean formal specifications in a vericoding benchmark and reports LLM verified-synthesis success rates on them.","repo":"https://github.com/Beneficial-AI-Foundation/vericoding-benchmark","thumb":"thumbs/2509.22908.jpg"},{"id":"2509.22819","title":"Hilbert: Recursively Building Formal Proofs with Informal Reasoning","authors":["Sumanth Varambally","Thomas Voice","Yanchao Sun","Zhifeng Chen","Rose Yu","Ke Ye"],"abstract":"Large Language Models (LLMs) demonstrate impressive mathematical reasoning abilities, but their solutions frequently contain errors that cannot be automatically checked. Formal theorem proving systems such as Lean 4 offer automated verification with complete accuracy, motivating recent efforts to build specialized prover LLMs that generate verifiable proofs in formal languages. However, a significant gap remains: current prover LLMs solve substantially fewer problems than general-purpose LLMs operating in natural language. We introduce Hilbert, an agentic framework that bridges this gap by combining the complementary strengths of informal reasoning and formal verification. Our system orchestrates four components: an informal LLM that excels at mathematical reasoning, a specialized prover LLM optimized for Lean 4 tactics, a formal verifier, and a semantic theorem retriever. Given a problem that the prover is unable to solve, Hilbert employs recursive decomposition to split the problem into subgoals that it solves with the prover or reasoner LLM. It leverages verifier feedback to refine incorrect proofs as necessary. Experimental results demonstrate that Hilbert substantially outperforms existing approaches on key benchmarks, achieving 99.2\\% on miniF2F, 6.6\\% points above the best publicly available method. Hilbert achieves the \\textbf{strongest known result} from a publicly available model on PutnamBench. It solves 462/660 problems (70.0\\%), outperforming proprietary approaches like SeedProver (50.4\\%) and achieving a 422\\% improvement over the best publicly available baseline. Thus, Hilbert effectively narrows the gap between informal reasoning and formal proof generation. Code is available at https://github.com/Rose-STL-Lab/ml-hilbert.","abs_url":"https://arxiv.org/abs/2509.22819","pdf_url":"https://arxiv.org/pdf/2509.22819v2","primary_category":"cs.AI","published":"2025-09-26","tags":["theorem-proving-ml","ai-agents","benchmark"],"lean_note":"Presents Hilbert, an agentic framework that generates and verifies Lean 4 proofs via recursive decomposition and verifier feedback.","repo":"https://github.com/Rose-STL-Lab/ml-hilbert","thumb":"thumbs/2509.22819.jpg"},{"id":"2509.20539","title":"Composition Direction of Seymour's Theorem for Regular Matroids -- Formally Verified","authors":["Martin Dvorak","Tristan Figueroa-Reid","Rida Hamadani","Byung-Hak Hwang","Evgenia Karunus","Vladimir Kolmogorov","Alexander Meiburg","Alexander Nelson","Peter Nelson","Mark Sandey","Ivan Sergeev"],"abstract":"Seymour's decomposition theorem is a hallmark result in matroid theory presenting a structural characterization of the class of regular matroids. Formalization of matroid theory faces many challenges, most importantly that only a limited number of notions and results have been implemented so far. In this work, we formalize the proof of the forward (composition) direction of Seymour's theorem for regular matroids. To this end, we develop a library in Lean 4 that implements definitions and results about totally unimodular matrices, vector matroids, their standard representations, regular matroids, and 1-, 2-, and 3-sums of matrices and binary matroids given by their standard representations. Using this framework, we formally state Seymour's decomposition theorem and implement a formally verified proof of the composition direction in the setting where the matroids have finite rank and may have infinite ground sets.","abs_url":"https://arxiv.org/abs/2509.20539","pdf_url":"https://arxiv.org/pdf/2509.20539v1","primary_category":"math.CO","published":"2025-09-24","tags":["formalization","combinatorics","verification"],"lean_note":"Develops a Lean 4 library on totally unimodular matrices and matroids to formally prove the composition direction of Seymour's decomposition theorem.","thumb":"thumbs/2509.20539.jpg"},{"id":"2509.19632","title":"Formalization of Harder-Narasimhan theory","authors":["Yijun Yuan"],"abstract":"The Harder-Narasimhan theory provides a canonical filtration of a vector bundle on a projective curve whose successive quotients are semistable with strictly decreasing slopes. In this article, we present the formalization of Harder-Narasimhan theory in the proof assistant Lean 4 with Mathlib. This formalization is based on a recent approach of Harder-Narasimhan theory by Chen and Jeannin, which reinterprets the theory in order-theoretic terms and avoids the classical dependence on algebraic geometry. As an application, we formalize the uniqueness of coprimary filtration of a finitely generated module over a noetherian ring, and the existence of the Jordan-Hölder filtration of a semistable Harder-Narasimhan game. Code available at: https://github.com/YijunYuan/HarderNarasimhan","abs_url":"https://arxiv.org/abs/2509.19632","pdf_url":"https://arxiv.org/pdf/2509.19632v3","primary_category":"math.AG","published":"2025-09-23","tags":["formalization","mathlib","algebra"],"lean_note":"Formalizes Harder-Narasimhan theory in Lean 4 with Mathlib via an order-theoretic reformulation avoiding algebraic geometry.","repo":"https://github.com/YijunYuan/HarderNarasimhan","thumb":"thumbs/2509.19632.jpg"},{"id":"2509.16370","title":"Dual-Regularized Riccati Recursions for Interior-Point Optimal Control","authors":["João Sousa-Pinto","Dominique Orban"],"abstract":"We derive closed-form extensions of the sequential and parallel Riccati recursions for solving dual-regularized linear-quadratic regulator (LQR) problems, with $O(N)$ sequential time and $O(\\log(N))$ parallel time, respectively. We show that these subproblems arise when using regularized primal-dual interior-point methods to solve smooth, constrained, non-convex, discrete-time optimal control problems via multiple-shooting, even in the presence of stagewise equality or inequality constraints, and without imposing any rank requirements on constraint Jacobians. We prove that, when certain inertia conditions on the Newton-KKT matrix are met, each nonzero primal step is a descent direction of an augmented barrier-Lagrangian merit function. We characterize these inertia conditions in terms of the positive-definiteness of the dual-regularized Riccati pivots (a weaker condition than the standard LQR positive-definiteness requirements), thereby yielding inexpensive certificates of the required inertia. We provide MIT-licensed implementations of our methods in C++ and in JAX, as well as a full formalization of our results in Lean. We benchmark our algorithm against leading optimal control and nonlinear programming solvers on complex trajectory optimization problems, establishing competitive performance on moderate problems and substantial gains as the horizon length, problem dimension, and constraint count increase.","abs_url":"https://arxiv.org/abs/2509.16370","pdf_url":"https://arxiv.org/pdf/2509.16370v6","primary_category":"math.OC","published":"2025-09-19","tags":["formalization","verification"],"lean_note":"Provides a full Lean formalization of dual-regularized Riccati recursions and their inertia/descent results for interior-point optimal control.","repo":"https://github.com/joaospinto/sip_optimal_control","thumb":"thumbs/2509.16370.jpg"},{"id":"2509.15116","title":"The mechanization of science illustrated by the Lean formalization of the multi-graded Proj construction","authors":["Arnaud Mayeux","Jujian Zhang"],"abstract":"We formalize the multi-graded Proj construction in Lean4, illustrating mechanized mathematics and formalization.","abs_url":"https://arxiv.org/abs/2509.15116","pdf_url":"https://arxiv.org/pdf/2509.15116v1","primary_category":"cs.LO","published":"2025-09-18","tags":["formalization","algebra","geometry"],"lean_note":"Formalizes the multi-graded Proj construction from algebraic geometry in Lean 4.","repo":"https://github.com/ProjConstruction/Proj","thumb":"thumbs/2509.15116.jpg"},{"id":"2509.16239","title":"Gödel Mirror: A Formal System For Contradiction-Driven Recursion","authors":["Jhet Chan"],"abstract":"We introduce the Gödel Mirror, a formal system defined in Lean 4 that treats contradiction as a control signal for recursive structural evolution. Inspired by Gödelian self-reference, our system's operational semantics encode symbolic paradoxes as deterministic transitions. Unlike systems designed to guarantee normalization, the Gödel Mirror is a minimal and verifiable architecture that leverages a controlled, non-terminating loop as a productive feature. Our Lean 4 mechanization proves that self-referential paradoxes are deterministically encapsulated and resolved into new structures without leading to logical explosion, yielding a paraconsistent inference loop: Paradox -> Encapsulate -> Reenter -> Node We argue that this calculus opens a new class of symbolic systems in which contradiction is metabolized into structure, providing a formal basis for agents capable of resolving internal inconsistencies.","abs_url":"https://arxiv.org/abs/2509.16239","pdf_url":"https://arxiv.org/pdf/2509.16239v1","primary_category":"cs.LO","published":"2025-09-16","tags":["formalization","logic","verification"],"lean_note":"Defines and mechanizes the Gödel Mirror, a contradiction-driven recursion calculus, in Lean 4 with machine-checked metatheory.","repo":"https://github.com/jhetchan/godel-mirror","thumb":"thumbs/2509.16239.jpg"},{"id":"2509.14274","title":"Discovering New Theorems via LLMs with In-Context Proof Learning in Lean","authors":["Kazumi Kasaura","Naoto Onda","Yuta Oriike","Masaya Taniguchi","Akiyoshi Sannai","Sho Sonoda"],"abstract":"Large Language Models (LLMs) have demonstrated significant promise in formal theorem proving. In this study, we investigate the ability of LLMs to discover novel theorems and produce verified proofs. We propose a pipeline called \\textit{Conjecturing-Proving Loop} (CPL), which iteratively generates mathematical conjectures and attempts to prove them in Lean 4. A key feature of CPL is that each iteration conditions the LLM on previously generated theorems and their formal proofs, enabling parameter-free improvement of proof strategies via in-context learning. We provide both theoretical and experimental evidence that CPL increases the discovery rate of hard-to-prove theorems compared to frameworks that generate statements and proofs simultaneously. Moreover, our experiments show that reusing the LLM's own formally verified outputs as context consistently improves subsequent proof success, demonstrating the effectiveness of self-generated in-context learning for neural theorem proving. The source code is available at https://github.com/auto-res/ConjecturingProvingLoop.","abs_url":"https://arxiv.org/abs/2509.14274","pdf_url":"https://arxiv.org/pdf/2509.14274v2","primary_category":"cs.LG","published":"2025-09-16","tags":["theorem-proving-ml","autoformalization","ai-agents"],"lean_note":"Iteratively generates and proves novel mathematical conjectures in Lean 4 via a Conjecturing-Proving Loop with in-context proof learning.","repo":"https://github.com/auto-res/ConjecturingProvingLoop","thumb":"thumbs/2509.14274.jpg"},{"id":"2509.13142","title":"Formalizing dimensional analysis using the Lean theorem prover","authors":["Maxwell P. Bobbin","Colin Jones","John Velkey","Tyler R. Josephson"],"abstract":"Dimensional analysis is fundamental to the formulation and validation of physical laws, ensuring that equations are dimensionally homogeneous and scientifically meaningful. In this work, we use Lean 4 to formalize the mathematics of dimensional analysis. We define physical dimensions as mappings from base dimensions to exponents, prove that they form an Abelian group under multiplication, and implement derived dimensions and dimensional homogeneity theorems. Building on this foundation, we introduce a definition of physical variables that combines numeric values with dimensions, extend the framework to incorporate SI base units and fundamental constants, and implement the Buckingham Pi Theorem. Finally, we demonstrate the approach on an example: the Lennard-Jones potential, where our framework enforces dimensional consistency and enables formal proofs of physical properties such as zero-energy separation and the force law. This work establishes a reusable, formally verified framework for dimensional analysis in Lean, providing a foundation for future libraries in formalized science and a pathway toward scientific computing environments with built-in guarantees of dimensional correctness.","abs_url":"https://arxiv.org/abs/2509.13142","pdf_url":"https://arxiv.org/pdf/2509.13142v1","primary_category":"physics.chem-ph","published":"2025-09-16","tags":["formalization","verification","physics"],"lean_note":"Uses Lean 4 to formalize dimensional analysis, proving physical dimensions form an Abelian group and implementing the Buckingham Pi Theorem.","repo":"https://github.com/ecyrbe/lean-units","thumb":"thumbs/2509.13142.jpg"},{"id":"2509.13020","title":"Łukasiewicz Logic with Actions for Neural Networks training","authors":["Ioana Leuştean","Bogdan Macovei"],"abstract":"Based on the already known connection between multilayer perceptrons and Lukasiewicz logic with rational coefficients, we take a step forward in analyzing its training process using a three-sorted hybrid modal logic: a multilayer perceptron is a logical formula; the actions of the training process are modal operators; the training process is a sequence of logical deductions. Using the proof assistant and the programming language Lean 4, the algorithmic implementation of the training process is certified by logical proofs.","abs_url":"https://arxiv.org/abs/2509.13020","pdf_url":"https://arxiv.org/pdf/2509.13020v1","primary_category":"cs.LO","published":"2025-09-16","tags":["formalization","verification","logic","ml-theory"],"lean_note":"Uses Lean 4 to certify a neural-network training process via logical proofs, treating perceptrons as Łukasiewicz logic formulas.","thumb":"thumbs/2509.13020.jpg"},{"id":"2509.10828","title":"Structuring Definitions in Mathematical Libraries","authors":["Alena Gusakov","Peter Nelson","Stephen Watt"],"abstract":"Codifying mathematical theories in a proof assistant or computer algebra system is a challenging task, of which the most difficult part is, counterintuitively, structuring definitions. This results in a steep learning curve for new users and slow progress in formalizing even undergraduate level mathematics. There are many considerations one has to make, such as level of generality, readability, and ease of use in the type system, and there are typically multiple equivalent or related definitions from which to choose. Often, a definition that is ultimately selected for formalization is settled on after a lengthy trial and error process. This process involves testing potential definitions for usability by formalizing standard theorems about them, and weeding out the definitions that are unwieldy. Inclusion of a formal definition in a centralized community-run mathematical library is typically an indication that the definition is \"good.\" For this reason, in this survey, we make some observations about what makes a definition \"good,\" and examine several case studies of the refining process for definitions that have ultimately been added to the Lean Theorem Prover community-run mathematical library, mathlib. We observe that some of the difficulties are shared with the design of libraries for computer algebra systems, and give examples of related issues in that context.","abs_url":"https://arxiv.org/abs/2509.10828","pdf_url":"https://arxiv.org/pdf/2509.10828v2","primary_category":"cs.SC","published":"2025-09-13","tags":["mathlib","formalization","meta-study"],"lean_note":"Surveys, with mathlib case studies, what makes a formal definition \"good\" when added to the Lean library.","thumb":"thumbs/2509.10828.jpg"},{"id":"2509.09726","title":"Natural Language Translation of Formal Proofs through Informalization of Proof Steps and Recursive Summarization along Proof Structure","authors":["Seiji Hattori","Takuya Matsuzaki","Makoto Fujiwara"],"abstract":"This paper proposes a natural language translation method for machine-verifiable formal proofs that leverages the informalization (verbalization of formal language proof steps) and summarization capabilities of LLMs. For evaluation, it was applied to formal proof data created in accordance with natural language proofs taken from an undergraduate-level textbook, and the quality of the generated natural language proofs was analyzed in comparison with the original natural language proofs. Furthermore, we will demonstrate that this method can output highly readable and accurate natural language proofs by applying it to existing formal proof library of the Lean proof assistant.","abs_url":"https://arxiv.org/abs/2509.09726","pdf_url":"https://arxiv.org/pdf/2509.09726v1","primary_category":"cs.CL","published":"2025-09-10","tags":["autoformalization","theorem-proving-ml"],"lean_note":"Translates formal proofs from the Lean proof assistant library into natural language via LLM informalization and recursive summarization.","repo":"https://github.com/hattori-matsuzakilab/AutoInformalizationWithTemplate","thumb":"thumbs/2509.09726.jpg"},{"id":"2509.07609","title":"What's in the Box: Ergonomic and Expressive Capture Tracking over Generic Data Structures (Extended Version)","authors":["Yichen Xu","Oliver Bračevac","Cao Nguyen Pham","Martin Odersky"],"abstract":"Capturing types in Scala unify static effect and resource tracking with object capabilities, enabling lightweight effect polymorphism with minimal notational overhead. However, their expressiveness has been insufficient for tracking capabilities embedded in generic data structures, preventing them from scaling to the standard collections library -- an essential prerequisite for broader adoption. This limitation stems from the inability to name capabilities within the system's notion of box types. This paper develops System Capless, a new foundation for capturing types that provides the theoretical basis for reach capabilities (rcaps), a novel mechanism for naming \"what's in the box.\" The calculus refines the universal capability notion into a new scheme with existential and universal capture set quantification. Intuitively, rcaps witness existentially quantified capture sets inside the boxes of generic types in a way that does not require exposing existential capture types in the surface language. We have fully mechanized the formal metatheory of System Capless in Lean, including proofs of type soundness and scope safety. System Capless supports the same lightweight notation of capturing types plus rcaps, as certified by a type-preserving translation, and also enables fully optional explicit capture-set quantification to increase expressiveness. Finally, we present a full reimplementation of capture checking in Scala 3 based on System Capless and migrate the entire Scala collections library and an asynchronous programming library to evaluate its practicality and ergonomics. Our results demonstrate that reach capabilities enable the adoption of capture checking in production code with minimal changes and minimal-to-zero notational overhead in a vast majority of cases.","abs_url":"https://arxiv.org/abs/2509.07609","pdf_url":"https://arxiv.org/pdf/2509.07609v1","primary_category":"cs.PL","published":"2025-09-09","tags":["type-theory","verification"],"lean_note":"Fully mechanizes the metatheory of System Capless, including type soundness and scope safety proofs, in Lean 4.","repo":"https://github.com/scala/scala3","thumb":"thumbs/2509.07609.jpg"},{"id":"2509.06614","title":"A Secure Sequencer and Data Availability Committee for Rollups (Extended Version)","authors":["Margarita Capretto","Martín Ceresa","Antonio Fernández Anta","Pedro Moreno-Sanchez","César Sánchez"],"abstract":"Blockchains face a scalability limitation, partly due to the throughput limitations of consensus protocols, especially when aiming to obtain a high degree of decentralization. Layer 2 Rollups (L2s) are a faster alternative to conventional blockchains. L2s perform most computations offchain using minimally blockchains (L1) under-the-hood to guarantee correctness. A sequencer is a service that receives offchain L2 transaction requests, batches these transactions, and commits compressed or hashed batches to L1. Using hashing needs less L1 space, which is beneficial for gas cost, but requires a data availability committee (DAC) service to translate hashes into their corresponding batches of transaction requests. The behavior of sequencers and DACs influence the evolution of the L2 blockchain, presenting a potential security threat and delaying L2 adoption. We propose in this paper fraud-proof mechanisms, arbitrated by L1 contracts, to detect and generate evidence of dishonest behavior of the sequencer and DAC. We study how these fraud-proofs limit the power of adversaries that control different number of sequencer and DACs members, and provide incentives for their honest behavior. We designed these fraud-proof mechanisms as two player games. Unlike the generic fraud-proofs in current L2s (designed to guarantee the correct execution of transactions), our fraud-proofs are over pred-etermined algorithms that verify the properties that determine the correctness of the DAC. Arbitrating over concrete algorithms makes our fraud-proofs more efficient, easier to understand, and simpler to prove correct. We provide as an artifact a mechanization in LEAN4 of our fraud-proof games, including (1) the verified strategies that honest players should play to win all games as well as (2) mechanisms to detect dishonest claims.","abs_url":"https://arxiv.org/abs/2509.06614","pdf_url":"https://arxiv.org/pdf/2509.06614v2","primary_category":"cs.CR","published":"2025-09-08","tags":["verification","cryptography"],"lean_note":"Mechanizes fraud-proof games for rollup sequencers and data-availability committees in LEAN4, including verified honest strategies.","repo":"https://github.com/OffchainLabs/nitro","thumb":"thumbs/2509.06614.jpg"},{"id":"2509.04922","title":"Higher order differential calculus in mathlib","authors":["Sébastien Gouëzel"],"abstract":"We report on the higher-order differential calculus library developed inside the Lean mathematical library mathlib. To support a broad range of applications, we depart in several ways from standard textbook definitions: we allow arbitrary fields of scalars, we work with functions defined on domains rather than full spaces, and we integrate analytic functions in the broader scale of smooth functions. These generalizations introduce significant challenges, which we address from both the mathematical and the formalization perspectives.","abs_url":"https://arxiv.org/abs/2509.04922","pdf_url":"https://arxiv.org/pdf/2509.04922v1","primary_category":"cs.LO","published":"2025-09-05","tags":["formalization","mathlib","analysis"],"lean_note":"Reports on the higher-order differential calculus library formalized inside Lean's mathematical library Mathlib.","thumb":"thumbs/2509.04922.jpg"},{"id":"ext-034","title":"Verifying Datalog Reasoning with Lean","authors":["Johannes Tantow","Lukas Gerlach","Stephan Mennicke","Markus Krötzsch"],"abstract":"This paper presents a framework for verifying Datalog reasoning results in the Lean 4 theorem prover. The authors develop a certified checker that can verify the output of Datalog engines, ensuring that derived facts are correct consequences of the input rules and facts. The approach bridges the gap between efficient Datalog evaluation and formally verified correctness guarantees.","abs_url":"https://iccl.inf.tu-dresden.de/web/Inproceedings3433","pdf_url":"https://iccl.inf.tu-dresden.de/web/Inproceedings3433","primary_category":"cs.LO","published":"2025-09-01","tags":["verification","logic","tooling"],"lean_note":"Builds a certified Datalog reasoning checker in Lean 4 for verifying rule-based inference results."},{"id":"2508.21593","title":"Growing Mathlib: maintenance of a large scale mathematical library","authors":["Anne Baanen","Matthew Robert Ballard","Johan Commelin","Bryan Gin-ge Chen","Michael Rothgang","Damiano Testa"],"abstract":"The Lean mathematical library Mathlib is one of the fastest-growing libraries of formalised mathematics. We describe various strategies to manage this growth, while allowing for change and avoiding maintainer overload. This includes dealing with breaking changes via a deprecation system, using code quality analysis tools (linters) to provide direct user feedback about common pitfalls, speeding up compilation times through conscious library (re-)design, dealing with technical debt as well as writing custom tooling to help with the review and triage of new contributions.","abs_url":"https://arxiv.org/abs/2508.21593","pdf_url":"https://arxiv.org/pdf/2508.21593v2","primary_category":"cs.PL","published":"2025-08-29","tags":["mathlib","tooling","meta-study","formalization"],"lean_note":"Describes strategies, linters, deprecation systems, and custom tooling for maintaining the Lean Mathlib library.","repo":"https://github.com/IPDSnelting/velcom","thumb":"thumbs/2508.21593.jpg"},{"id":"2508.20671","title":"A Unifying Framework for Global Optimization: From Theory to Formalization","authors":["Gaëtan Serré","Argyris Kalogeratos","Nicolas Vayatis"],"abstract":"We introduce an abstract measure___theoretic framework that serves as a tool to rigorously study stochastic iterative global optimization algorithms as a unified class. The framework is formulated in terms of probability kernels, which, via the Ionescu--Tulcea theorem, induce probability measures on the space of sequences of algorithm iterations, endowed with two intuitive properties. This framework answers the need for a general, implementation___independent formalism in the analysis of such algorithms, providing a starting point for formalizing global optimization results in proof-assistants. To illustrate the relevance of our tool, we show that common algorithms fit naturally in the framework, and we also use it to give a rigorous proof of a general consistency theorem for stochastic iterative global optimization algorithms (Proposition 3 of (Malherbe, et al., 2017). This proof and the entire framework are formalized in the Lean proof assistant. This formalization both ensures the correctness of the definitions and proofs, and provides a basis for future machine-assisted formalizations in the field.","abs_url":"https://arxiv.org/abs/2508.20671","pdf_url":"https://arxiv.org/pdf/2508.20671v3","primary_category":"cs.FL","published":"2025-08-28","tags":["formalization","probability","analysis"],"lean_note":"Formalizes a measure-theoretic framework and a consistency theorem for stochastic global optimization algorithms in Lean.","thumb":"thumbs/2508.20671.jpg"},{"id":"2508.18914","title":"FormaRL: Enhancing Autoformalization with no Labeled Data","authors":["Yanxing Huang","Xinling Jin","Sijie Liang","Peng Li","Yang Liu"],"abstract":"Autoformalization is one of the central tasks in formal verification, while its advancement remains hindered due to the data scarcity and the absence efficient methods. In this work we propose \\textbf{FormaRL}, a simple yet efficient reinforcement learning framework for autoformalization which only requires a small amount of unlabeled data. FormaRL integrates syntax check from Lean compiler and consistency check from large language model to calculate the reward, and adopts GRPO algorithm to update the formalizer. We also curated a proof problem dataset from undergraduate-level math materials, named \\textbf{uproof}, in the hope to facilitate the exploration of autoformalization and theorem proving in advanced math. Experiments show that FormaRL can increase the pass@1 autoformalization accuracy of Qwen2.5-Coder-7B-Instruct by 4 $\\sim$ 6x (4.04\\% $\\to$ 26.15\\% on ProofNet and 2.4\\% $\\to$ 9.6\\% on uproof) with merely 859 unlabeled data. And on uproof our method also achieved a strong improvement in out-of-distribution performance compared to existing open-source state-of-the-art autoformalizers on both pass@1 accuracy (6.2\\% $\\to$ 9.6\\%) and pass@16 accuracy (24.4\\% $\\to$ 33.6\\%). Training code of FormaRL is open-sourced at https://github.com/THUNLP-MT/FormaRL.","abs_url":"https://arxiv.org/abs/2508.18914","pdf_url":"https://arxiv.org/pdf/2508.18914v1","primary_category":"cs.AI","published":"2025-08-26","tags":["autoformalization","theorem-proving-ml","dataset"],"lean_note":"FormaRL uses the Lean compiler's syntax check plus an LLM consistency check as RL reward for autoformalization into Lean.","repo":"https://github.com/THUNLP-MT/FormaRL","thumb":"thumbs/2508.18914.jpg"},{"id":"2508.15878","title":"Lean Meets Theoretical Computer Science: Scalable Synthesis of Theorem Proving Challenges in Formal-Informal Pairs","authors":["Terry Jingchen Zhang","Wenyuan Jiang","Rongchuan Liu","Yisong Wang","Junran Yang","Ning Wang","Nicole Ni","Yinya Huang","Mrinmaya Sachan"],"abstract":"Formal theorem proving (FTP) has emerged as a critical foundation for evaluating the reasoning capabilities of large language models, enabling automated verification of mathematical proofs at scale. However, progress has been constrained by limited datasets due to the high cost of manual curation and the scarcity of challenging problems with verified formal-informal correspondences. We propose leveraging theoretical computer science (TCS) as a scalable source of rigorous proof problems, where algorithmic definitions enable automated generation of arbitrarily many challenging theorem-proof pairs. We demonstrate this approach on two TCS domains: Busy Beaver problems, which involve proving bounds on Turing machine halting behavior, and Mixed Boolean Arithmetic problems, which combine logical and arithmetic reasoning. Our framework automatically synthesizes problems with parallel formal (Lean4) and informal (Markdown) specifications, creating a scalable pipeline for generating verified proof challenges. Evaluation on frontier models reveals substantial gaps in automated theorem proving: while DeepSeekProver-V2-671B achieves 57.5\\% success on Busy Beaver problems, it manages only 12\\% on Mixed Boolean Arithmetic problems. These results highlight the difficulty of long-form proof generation even for problems that are computationally easy to verify, demonstrating the value of TCS domains for advancing automated reasoning research.","abs_url":"https://arxiv.org/abs/2508.15878","pdf_url":"https://arxiv.org/pdf/2508.15878v2","primary_category":"cs.LO","published":"2025-08-21","tags":["benchmark","theorem-proving-ml","dataset","formalization"],"lean_note":"Synthesizes theorem-proving challenges with parallel Lean 4 formal and Markdown informal specifications from theoretical computer science problems.","thumb":"thumbs/2508.15878.jpg"},{"id":"2508.14644","title":"LeanGeo: Formalizing Competitional Geometry problems in Lean","authors":["Chendong Song","Zihan Wang","Frederick Pu","Haiming Wang","Xiaohan Lin","Junqi Liu","Jia Li","Zhengying Liu"],"abstract":"Geometry problems are a crucial testbed for AI reasoning capabilities. Most existing geometry solving systems cannot express problems within a unified framework, thus are difficult to integrate with other mathematical fields. Besides, since most geometric proofs rely on intuitive diagrams, verifying geometry problems is particularly challenging. To address these gaps, we introduce LeanGeo, a unified formal system for formalizing and solving competition-level geometry problems within the Lean 4 theorem prover. LeanGeo features a comprehensive library of high-level geometric theorems with Lean's foundational logic, enabling rigorous proof verification and seamless integration with Mathlib. We also present LeanGeo-Bench, a formal geometry benchmark in LeanGeo, comprising problems from the International Mathematical Olympiad (IMO) and other advanced sources. Our evaluation demonstrates the capabilities and limitations of state-of-the-art Large Language Models on this benchmark, highlighting the need for further advancements in automated geometric reasoning. We open source the theorem library and the benchmark of LeanGeo at https://github.com/project-numina/LeanGeo/tree/master.","abs_url":"https://arxiv.org/abs/2508.14644","pdf_url":"https://arxiv.org/pdf/2508.14644v1","primary_category":"cs.AI","published":"2025-08-20","tags":["formalization","benchmark","geometry","theorem-proving-ml"],"lean_note":"Introduces LeanGeo, a Lean 4 formal system and theorem library for competition geometry integrated with Mathlib, plus the LeanGeo-Bench benchmark.","repo":"https://github.com/project-numina/LeanGeo","thumb":"thumbs/2508.14644.jpg"},{"id":"2508.06931","title":"Automated Formalization via Conceptual Retrieval-Augmented LLMs","authors":["Wangyue Lu","Lun Du","Sirui Li","Ke Weng","Haozhe Sun","Hengyu Liu","Minghe Yu","Tiancheng Zhang","Ge Yu"],"abstract":"Interactive theorem provers (ITPs) require manual formalization, which is labor-intensive and demands expert knowledge. While automated formalization offers a potential solution, it faces two major challenges: model hallucination (e.g., undefined predicates, symbol misuse, and version incompatibility) and the semantic gap caused by ambiguous or missing premises in natural language descriptions. To address these issues, we propose CRAMF, a Concept-driven Retrieval-Augmented Mathematical Formalization framework. CRAMF enhances LLM-based autoformalization by retrieving formal definitions of core mathematical concepts, providing contextual grounding during code generation. However, applying retrieval-augmented generation (RAG) in this setting is non-trivial due to the lack of structured knowledge bases, the polymorphic nature of mathematical concepts, and the high precision required in formal retrieval. We introduce a framework for automatically constructing a concept-definition knowledge base from Mathlib4, the standard mathematical library for the Lean 4 theorem prover, indexing over 26,000 formal definitions and 1,000+ core mathematical concepts. To address conceptual polymorphism, we propose contextual query augmentation with domain- and application-level signals. In addition, we design a dual-channel hybrid retrieval strategy with reranking to ensure accurate and relevant definition retrieval. Experiments on miniF2F, ProofNet, and our newly proposed AdvancedMath benchmark show that CRAMF can be seamlessly integrated into LLM-based autoformalizers, yielding consistent improvements in translation accuracy, achieving up to 62.1% and an average of 29.9% relative improvement.","abs_url":"https://arxiv.org/abs/2508.06931","pdf_url":"https://arxiv.org/pdf/2508.06931v2","primary_category":"cs.AI","published":"2025-08-09","tags":["autoformalization","mathlib","theorem-proving-ml"],"lean_note":"CRAMF builds a concept-definition knowledge base from Mathlib4 to retrieval-augment LLM autoformalization of math into Lean 4.","repo":"https://github.com/kahvia0526/CRAMF","thumb":"thumbs/2508.06931.jpg"},{"id":"2508.05563","title":"Carleson operators on doubling metric measure spaces","authors":["Lars Becker","Floris van Doorn","Asgar Jamneshan","Rajula Srivastava","Christoph Thiele"],"abstract":"Doubling metric measure spaces provide a natural framework for singular integral operators. In contrast, the study of maximally modulated singular integral operators, the so-called Carleson operators, has largely been limited to Euclidean space with modulation functions such as polynomials defined by algebraic means. We present a general axiomatic approach to modulation functions on doubling metric measure spaces and prove $L^p$ bounds for the corresponding Carleson operators in Theorem 1.1 and Theorem 1.2. This generalizes classical and modern results on Carleson operators. In addition to the proofs presented here, our main results have been computer verified using the language Lean and the library mathlib, as documented in the sibling communication arXiv:2405.06423.","abs_url":"https://arxiv.org/abs/2508.05563","pdf_url":"https://arxiv.org/pdf/2508.05563v1","primary_category":"math.CA","published":"2025-08-07","tags":["formalization","verification","mathlib","analysis"],"lean_note":"Main Lp-bound theorems for Carleson operators on doubling metric measure spaces were computer-verified in Lean using Mathlib.","thumb":"thumbs/2508.05563.jpg"},{"id":"2508.03613","title":"Goedel-Prover-V2: Scaling Formal Theorem Proving with Scaffolded Data Synthesis and Self-Correction","authors":["Yong Lin","Shange Tang","Bohan Lyu","Ziran Yang","Jui-Hui Chung","Haoyu Zhao","Lai Jiang","Yihan Geng","Jiawei Ge","Jingruo Sun","Jiayun Wu","Jiri Gesi","Ximing Lu","David Acuna","Kaiyu Yang","Hongzhou Lin","Yejin Choi","Danqi Chen","Sanjeev Arora","Chi Jin"],"abstract":"We introduce Goedel-Prover-V2, a series of open-source language models that set a new state-of-the-art in automated theorem proving. Built on the standard expert iteration and reinforcement learning pipeline, our approach incorporates three key innovations: (1) Scaffolded data synthesis: We generate synthetic tasks of increasing difficulty to train the model to master increasingly complex theorems; (2) Verifier-guided self-correction: We enable the model to iteratively revise its proofs by leveraging feedback from the Lean compiler; (3) Model averaging: We merge model checkpoints to mitigate the decrease in model output diversity in later stages of training. Our small model, Goedel-Prover-V2-8B, reaches 84.6% pass@32 on MiniF2F and outperforms DeepSeek-Prover-V2-671B under the same metric, despite being 80X smaller. Our flagship model, Goedel-Prover-V2-32B, achieves 88.1% on MiniF2F at pass@32 in standard mode and 90.4% in self-correction mode, outperforming prior SOTA by a large margin. Additionally, our flagship model solves 86 problems on PutnamBench at pass@184, securing the first place among open-source models on the leaderboard, surpassing DeepSeek-Prover-V2-671B's record of solving 47 problems by pass@1024 with a significantly smaller model size and compute budget. At the time of its release (July-August 2025), Goedel-Prover-V2 achieves the strongest overall performance among all open-source theorem provers. It also ranks among the top-performing models--including closed-source systems with publicly reported performance--under a constrained test-time compute budget. Our models, code, and data are released at https://github.com/Goedel-LM/Goedel-Prover-V2.","abs_url":"https://arxiv.org/abs/2508.03613","pdf_url":"https://arxiv.org/pdf/2508.03613v1","primary_category":"cs.LG","published":"2025-08-05","tags":["theorem-proving-ml","benchmark"],"lean_note":"Trains open-source theorem-proving LLMs using Lean compiler feedback for self-correction, evaluated on MiniF2F and PutnamBench.","repo":"https://github.com/Goedel-LM/Goedel-Prover-V2","thumb":"thumbs/2508.03613.jpg"},{"id":"2508.03173","title":"Geoint-R1: Formalizing Multimodal Geometric Reasoning with Dynamic Auxiliary Constructions","authors":["Jingxuan Wei","Caijun Jia","Qi Chen","Honghao He","Linzhuang Sun","Conghui He","Lijun Wu","Bihui Yu","Cheng Tan"],"abstract":"Mathematical geometric reasoning is essential for scientific discovery and educational development, requiring precise logic and rigorous formal verification. While recent advances in Multimodal Large Language Models (MLLMs) have improved reasoning tasks, existing models typically struggle with formal geometric reasoning, particularly when dynamically constructing and verifying auxiliary geometric elements. To address these challenges, we introduce Geoint-R1, a multimodal reasoning framework designed to generate formally verifiable geometric solutions from textual descriptions and visual diagrams. Geoint-R1 uniquely integrates auxiliary elements construction, formal reasoning represented via Lean4, and interactive visualization. To systematically evaluate and advance formal geometric reasoning, we propose the Geoint benchmark, comprising 1,885 rigorously annotated geometry problems across diverse topics such as plane, spatial, and solid geometry. Each problem includes structured textual annotations, precise Lean4 code for auxiliary constructions, and detailed solution steps verified by experts. Extensive experiments demonstrate that Geoint-R1 significantly surpasses existing multimodal and math-specific reasoning models, particularly on challenging problems requiring explicit auxiliary element constructions.","abs_url":"https://arxiv.org/abs/2508.03173","pdf_url":"https://arxiv.org/pdf/2508.03173v1","primary_category":"cs.AI","published":"2025-08-05","tags":["theorem-proving-ml","geometry"],"lean_note":"Geoint-R1 generates Lean4 code for auxiliary geometric constructions, with a 1,885-problem benchmark of Lean4-annotated geometry problems.","repo":"https://github.com/LengSicong/MMR1","thumb":"thumbs/2508.03173.jpg"},{"id":"2507.23726","title":"Seed-Prover: Deep and Broad Reasoning for Automated Theorem Proving","authors":["Luoxin Chen","Jinming Gu","Liankai Huang","Wenhao Huang","Zhicheng Jiang","Allan Jie","Xiaoran Jin","Xing Jin","Chenggang Li","Kaijing Ma","Cheng Ren","Jiawei Shen","Wenlei Shi","Tong Sun","He Sun","Jiahui Wang","Siran Wang","Zhihong Wang","Chenrui Wei","Shufa Wei","Yonghui Wu","Yuchen Wu","Yihang Xia","Huajian Xin","Fan Yang","Huaiyuan Ying","Hongyi Yuan","Zheng Yuan","Tianyang Zhan","Chi Zhang","Yue Zhang","Ge Zhang","Tianyun Zhao","Jianqiu Zhao","Yichi Zhou","Thomas Hanwen Zhu"],"abstract":"LLMs have demonstrated strong mathematical reasoning abilities by leveraging reinforcement learning with long chain-of-thought, yet they continue to struggle with theorem proving due to the lack of clear supervision signals when solely using natural language. Dedicated domain-specific languages like Lean provide clear supervision via formal verification of proofs, enabling effective training through reinforcement learning. In this work, we propose \\textbf{Seed-Prover}, a lemma-style whole-proof reasoning model. Seed-Prover can iteratively refine its proof based on Lean feedback, proved lemmas, and self-summarization. To solve IMO-level contest problems, we design three test-time inference strategies that enable both deep and broad reasoning. Seed-Prover proves $78.1\\%$ of formalized past IMO problems, saturates MiniF2F, and achieves over 50\\% on PutnamBench, outperforming the previous state-of-the-art by a large margin. To address the lack of geometry support in Lean, we introduce a geometry reasoning engine \\textbf{Seed-Geometry}, which outperforms previous formal geometry engines. We use these two systems to participate in IMO 2025 and fully prove 5 out of 6 problems. This work represents a significant advancement in automated mathematical reasoning, demonstrating the effectiveness of formal verification with long chain-of-thought reasoning.","abs_url":"https://arxiv.org/abs/2507.23726","pdf_url":"https://arxiv.org/pdf/2507.23726v2","primary_category":"cs.AI","published":"2025-07-31","tags":["theorem-proving-ml","benchmark","geometry"],"lean_note":"Seed-Prover is a whole-proof LLM that iteratively refines Lean proofs using Lean feedback, proved lemmas, and self-summarization.","thumb":"thumbs/2507.23726.jpg"},{"id":"2507.20199","title":"StepFun-Prover Preview: Let's Think and Verify Step by Step","authors":["Shijie Shang","Ruosi Wan","Yue Peng","Yutong Wu","Xiong-hui Chen","Jie Yan","Xiangyu Zhang"],"abstract":"We present StepFun-Prover Preview, a large language model designed for formal theorem proving through tool-integrated reasoning. Using a reinforcement learning pipeline that incorporates tool-based interactions, StepFun-Prover can achieve strong performance in generating Lean 4 proofs with minimal sampling. Our approach enables the model to emulate human-like problem-solving strategies by iteratively refining proofs based on real-time environment feedback. On the miniF2F-test benchmark, StepFun-Prover achieves a pass@1 success rate of $70.0\\%$. Beyond advancing benchmark performance, we introduce an end-to-end training framework for developing tool-integrated reasoning models, offering a promising direction for automated theorem proving and Math AI assistant.","abs_url":"https://arxiv.org/abs/2507.20199","pdf_url":"https://arxiv.org/pdf/2507.20199v3","primary_category":"cs.AI","published":"2025-07-27","tags":["theorem-proving-ml","ai-agents","benchmark"],"lean_note":"StepFun-Prover is an LLM trained with tool-integrated RL to generate Lean 4 proofs, evaluated on miniF2F-test.","repo":"https://github.com/stepfun-ai/StepFun-Prover-Preview","thumb":"thumbs/2507.20199.jpg"},{"id":"2507.16086","title":"Understanding Haskell-style Overloading via Open Data and Open Functions","authors":["Andrew Marmaduke","Apoorv Ingle","J. Garrett Morris"],"abstract":"We present a new, uniform semantics for Haskell-style overloading. We realize our approach in a new core language, System F$_\\mathrm{D}$, whose metatheory we mechanize in the Lean4 interactive theorem prover. System F$_\\mathrm{D}$ is distinguished by its open data types and open functions, each given by a collection of instances rather than by a single definition. We show that System F$_\\mathrm{D}$ can encode advanced features of Haskell's of type class systems, more expressively than current semantics of these features, and without assuming additional type equality axioms.","abs_url":"https://arxiv.org/abs/2507.16086","pdf_url":"https://arxiv.org/pdf/2507.16086v1","primary_category":"cs.PL","published":"2025-07-21","tags":["formalization","verification","type-theory"],"lean_note":"Mechanizes the metatheory of System F_D, a core language for Haskell-style overloading, in the Lean4 theorem prover.","thumb":"thumbs/2507.16086.jpg"},{"id":"2507.15225","title":"Solving Formal Math Problems by Decomposition and Iterative Reflection","authors":["Yichi Zhou","Jianqiu Zhao","Yongxin Zhang","Bohan Wang","Siran Wang","Luoxin Chen","Jiahui Wang","Haowei Chen","Allan Jie","Xinbo Zhang","Haocheng Wang","Luong Trung","Rong Ye","Phan Nhat Hoang","Huishuai Zhang","Peng Sun","Hang Li"],"abstract":"General-purpose Large Language Models (LLMs) have achieved remarkable success in intelligence, performing comparably to human experts on complex reasoning tasks such as coding and mathematical reasoning. However, generating formal proofs in specialized languages like Lean 4 remains a significant challenge for these models, limiting their application in complex theorem proving and automated verification. Current approaches typically require specializing models through fine-tuning on dedicated formal corpora, incurring high costs for data collection and training. In this work, we introduce \\textbf{Delta Prover}, an agent-based framework that orchestrates the interaction between a general-purpose LLM and the Lean 4 proof environment. Delta Prover leverages the reflection and reasoning capabilities of general-purpose LLMs to interactively construct formal proofs in Lean 4, circumventing the need for model specialization. At its core, the agent integrates two novel, interdependent components: an algorithmic framework for reflective decomposition and iterative proof repair, and a custom Domain-Specific Language (DSL) built upon Lean 4 for streamlined subproblem management. \\textbf{Delta Prover achieves a state-of-the-art 95.9\\% success rate on the miniF2F-test benchmark, surpassing all existing approaches, including those requiring model specialization.} Furthermore, Delta Prover exhibits a significantly stronger test-time scaling law compared to standard Best-of-N proof strategies. Crucially, our findings demonstrate that general-purpose LLMs, when guided by an effective agentic structure, possess substantial untapped theorem-proving capabilities. This presents a computationally efficient alternative to specialized models for robust automated reasoning in formal environments.","abs_url":"https://arxiv.org/abs/2507.15225","pdf_url":"https://arxiv.org/pdf/2507.15225v1","primary_category":"cs.AI","published":"2025-07-21","tags":["theorem-proving-ml","ai-agents","benchmark"],"lean_note":"Delta Prover is an agent framework that interactively constructs Lean 4 proofs using a general-purpose LLM and a custom Lean 4 DSL.","repo":"https://github.com/ByteDance-Seed/lean4-agent","thumb":"thumbs/2507.15225.jpg"},{"id":"2507.14722","title":"LeanTree: Accelerating White-Box Proof Search with Factorized States in Lean 4","authors":["Matěj Kripner","Michal Šustr","Milan Straka"],"abstract":"Automated theorem proving (ATP) has been a classical problem in artificial intelligence since its inception, yet it remains challenging due to its vast state and action space. Large language models (LLMs) have recently emerged as a promising heuristic for ATP, but they lack correctness guarantees and thus require interaction with a proof verifier. Such interactions typically follow one of two approaches: black-box interaction, which does not utilize intermediate proof states, or white-box approaches, which allow for incremental proof construction and examination of intermediate states. While black-box approaches have directly benefited from recent LLM advances, white-box methods have comparatively lagged behind. In this paper, we address this gap by introducing LeanTree, which consists of (i) a tool built in the Lean 4 language that factorizes complex proof states into simpler, independent branches, and (ii) a dataset of these factorized intermediate states. Our white-box tooling offers several advantages over black-box approaches: it simplifies evaluation, reduces necessary context, generates richer training data, enables parallel search across multiple states, supports efficient reuse of states, and provides feedback in case of errors. Our preliminary results hint that white-box approaches outperform black-box alternatives in some settings.","abs_url":"https://arxiv.org/abs/2507.14722","pdf_url":"https://arxiv.org/pdf/2507.14722v1","primary_category":"cs.LG","published":"2025-07-19","tags":["theorem-proving-ml","tooling","dataset","tactic-automation"],"lean_note":"LeanTree is a Lean 4 tool factorizing proof states into independent branches, plus a dataset for white-box proof search.","repo":"https://github.com/Kripner/leantree","thumb":"thumbs/2507.14722.jpg"},{"id":"2507.11275","title":"FMC: Formalization of Natural Language Mathematical Competition Problems","authors":["Jiaxuan Xie","Chengwu Liu","Ye Yuan","Siqi Li","Zhiping Xiao","Ming Zhang"],"abstract":"Efficient and accurate autoformalization methods, which leverage large-scale datasets of extensive natural language mathematical problems to construct formal language datasets, are key to advancing formal mathematical reasoning. In this paper, we propose an autoformalization pipeline based on large language models with error feedback, achieving a fully automatic and training-free formalization approach. Using this pipeline, we curate an Olympiad-level dataset aligning natural language problems with Lean formalizations. The dataset comprises $3,922$ mathematical problems in natural language and $9,787$ in Lean, of which $64.46\\%$ were assessed as at least above-average quality, making it suitable as a benchmark for automated theorem provers. Additionally, we investigate the formalization and reasoning capabilities of various LLMs and empirically demonstrate that few-shot learning, error feedback, and increasing sampling numbers enhance the autoformalization process. Experiments of three automated theorem provers on the \\dataset\\ dataset also highlight its challenging nature and its value as a benchmark for formal reasoning tasks.","abs_url":"https://arxiv.org/abs/2507.11275","pdf_url":"https://arxiv.org/pdf/2507.11275v1","primary_category":"cs.CL","published":"2025-07-15","tags":["autoformalization","dataset","benchmark","theorem-proving-ml"],"lean_note":"A training-free LLM pipeline with error feedback autoformalizes competition problems into a Lean dataset usable as a prover benchmark.","thumb":"thumbs/2507.11275.jpg"},{"id":"2507.08665","title":"KELPS: A Framework for Verified Multi-Language Autoformalization via Semantic-Syntactic Alignment","authors":["Jiyao Zhang","Chengli Zhong","Hui Xu","Qige Li","Yi Zhou"],"abstract":"Modern large language models (LLMs) show promising progress in formalizing informal mathematics into machine-verifiable theorems. However, these methods still face bottlenecks due to the limited quantity and quality of multilingual parallel corpora. In this paper, we propose a novel neuro-symbolic framework KELPS (Knowledge-Equation based Logical Processing System) to address these problems. KELPS is an iterative framework for translating, synthesizing, and filtering informal data into multiple formal languages (Lean, Coq, and Isabelle). First, we translate natural language into Knowledge Equations (KEs), a novel language that we designed, theoretically grounded in assertional logic. Next, we convert them to target languages through rigorously defined rules that preserve both syntactic structure and semantic meaning. This process yielded a parallel corpus of over 60,000 problems. Our framework achieves 88.9% syntactic accuracy (pass@1) on MiniF2F, outperforming SOTA models such as Deepseek-V3 (81%) and Herald (81.3%) across multiple datasets. All datasets and codes are available in the supplementary materials.","abs_url":"https://arxiv.org/abs/2507.08665","pdf_url":"https://arxiv.org/pdf/2507.08665v1","primary_category":"cs.CL","published":"2025-07-11","tags":["autoformalization","theorem-proving-ml","dataset"],"lean_note":"KELPS translates informal math into Lean, Coq, and Isabelle via Knowledge Equations, producing a verified multilingual parallel corpus.","thumb":"thumbs/2507.08665.jpg"},{"id":"2507.08649","title":"Leanabell-Prover-V2: Verifier-integrated Reasoning for Formal Theorem Proving via Reinforcement Learning","authors":["Xingguang Ji","Yahui Liu","Qi Wang","Jingyuan Zhang","Yang Yue","Rui Shi","Chenxi Sun","Fuzheng Zhang","Guorui Zhou","Kun Gai"],"abstract":"We introduce our Leanabell-Prover-V2, a 7B large language models (LLMs) that can produce formal theorem proofs in Lean 4, with verifier-integrated Long Chain-of-Thoughts (CoT). Following our previous work Leanabell-Prover-V1, we continual to choose to posttrain existing strong prover models for further performance improvement. In our V2 version, we mainly upgrade the Reinforcement Learning (RL) with feedback provided by the Lean 4 verifier. Crucially, verifier feedback, such as indicating success or detailing specific errors, allows the LLM to become ``self-aware'' of the correctness of its own reasoning process and learn to reflexively correct errors. Leanabell-Prover-V2 directly optimizes LLM reasoning trajectories with multi-turn verifier interactions, together with feedback token masking for stable RL training and a simple reward strategy. Experiments show that Leanabell-Prover-V2 improves performance by 3.2% (pass@128) with Kimina-Prover-Preview-Distill-7B and 2.0% (pass@128) with DeepSeek-Prover-V2-7B on the MiniF2F test set. The source codes, curated data and models are available at: https://github.com/Leanabell-LM/Leanabell-Prover-V2.","abs_url":"https://arxiv.org/abs/2507.08649","pdf_url":"https://arxiv.org/pdf/2507.08649v1","primary_category":"cs.AI","published":"2025-07-11","tags":["theorem-proving-ml","ai-agents","benchmark","formalization"],"lean_note":"Post-trains a 7B LLM with reinforcement learning using Lean 4 verifier feedback to produce formal proofs.","repo":"https://github.com/Leanabell-LM/Leanabell-Prover-V2","thumb":"thumbs/2507.08649.jpg"},{"id":"2507.06456","title":"Fast Collection Operations from Indexed Stream Fusion","authors":["Scott Kovach","Praneeth Kolichala","Kyle A. Miller","David Broman","Fredrik Kjolstad"],"abstract":"We present a system of efficient methods for traversing and combining associative collection data structures. A distinguishing feature of the system is that, like traditional sequential iterator libraries, it does not require specialized compiler infrastructure or staged compilation for efficiency and composability. By using a representation based on indexed streams, the library can express complex joins over input collections while using no intermediate allocations. We implement the library for the Lean, Morphic, and Rust programming languages and provide a mechanized proof of functional correctness in Lean.","abs_url":"https://arxiv.org/abs/2507.06456","pdf_url":"https://arxiv.org/pdf/2507.06456v1","primary_category":"cs.PL","published":"2025-07-08","tags":["verification","tooling"],"lean_note":"Implements an indexed-stream-fusion collection library in Lean and provides a mechanized Lean proof of its functional correctness.","thumb":"thumbs/2507.06456.jpg"},{"id":"2507.06181","title":"CriticLean: Critic-Guided Reinforcement Learning for Mathematical Formalization","authors":["Zhongyuan Peng","Yifan Yao","Kaijing Ma","Shuyue Guo","Yizhe Li","Yichi Zhang","Chenchen Zhang","Yifan Zhang","Zhouliang Yu","Luming Li","Minghao Liu","Yihang Xia","Jiawei Shen","Yuchen Wu","Yixin Cao","Zhaoxiang Zhang","Wenhao Huang","Jiaheng Liu","Ge Zhang"],"abstract":"Translating natural language mathematical statements into formal, executable code is a fundamental challenge in automated theorem proving. While prior work has focused on generation and compilation success, little attention has been paid to the critic phase-the evaluation of whether generated formalizations truly capture the semantic intent of the original problem. In this paper, we introduce CriticLean, a novel critic-guided reinforcement learning framework that elevates the role of the critic from a passive validator to an active learning component. Specifically, first, we propose the CriticLeanGPT, trained via supervised fine-tuning and reinforcement learning, to rigorously assess the semantic fidelity of Lean 4 formalizations. Then, we introduce CriticLeanBench, a benchmark designed to measure models' ability to distinguish semantically correct from incorrect formalizations, and demonstrate that our trained CriticLeanGPT models can significantly outperform strong open- and closed-source baselines. Building on the CriticLean framework, we construct FineLeanCorpus, a dataset comprising over 285K problems that exhibits rich domain diversity, broad difficulty coverage, and high correctness based on human evaluation. Overall, our findings highlight that optimizing the critic phase is essential for producing reliable formalizations, and we hope our CriticLean will provide valuable insights for future advances in formal mathematical reasoning.","abs_url":"https://arxiv.org/abs/2507.06181","pdf_url":"https://arxiv.org/pdf/2507.06181v1","primary_category":"cs.CL","published":"2025-07-08","tags":["autoformalization","theorem-proving-ml","benchmark","dataset"],"lean_note":"Builds a critic, benchmark, and 285K-problem dataset for evaluating semantic fidelity of Lean 4 autoformalizations.","thumb":"thumbs/2507.06181.jpg"},{"id":"2507.05327","title":"A Formalization of Divided Powers in Lean","authors":["Antoine Chambert-Loir","María Inés de Frutos-Fernández"],"abstract":"Given an ideal $I$ in a commutative ring $A$, a divided power structure on $I$ is a collection of maps $\\{γ_n \\colon I \\to A\\}_{n \\in \\mathbb{N}}$, subject to axioms that imply that it behaves like the family $\\{x \\mapsto \\frac{x^n}{n!}\\}_{n \\in \\mathbb{N}}$, but which can be defined even when division by factorials is not possible in $A$. Divided power structures have important applications in diverse areas of mathematics, including algebraic topology, number theory and algebraic geometry. In this article we describe a formalization in Lean 4 of the basic theory of divided power structures, including divided power morphisms and sub-divided power ideals, and we provide several fundamental constructions, in particular quotients and sums. This constitutes the first formalization of this theory in any theorem prover. As a prerequisite of general interest, we expand the formalized theory of multivariate power series rings, endowing them with a topology and defining evaluation and substitution of power series.","abs_url":"https://arxiv.org/abs/2507.05327","pdf_url":"https://arxiv.org/pdf/2507.05327v1","primary_category":"cs.LO","published":"2025-07-07","tags":["formalization","mathlib","algebra","number-theory"],"lean_note":"First formalization in Lean 4 of divided power structures, including morphisms, sub-ideals, quotients, and sums.","repo":"https://github.com/mariainesdff/divided_powers_journal","thumb":"thumbs/2507.05327.jpg"},{"id":"2507.07052","title":"Quantifying Bounded Rationality: Formal Verification of Simon's Satisficing Through Flexible Stochastic Dominance","authors":["Jingyuan Li","Zhou Lin"],"abstract":"This paper introduces Flexible First-Order Stochastic Dominance (FFSD), a mathematically rigorous framework that formalizes Herbert Simon's concept of bounded rationality using the Lean 4 theorem prover. We develop machine-verified proofs demonstrating that FFSD bridges classical expected utility theory with Simon's satisficing behavior through parameterized tolerance thresholds. Our approach yields several key results: (1) a critical threshold $\\varepsilon < 1/2$ that guarantees uniqueness of reference points, (2) an equivalence theorem linking FFSD to expected utility maximization for approximate indicator functions, and (3) extensions to multi-dimensional decision settings. By encoding these concepts in Lean 4's dependent type theory, we provide the first machine-checked formalization of Simon's bounded rationality, creating a foundation for mechanized reasoning about economic decision-making under uncertainty with cognitive limitations. This work contributes to the growing intersection between formal mathematics and economic theory, demonstrating how interactive theorem proving can advance our understanding of behavioral economics concepts that have traditionally been expressed only qualitatively.","abs_url":"https://arxiv.org/abs/2507.07052","pdf_url":"https://arxiv.org/pdf/2507.07052v1","primary_category":"q-fin.MF","published":"2025-07-02","tags":["formalization","economics","verification"],"lean_note":"Formalizes Flexible First-Order Stochastic Dominance and Simon's bounded rationality with machine-checked proofs in Lean 4.","repo":"https://github.com/jingyuanli-hk/Quantifying-Bounded-Rationality","thumb":"thumbs/2507.07052.jpg"},{"id":"2507.01780","title":"LeanLTL: A unifying framework for linear temporal logics in Lean","authors":["Eric Vin","Kyle A. Miller","Daniel J. Fremont"],"abstract":"We propose LeanLTL, a unifying framework for linear temporal logics in Lean 4. LeanLTL supports reasoning about traces that represent either infinite or finite linear time. The library allows traditional LTL syntax to be combined with arbitrary Lean expressions, making it straightforward to define properties involving numerical or other types. We prove that standard flavors of LTL can be embedded in our framework. The library also provides automation for reasoning about LeanLTL formulas in a way that facilitates using Lean's existing tactics. Finally, we provide examples illustrating the utility of the library in reasoning about systems that come from applications.","abs_url":"https://arxiv.org/abs/2507.01780","pdf_url":"https://arxiv.org/pdf/2507.01780v1","primary_category":"cs.LO","published":"2025-07-02","tags":["formalization","logic","tactic-automation"],"lean_note":"Presents LeanLTL, a Lean 4 library unifying linear temporal logics with proof automation over finite and infinite traces.","thumb":"thumbs/2507.01780.jpg"},{"id":"2506.22005","title":"LeanConjecturer: Automatic Generation of Mathematical Conjectures for Theorem Proving","authors":["Naoto Onda","Kazumi Kasaura","Yuta Oriike","Masaya Taniguchi","Akiyoshi Sannai","Sho Sonoda"],"abstract":"We introduce LeanConjecturer, a pipeline for automatically generating university-level mathematical conjectures in Lean 4 using Large Language Models (LLMs). Our hybrid approach combines rule-based context extraction with LLM-based theorem statement generation, addressing the data scarcity challenge in formal theorem proving. Through iterative generation and evaluation, LeanConjecturer produced 12,289 conjectures from 40 Mathlib seed files, with 3,776 identified as syntactically valid and non-trivial, that is, cannot be proven by \\texttt{aesop} tactic. We demonstrate the utility of these generated conjectures for reinforcement learning through Group Relative Policy Optimization (GRPO), showing that targeted training on domain-specific conjectures can enhance theorem proving capabilities. Our approach generates 103.25 novel conjectures per seed file on average, providing a scalable solution for creating training data for theorem proving systems. Our system successfully verified several non-trivial theorems in topology, including properties of semi-open, alpha-open, and pre-open sets, demonstrating its potential for mathematical discovery beyond simple variations of existing results.","abs_url":"https://arxiv.org/abs/2506.22005","pdf_url":"https://arxiv.org/pdf/2506.22005v1","primary_category":"cs.AI","published":"2025-06-27","tags":["autoformalization","theorem-proving-ml","dataset","mathlib","topology"],"lean_note":"LeanConjecturer generates university-level mathematical conjectures as Lean 4 statements from Mathlib seed files for theorem-proving training.","repo":"https://github.com/auto-res/LeanConjecturer","thumb":"thumbs/2506.22005.jpg"},{"id":"2506.19923","title":"Prover Agent: An Agent-Based Framework for Formal Mathematical Proofs","authors":["Kaito Baba","Chaoran Liu","Shuhei Kurita","Akiyoshi Sannai"],"abstract":"We present Prover Agent, a novel AI agent for automated theorem proving that integrates large language models (LLMs) with a formal proof assistant, Lean. Prover Agent coordinates an informal reasoning LLM, a formal prover model, and feedback from Lean while also generating auxiliary lemmas. These auxiliary lemmas are not limited to subgoals in the formal proof but can also include special cases or potentially useful facts derived from the assumptions, which help in discovering a viable proof strategy. It achieves an 88.1% success rate on MiniF2F and solves 25 problems on the PutnamBench with a smaller sample budget than previous approaches, establishing a new state-of-the-art on both benchmarks among methods using small language models (SLMs). We also present theoretical analyses and case studies that illustrate how these generated lemmas contribute to solving challenging problems. Our code is publicly available at https://github.com/kAIto47802/Prover-Agent.","abs_url":"https://arxiv.org/abs/2506.19923","pdf_url":"https://arxiv.org/pdf/2506.19923v5","primary_category":"cs.AI","published":"2025-06-24","tags":["theorem-proving-ml","ai-agents","benchmark"],"lean_note":"Prover Agent couples LLMs with the Lean proof assistant and auxiliary-lemma generation for automated theorem proving on MiniF2F and PutnamBench.","repo":"https://github.com/kAIto47802/Prover-Agent","thumb":"thumbs/2506.19923.jpg"},{"id":"2506.18616","title":"A Formalization of the Ionescu-Tulcea Theorem in Mathlib","authors":["Etienne Marion"],"abstract":"We describe the formalization of the Ionescu-Tulcea theorem, showing the existence of a probability measure on the space of trajectories of a Markov chain, in the proof assistant Lean using the integrated library Mathlib. We first present a mathematical proof before exposing the difficulties which arise when trying to formalize it, and how they were overcome. We then build on this work to formalize the construction of the product of an arbitrary family of probability measures.","abs_url":"https://arxiv.org/abs/2506.18616","pdf_url":"https://arxiv.org/pdf/2506.18616v5","primary_category":"math.PR","published":"2025-06-23","tags":["formalization","mathlib","probability"],"lean_note":"Formalizes the Ionescu-Tulcea theorem in Lean using Mathlib and builds the product of an arbitrary family of probability measures.","thumb":"thumbs/2506.18616.jpg"},{"id":"2506.17104","title":"Towards Advanced Mathematical Reasoning for LLMs via First-Order Logic Theorem Proving","authors":["Chuxue Cao","Mengze Li","Juntao Dai","Jinluan Yang","Zijian Zhao","Shengyu Zhang","Weijie Shi","Chengzhong Liu","Sirui Han","Yike Guo"],"abstract":"Large language models (LLMs) have shown promising first-order logic (FOL) reasoning capabilities with applications in various areas. However, their effectiveness in complex mathematical reasoning involving multi-step FOL deductions is still under-researched. While LLMs perform competitively on established mathematical reasoning benchmarks, they struggle with multi-step FOL tasks, as demonstrated by Deepseek-Prover-V2-7B's low accuracy (4.2%) on our proposed theorem proving dataset. This issue arises from the limited exploration of diverse proof strategies and the potential for early reasoning mistakes to undermine entire proofs. To address these issues, we propose DREAM, a self-adaptive solution that enhances the Diversity and REAsonability of LLMs' generation strategies. DREAM incorporates an Axiom-Driven Strategy Diversification mechanism to promote varied strategic outcomes and a Sub-Proposition Error Feedback to help LLMs reflect on and correct their proofs. Our contributions include pioneering advancements in LLMs' mathematical reasoning through FOL theorem proving, introducing a novel inference stage solution that improves performance by 0.6% to 6.4%, and providing a curated dataset of 447 mathematical theorems in Lean 4 format for evaluation.","abs_url":"https://arxiv.org/abs/2506.17104","pdf_url":"https://arxiv.org/pdf/2506.17104v1","primary_category":"cs.AI","published":"2025-06-20","tags":["theorem-proving-ml","dataset","logic"],"lean_note":"Releases a curated dataset of 447 mathematical theorems in Lean 4 format for evaluating LLM first-order-logic theorem proving.","thumb":"thumbs/2506.17104.jpg"},{"id":"2506.12859","title":"A Complete Derivation of the Fermion Spectrum from the Recognition Composition Law","authors":["Jonathan Washburn","Elshad Allahyarov"],"abstract":"We present a first-principles derivation of the masses of all twelve known fermions -- three charged leptons, six quarks, and three neutrinos -- and the fine-structure constant $α^{-1}$, from a single discrete functional equation, the Recognition Composition Law (RCL), with \\textbf{zero continuously adjustable parameters}. The mass spectrum follows from the RCL supplemented by four regularity conditions and eight structural theorems (T1--T8): the golden ratio $\\varphi=(1+\\sqrt{5})/2$ emerges as the unique hierarchy base (T6); an 8-step period is fixed by the 3-cube Hamiltonian cycle (T7); three spatial dimensions are selected by a unique combinatorial identity (T8). All integers entering the mass formula are the six combinatorial invariants of the 3-cube $Q_3$; none is fitted. The sole empirical input is the electron mass, which fixes an irreducible unit-conversion constant~$τ_0$. Predictions are confronted with PDG measurements. Charged-lepton masses are reproduced at sub-ppm accuracy for the muon and $\\sim\\!10^{-4}$ for the tau (Table~\\ref{tab:lepton_validation}). All six quark masses are predicted at integer level; first-generation quarks agree to better than $1\\%$, while second/third-generation residuals of $2$--$16\\%$ are expected integer-precision effects (Table~\\ref{tab:quark_validation}). Neutrino mass-squared splittings agree with NuFIT~5.3 within $1$--$2σ$, normal ordering is predicted, and $Σm_ν\\approx 0.063$~eV satisfies cosmological bounds. All structural claims are machine-verified in Lean~4 (179 files, 0~\\texttt{sorry}; \\texttt{github.com/\\allowbreak jonwashburn/\\allowbreak recognition-science}).","abs_url":"https://arxiv.org/abs/2506.12859","pdf_url":"https://arxiv.org/pdf/2506.12859v3","primary_category":"physics.gen-ph","published":"2025-06-15","tags":["formalization","verification","physics"],"lean_note":"All structural theorems behind the claimed fermion-mass derivation are machine-verified in Lean 4 across 179 files with zero sorry.","thumb":"thumbs/2506.12859.jpg"},{"id":"2506.08321","title":"LeanTutor: Towards a Verified AI Mathematical Proof Tutor","authors":["Manooshree Patel","Rayna Bhattacharyya","Thomas Lu","Arnav Mehta","Niels Voss","Narges Norouzi","Gireeja Ranade"],"abstract":"This paper considers the development of an AI-based provably-correct mathematical proof tutor. While Large Language Models (LLMs) allow seamless communication in natural language, they are error prone. Theorem provers such as Lean allow for provable-correctness, but these are hard for students to learn. We present a proof-of-concept system (LeanTutor) by combining the complementary strengths of LLMs and theorem provers. LeanTutor is composed of three modules: (i) an autoformalizer/proof-checker, (ii) a next-step generator, and (iii) a natural language feedback generator. To evaluate the system, we introduce PeanoBench, a dataset of 371 Peano Arithmetic proofs in human-written natural language and formal language, derived from the Natural Numbers Game.","abs_url":"https://arxiv.org/abs/2506.08321","pdf_url":"https://arxiv.org/pdf/2506.08321v2","primary_category":"cs.AI","published":"2025-06-10","tags":["autoformalization","ai-agents","benchmark","dataset"],"lean_note":"LeanTutor uses Lean as a proof checker, autoformalizing student natural-language proofs into Lean and generating verified next steps.","thumb":"thumbs/2506.08321.jpg"},{"id":"2506.07477","title":"Premise Selection for a Lean Hammer","authors":["Thomas Zhu","Joshua Clune","Jeremy Avigad","Albert Qiaochu Jiang","Sean Welleck"],"abstract":"Neural methods are transforming automated reasoning for proof assistants, yet integrating these advances into practical verification workflows remains challenging. A hammer is a tool that integrates premise selection, translation to external automatic theorem provers, and proof reconstruction into one overarching tool to automate tedious reasoning steps. We present LeanPremise, a novel neural premise selection system, and we combine it with existing translation and proof reconstruction components to create LeanHammer, the first end-to-end domain general hammer for the Lean proof assistant. Unlike existing Lean premise selectors, LeanPremise is specifically trained for use with a hammer in dependent type theory. It also dynamically adapts to user-specific contexts, enabling it to effectively recommend premises from libraries outside LeanPremise's training data as well as lemmas defined by the user locally. With comprehensive evaluations, we show that LeanPremise enables LeanHammer to solve 21% more goals than existing premise selectors and generalizes well to diverse domains. Our work helps bridge the gap between neural retrieval and symbolic reasoning, making formal verification more accessible to researchers and practitioners.","abs_url":"https://arxiv.org/abs/2506.07477","pdf_url":"https://arxiv.org/pdf/2506.07477v2","primary_category":"cs.LG","published":"2025-06-09","tags":["theorem-proving-ml","tactic-automation","tooling","mathlib"],"lean_note":"Presents LeanPremise neural premise selection and LeanHammer, an end-to-end hammer for the Lean 4 proof assistant.","repo":"https://github.com/hanwenzhu/premise-selection","thumb":"thumbs/2506.07477.jpg"},{"id":"2506.07066","title":"From Axioms to Algorithms: Mechanized Proofs of the vNM Utility Theorem","authors":["Li Jingyuan"],"abstract":"This paper presents a comprehensive formalization of the von Neumann-Morgenstern (vNM) expected utility theorem using the Lean 4 interactive theorem prover. We implement the classical axioms of preference-completeness, transitivity, continuity, and independence-enabling machine-verified proofs of both the existence and uniqueness of utility representations. Our formalization captures the mathematical structure of preference relations over lotteries, verifying that preferences satisfying the vNM axioms can be represented by expected utility maximization. Our contributions include a granular implementation of the independence axiom, formally verified proofs of fundamental claims about mixture lotteries, constructive demonstrations of utility existence, and computational experiments validating the results. We prove equivalence to classical presentations while offering greater precision at decision boundaries. This formalization provides a rigorous foundation for applications in economic modeling, AI alignment, and management decision systems, bridging the gap between theoretical decision theory and computational implementation.","abs_url":"https://arxiv.org/abs/2506.07066","pdf_url":"https://arxiv.org/pdf/2506.07066v1","primary_category":"econ.TH","published":"2025-06-08","tags":["formalization","economics","logic"],"lean_note":"Formalizes the von Neumann-Morgenstern expected utility theorem, including existence and uniqueness, in Lean 4.","repo":"https://github.com/jingyuanli-hk/vNM-Theorem-pub","thumb":"thumbs/2506.07066.jpg"},{"id":"2506.06034","title":"MATP-BENCH: Can MLLM Be a Good Automated Theorem Prover for Multimodal Problems?","authors":["Zhitao He","Zongwei Lyu","Dazhong Chen","Dadi Guo","Yi R. Fung"],"abstract":"Numerous theorems, such as those in geometry, are often presented in multimodal forms (e.g., diagrams). Humans benefit from visual reasoning in such settings, using diagrams to gain intuition and guide the proof process. Modern Multimodal Large Language Models (MLLMs) have demonstrated remarkable capabilities in solving a wide range of mathematical problems. However, the potential of MLLMs as Automated Theorem Provers (ATPs), specifically in the multimodal domain, remains underexplored. In this paper, we introduce the Multimodal Automated Theorem Proving benchmark (MATP-BENCH), a new Multimodal, Multi-level, and Multi-language benchmark designed to evaluate MLLMs in this role as multimodal automated theorem provers. MATP-BENCH consists of 1056 multimodal theorems drawn from high school, university, and competition-level mathematics. All these multimodal problems are accompanied by formalizations in Lean 4, Coq and Isabelle, thus making the benchmark compatible with a wide range of theorem-proving frameworks. MATP-BENCH requires models to integrate sophisticated visual understanding with mastery of a broad spectrum of mathematical knowledge and rigorous symbolic reasoning to generate formal proofs. We use MATP-BENCH to evaluate a variety of advanced multimodal language models. Existing methods can only solve a limited number of the MATP-BENCH problems, indicating that this benchmark poses an open challenge for research on automated theorem proving.","abs_url":"https://arxiv.org/abs/2506.06034","pdf_url":"https://arxiv.org/pdf/2506.06034v1","primary_category":"cs.CL","published":"2025-06-06","tags":["benchmark","theorem-proving-ml","autoformalization","geometry"],"lean_note":"Introduces MATP-BENCH whose 1056 multimodal theorems are formalized in Lean 4 (alongside Coq and Isabelle) to evaluate MLLM provers.","repo":"https://github.com/dwrensha/compfiles","thumb":"thumbs/2506.06034.jpg"},{"id":"2506.04592","title":"Safe: Enhancing Mathematical Reasoning in Large Language Models via Retrospective Step-aware Formal Verification","authors":["Chengwu Liu","Ye Yuan","Yichun Yin","Yan Xu","Xin Xu","Zaoyu Chen","Yasheng Wang","Lifeng Shang","Qun Liu","Ming Zhang"],"abstract":"Chain-of-Thought (CoT) prompting has become the de facto method to elicit reasoning capabilities from large language models (LLMs). However, to mitigate hallucinations in CoT that are notoriously difficult to detect, current methods such as process reward models (PRMs) or self-consistency operate as opaque boxes and do not provide checkable evidence for their judgments, possibly limiting their effectiveness. To address this issue, we draw inspiration from the idea that \"the gold standard for supporting a mathematical claim is to provide a proof\". We propose a retrospective, step-aware formal verification framework $Safe$. Rather than assigning arbitrary scores, we strive to articulate mathematical claims in formal mathematical language Lean 4 at each reasoning step and provide formal proofs to identify hallucinations. We evaluate our framework $Safe$ across multiple language models and various mathematical datasets, demonstrating a significant performance improvement while offering interpretable and verifiable evidence. We also propose $FormalStep$ as a benchmark for step correctness theorem proving with $30,809$ formal statements. To the best of our knowledge, our work represents the first endeavor to utilize formal mathematical language Lean 4 for verifying natural language content generated by LLMs, aligning with the reason why formal mathematical languages were created in the first place: to provide a robust foundation for hallucination-prone human-written proofs.","abs_url":"https://arxiv.org/abs/2506.04592","pdf_url":"https://arxiv.org/pdf/2506.04592v1","primary_category":"cs.CL","published":"2025-06-05","tags":["autoformalization","verification","benchmark","theorem-proving-ml"],"lean_note":"Articulates each LLM reasoning step as a Lean 4 formal statement and provides formal proofs to detect hallucinations, with the FormalStep benchmark.","repo":"https://github.com/liuchengwucn/Safe","thumb":"thumbs/2506.04592.jpg"},{"id":"2506.11085","title":"LeanExplore: A search engine for Lean 4 declarations","authors":["Justin Asher"],"abstract":"The expanding Lean 4 ecosystem poses challenges for navigating its vast libraries. This paper introduces LeanExplore, a search engine for Lean 4 declarations. LeanExplore enables users to semantically search for statements, both formally and informally, across select Lean 4 packages (including Batteries, Init, Lean, Mathlib, PhysLean, and Std). This search capability is powered by a hybrid ranking strategy, integrating scores from a multi-source semantic embedding model (capturing conceptual meaning from formal Lean code, docstrings, AI-generated informal translations, and declaration titles), BM25+ for keyword-based lexical relevance, and a PageRank-based score reflecting declaration importance and interconnectedness. The search engine is accessible via a dedicated website (https://www.leanexplore.com/) and a Python API (https://github.com/justincasher/lean-explore). Furthermore, the database can be downloaded, allowing users to self-host the service. LeanExplore integrates easily with LLMs via the model context protocol (MCP), enabling users to chat with an AI assistant about Lean declarations or utilize the search engine for building theorem-proving agents. This work details LeanExplore's architecture, data processing, functionalities, and its potential to enhance Lean 4 workflows and AI-driven mathematical research","abs_url":"https://arxiv.org/abs/2506.11085","pdf_url":"https://arxiv.org/pdf/2506.11085v1","primary_category":"cs.SE","published":"2025-06-04","tags":["tooling","mathlib","theorem-proving-ml"],"lean_note":"LeanExplore is a semantic search engine indexing Lean 4 declarations across Batteries, Init, Lean, Mathlib, PhysLean, and Std.","repo":"https://github.com/justincasher/lean-explore","thumb":"thumbs/2506.11085.jpg"},{"id":"2505.24230","title":"ProofNet++: A Neuro-Symbolic System for Formal Proof Verification with Self-Correction","authors":["Murari Ambati"],"abstract":"We propose ProofNet++, a neuro-symbolic framework that enhances automated theorem proving by combining large language models (LLMs) with formal proof verification and self-correction mechanisms. Current LLM-based systems suffer from hallucinated logical steps and unverifiable reasoning. ProofNet++ mitigates these limitations by integrating symbolic proof tree supervision, a reinforcement learning loop using verifiers as reward functions, and an iterative self-correction module. Our experiments on miniF2F, Lean's mathlib, and HOL Light show that ProofNet++ significantly improves proof accuracy, correctness, and formal verifiability over prior models. We provide theoretical analysis of the convergence and stability of the verifier-guided RL framework and release our datasets and codebase for future research.","abs_url":"https://arxiv.org/abs/2505.24230","pdf_url":"https://arxiv.org/pdf/2505.24230v1","primary_category":"cs.AI","published":"2025-05-30","tags":["theorem-proving-ml","ai-agents","verification","dataset"],"lean_note":"Uses Lean 4 as a backend proof verifier and reward source in an RL theorem-proving loop, evaluating on miniF2F and a Lean mathlib-derived dataset.","thumb":"thumbs/2505.24230.jpg"},{"id":"2505.23135","title":"VERINA: Benchmarking Verifiable Code Generation","authors":["Zhe Ye","Zhengxu Yan","Jingxuan He","Timothe Kasriel","Kaiyu Yang","Dawn Song"],"abstract":"Large language models (LLMs) are increasingly integrated in software development, but ensuring correctness in LLM-generated code remains challenging and often requires costly manual review. Verifiable code generation -- jointly generating code, specifications, and proofs of code-specification alignment -- offers a promising path to address this limitation and further unleash LLMs' benefits in coding. Yet, there exists a significant gap in evaluation: current benchmarks often focus on only individual components rather than providing a holistic evaluation framework of all tasks. In this paper, we introduce VERINA (Verifiable Code Generation Arena), a high-quality benchmark enabling a comprehensive and modular evaluation of code, specification, and proof generation as well as their compositions. VERINA consists of 189 manually curated coding tasks in Lean, with detailed problem descriptions, reference implementations, formal specifications, and extensive test suites. Our extensive evaluation of state-of-the-art LLMs reveals significant challenges in verifiable code generation, especially in proof generation, underscoring the need for improving LLM-based theorem provers in verification domains. The best model, OpenAI o3, achieves a 72.6\\% code correctness rate, 52.3\\% for specification soundness and completeness, and a mere 4.9\\% proof success rate (based on one trial per task). We hope VERINA will catalyze progress in verifiable code generation by providing a rigorous and comprehensive benchmark. We release our dataset on https://huggingface.co/datasets/sunblaze-ucb/verina and our evaluation code on https://github.com/sunblaze-ucb/verina.","abs_url":"https://arxiv.org/abs/2505.23135","pdf_url":"https://arxiv.org/pdf/2505.23135v3","primary_category":"cs.LG","published":"2025-05-29","tags":["benchmark","verification","theorem-proving-ml","dataset"],"lean_note":"Provides VERINA, a benchmark of 189 curated Lean coding tasks evaluating LLM code, specification, and proof generation.","repo":"https://github.com/sunblaze-ucb/verina","thumb":"thumbs/2505.23135.jpg"},{"id":"2505.20613","title":"REAL-Prover: Retrieval Augmented Lean Prover for Mathematical Reasoning","authors":["Ziju Shen","Naohao Huang","Fanyi Yang","Yutong Wang","Guoxiong Gao","Tianyi Xu","Jiedong Jiang","Wanyi He","Pu Yang","Mengzhou Sun","Haocheng Ju","Peihao Wu","Bryan Dai","Bin Dong"],"abstract":"Nowadays, formal theorem provers have made monumental progress on high-school and competition-level mathematics, but few of them generalize to more advanced mathematics. In this paper, we present REAL-Prover, a new open-source stepwise theorem prover for Lean 4 to push this boundary. This prover, based on our fine-tuned large language model (REAL-Prover-v1) and integrated with a retrieval system (Leansearch-PS), notably boosts performance on solving college-level mathematics problems. To train REAL-Prover-v1, we developed HERALD-AF, a data extraction pipeline that converts natural language math problems into formal statements, and a new open-source Lean 4 interactive environment (Jixia-interactive) to facilitate synthesis data collection. In our experiments, our prover using only supervised fine-tune achieves competitive results with a 23.7% success rate (Pass@64) on the ProofNet dataset-comparable to state-of-the-art (SOTA) models. To further evaluate our approach, we introduce FATE-M, a new benchmark focused on algebraic problems, where our prover achieves a SOTA success rate of 56.7% (Pass@64).","abs_url":"https://arxiv.org/abs/2505.20613","pdf_url":"https://arxiv.org/pdf/2505.20613v3","primary_category":"cs.CL","published":"2025-05-27","tags":["theorem-proving-ml","tactic-automation","benchmark","algebra"],"lean_note":"A fine-tuned stepwise LLM theorem prover for Lean 4 with a Mathlib retrieval system for college-level mathematics.","repo":"https://github.com/frenzymath/REAL-Prover","thumb":"thumbs/2505.20613.jpg"},{"id":"2505.19975","title":"Formalizing a classification theorem for low-dimensional solvable Lie algebras in Lean","authors":["Viviana del Barco","Gustavo Infanti","Exequiel Rivas","Paul Schwahn"],"abstract":"We present a formalization, in the theorem prover Lean, of the classification of solvable Lie algebras of dimension at most three over arbitrary fields. Lie algebras are algebraic objects which encode infinitesimal symmetries, and as such ubiquitous in geometry and physics. Our work involves explicit calculations on the level of the underlying vector spaces and provides a use case for the linear algebra and Lie theory routines in Lean's mathematical library mathlib. Along the way we formalize results about Lie algebras, define the semidirect product within this setting and add API for bases of vector spaces. In a wider context, this project aims to provide a complete mechanization of a classification theorem, covering both the statement and its full formal proof, and contribute to the development and broader adoption of such results in formalized mathematics.","abs_url":"https://arxiv.org/abs/2505.19975","pdf_url":"https://arxiv.org/pdf/2505.19975v1","primary_category":"cs.LO","published":"2025-05-26","tags":["formalization","algebra","mathlib"],"lean_note":"Formalizes in Lean the classification of solvable Lie algebras of dimension at most three over arbitrary fields, extending mathlib.","repo":"https://github.com/LieLean/LowDimSolvClassification","thumb":"thumbs/2505.19975.jpg"},{"id":"2505.18492","title":"Formally Solving Answer-Construction Problems in Lean","authors":["Jialiang Sun","Yuzhi Tang","Ao Li","Chris J. Maddison","Kuldeep S. Meel"],"abstract":"Mathematical competition problems fall into two broad types: theorem proving, which asks for a proof of a given statement, and answer construction, which requires constructing a property-satifying object with proofs. With recent advances in large language models (LLMs), formal theorem-proving techniques have made substantial progress on theorem-proving problems, yet formal answer construction remains less studied. This exposes a mismatch between current LLM model families: general LLMs are strong at informal conjecturing but are expensive and unreliable at formal proof generation, whereas prover LLMs are cheap and optimized for formal proofs but weak at mathematical reasoning for proposing candidate answers. Moreover, Lean proof checking alone does not enforce that a constructed witness is a canonical answer: circular or non-closed-form witnesses can eliminate the existential quantifier while failing to constitute an admissible contest answer. To close this gap, we introduce \\textit{Enumerate-Conjecture-Prove} (ECP), a neuro-symbolic framework in Lean for end-to-end answer construction with formal proofs. ECP leverages tool-assisted general LLMs to enumerate evidence and construct candidate answers, and invokes prover LLMs to produce machine-checked proofs. On PutnamBench's and autoformalized MathArena's answer-construction problems, ECP formally solves 17/346 and 18/75 instances with admissible answers and proofs, respectively, which outperform LLM baselines at aligned inference budgets. Our code is available at https://github.com/sunjia72/ecp-lpar.","abs_url":"https://arxiv.org/abs/2505.18492","pdf_url":"https://arxiv.org/pdf/2505.18492v5","primary_category":"cs.AI","published":"2025-05-24","tags":["theorem-proving-ml","autoformalization","ai-agents","benchmark"],"lean_note":"Enumerate-Conjecture-Prove is a neuro-symbolic framework in Lean 4 for answer-construction problems, using prover LLMs and Lean automation for machine-checked proofs.","repo":"https://github.com/sunjia72/ecp-lpar","thumb":"thumbs/2505.18492.jpg"},{"id":"2505.15796","title":"Lean-SMT: An SMT tactic for discharging proof goals in Lean","authors":["Abdalrhman Mohamed","Tomaz Mascarenhas","Harun Khan","Haniel Barbosa","Andrew Reynolds","Yicheng Qian","Cesare Tinelli","Clark Barrett"],"abstract":"Lean is an increasingly popular proof assistant based on dependent type theory. Despite its success, it still lacks important automation features present in more seasoned proof assistants, such as the Sledgehammer tactic in Isabelle/HOL. A key aspect of Sledgehammer is the use of proof-producing SMT solvers to prove a translated proof goal and the reconstruction of the resulting proof into valid justifications for the original goal. We present Lean-SMT, a tactic providing this functionality in Lean. We detail how the tactic converts Lean goals into SMT problems and, more importantly, how it reconstructs SMT proofs into native Lean proofs. We evaluate the tactic on established benchmarks used to evaluate Sledgehammer's SMT integration, with promising results. We also evaluate Lean-SMT as a standalone proof checker for proofs of SMT-LIB problems. We show that Lean-SMT offers a smaller trusted core without sacrificing too much performance.","abs_url":"https://arxiv.org/abs/2505.15796","pdf_url":"https://arxiv.org/pdf/2505.15796v1","primary_category":"cs.LO","published":"2025-05-21","tags":["tactic-automation","tooling","verification","logic"],"lean_note":"Lean-SMT is a Lean tactic that discharges goals via proof-producing SMT solvers and reconstructs SMT proofs into native Lean proofs.","repo":"https://github.com/ufmg-smite/lean-smt","thumb":"thumbs/2505.15796.jpg"},{"id":"2505.14929","title":"Lean-auto: An Interface between Lean 4 and Automated Theorem Provers","authors":["Yicheng Qian","Joshua Clune","Clark Barrett","Jeremy Avigad"],"abstract":"Proof automation is crucial to large-scale formal mathematics and software/hardware verification projects in ITPs. Sophisticated tools called hammers have been developed to provide general-purpose proof automation in ITPs such as Coq and Isabelle, leveraging the power of ATPs. An important component of a hammer is the translation algorithm from the ITP's logical system to the ATP's logical system. In this paper, we propose a novel translation algorithm for ITPs based on dependent type theory. The algorithm is implemented in Lean 4 under the name Lean-auto. When combined with ATPs, Lean-auto provides general-purpose, ATP-based proof automation in Lean 4 for the first time. Soundness of the main translation procedure is guaranteed, and experimental results suggest that our algorithm is sufficiently complete to automate the proof of many problems that arise in practical uses of Lean 4. We also find that Lean-auto solves more problems than existing tools on Lean 4's math library Mathlib4.","abs_url":"https://arxiv.org/abs/2505.14929","pdf_url":"https://arxiv.org/pdf/2505.14929v2","primary_category":"cs.LO","published":"2025-05-20","tags":["tactic-automation","tooling","verification","mathlib"],"lean_note":"Lean-auto translates Lean 4's dependent type theory to automated theorem provers, enabling general-purpose ATP-based proof automation.","repo":"https://github.com/ufmg-smite/lean-smt","thumb":"thumbs/2505.14929.jpg"},{"id":"2505.13938","title":"CLEVER: A Curated Benchmark for Formally Verified Code Generation","authors":["Amitayush Thakur","Jasper Lee","George Tsoukalas","Meghana Sistla","Matthew Zhao","Stefan Zetzsche","Greg Durrett","Yisong Yue","Swarat Chaudhuri"],"abstract":"We introduce ${\\rm C{\\small LEVER}}$, a high-quality, curated benchmark of 161 problems for end-to-end verified code generation in Lean. Each problem consists of (1) the task of generating a specification that matches a held-out ground-truth specification, and (2) the task of generating a Lean implementation that provably satisfies this specification. Unlike prior benchmarks, ${\\rm C{\\small LEVER}}$ avoids test-case supervision, LLM-generated annotations, and specifications that leak implementation logic or allow vacuous solutions. All outputs are verified post-hoc using Lean's type checker to ensure machine-checkable correctness. We use ${\\rm C{\\small LEVER}}$ to evaluate several few-shot and agentic approaches based on state-of-the-art language models. These methods all struggle to achieve full verification, establishing it as a challenging frontier benchmark for program synthesis and formal reasoning. Our benchmark can be found on GitHub(https://github.com/trishullab/clever) as well as HuggingFace(https://huggingface.co/datasets/amitayusht/clever). All our evaluation code is also available online(https://github.com/trishullab/clever-prover).","abs_url":"https://arxiv.org/abs/2505.13938","pdf_url":"https://arxiv.org/pdf/2505.13938v4","primary_category":"cs.LG","published":"2025-05-20","tags":["benchmark","verification","theorem-proving-ml","dataset"],"lean_note":"Introduces CLEVER, a curated 161-problem Lean benchmark for end-to-end formally verified specification and code generation.","repo":"https://github.com/trishullab/clever","thumb":"thumbs/2505.13938.jpg"},{"id":"2505.12933","title":"Formalising the Bruhat-Tits Tree","authors":["Judith Ludwig","Christian Merten"],"abstract":"In this article we describe the formalisation of the Bruhat-Tits tree - an important tool in modern number theory - in the Lean Theorem Prover. Motivated by the goal of connecting to ongoing research, we apply our formalisation to verify a result about harmonic cochains on the tree.","abs_url":"https://arxiv.org/abs/2505.12933","pdf_url":"https://arxiv.org/pdf/2505.12933v4","primary_category":"math.NT","published":"2025-05-19","tags":["formalization","number-theory","mathlib"],"lean_note":"Formalizes the Bruhat-Tits tree in the Lean theorem prover and verifies a result on harmonic cochains.","repo":"https://github.com/chrisflav/bruhat-tits","thumb":"thumbs/2505.12933.jpg"},{"id":"2505.12840","title":"Geometric Formalization of First-Order Stochastic Dominance in $N$ Dimensions: A Tractable Path to Multi-Dimensional Economic Decision Analysis","authors":["Jingyuan Li"],"abstract":"This paper introduces and formally verifies a novel geometric framework for first-order stochastic dominance (FSD) in $N$ dimensions using the Lean 4 theorem prover. Traditional analytical approaches to multi-dimensional stochastic dominance rely heavily on complex measure theory and multivariate calculus, creating significant barriers to formalization in proof assistants. Our geometric approach characterizes $N$-dimensional FSD through direct comparison of survival probabilities in upper-right orthants, bypassing the need for complex integration theory. We formalize key definitions and prove the equivalence between traditional FSD requirements and our geometric characterization. This approach achieves a more tractable and intuitive path to formal verification while maintaining mathematical rigor. We demonstrate how this framework directly enables formal analysis of multi-dimensional economic problems in portfolio selection, risk management, and welfare analysis. The work establishes a foundation for further development of verified decision-making tools in economics and finance, particularly for high-stakes domains requiring rigorous guarantees.","abs_url":"https://arxiv.org/abs/2505.12840","pdf_url":"https://arxiv.org/pdf/2505.12840v1","primary_category":"cs.LO","published":"2025-05-19","tags":["formalization","verification","economics","probability"],"lean_note":"Formally verifies a geometric characterization of N-dimensional first-order stochastic dominance using the Lean 4 theorem prover.","thumb":"thumbs/2505.12840.jpg"},{"id":"2505.12680","title":"Ineq-Comp: Benchmarking Human-Intuitive Compositional Reasoning in Automated Theorem Proving on Inequalities","authors":["Haoyu Zhao","Yihan Geng","Shange Tang","Yong Lin","Bohan Lyu","Hongzhou Lin","Chi Jin","Sanjeev Arora"],"abstract":"LLM-based formal proof assistants (e.g., in Lean) hold great promise for automating mathematical discovery. But beyond syntactic correctness, do these systems truly understand mathematical structure as humans do? We investigate this question in context of mathematical inequalities -- specifically the prover's ability to recognize that the given problem simplifies by applying a known inequality such as AM/GM. Specifically, we are interested in their ability to do this in a compositional setting where multiple inequalities must be applied as part of a solution. We introduce Ineq-Comp, a benchmark built from elementary inequalities through systematic transformations, including variable duplication, algebraic rewriting, and multi-step composition. Although these problems remain easy for humans, we find that most provers -- including Goedel, STP, and Kimina-7B -- struggle significantly. DeepSeek-Prover-V2-7B shows relative robustness, but still suffers a 20% performance drop (pass@32). Even for DeepSeek-Prover-V2-671B model, the gap between compositional variants and seed problems exists, implying that simply scaling up the model size alone does not fully solve the compositional weakness. Strikingly, performance remains poor for all models even when formal proofs of the constituent parts are provided in context, revealing that the source of weakness is indeed in compositional reasoning. Our results expose a persisting gap between the generalization behavior of current AI provers and human mathematical intuition. All data and evaluation code can be found at https://github.com/haoyuzhao123/LeanIneqComp.","abs_url":"https://arxiv.org/abs/2505.12680","pdf_url":"https://arxiv.org/pdf/2505.12680v2","primary_category":"cs.AI","published":"2025-05-19","tags":["benchmark","theorem-proving-ml","dataset","algebra"],"lean_note":"Introduces Ineq-Comp, a Lean benchmark of transformed inequalities that evaluates LLM-based Lean provers' compositional reasoning.","repo":"https://github.com/haoyuzhao123/LeanIneqComp","thumb":"thumbs/2505.12680.jpg"},{"id":"2505.09095","title":"Benchmarking Energy Calculations Using Formal Proofs","authors":["Ejike D. Ugwuanyi","Colin T. Jones","John Velkey","Tyler R. Josephson"],"abstract":"Traditional approaches for validating molecular simulations rely on making software open source and transparent, incorporating unit testing, and generally employing human oversight. We propose an approach that eliminates software errors using formal logic, providing proofs of correctness. We use the Lean theorem prover and programming language to create a rigorous, mathematically verified framework for computing molecular interaction energies. We demonstrate this in LeanLJ, a package of functions, proofs, and code execution software that implements Lennard Jones energy calculations in periodic boundaries. We introduce a strategy that uses polymorphic functions and typeclasses to bridge formal proofs (about idealized Real numbers) and executable programs (over floating point numbers). Execution of LeanLJ matches the current gold standard NIST benchmarks, while providing even stronger guarantees, given LeanLJ's grounding in formal mathematics. This approach can be extended to formally verified molecular simulations, in particular, and formally verified scientific computing software, in general. Keywords: Formal verification, Lean 4, molecular simulations, functional programming.","abs_url":"https://arxiv.org/abs/2505.09095","pdf_url":"https://arxiv.org/pdf/2505.09095v2","primary_category":"cond-mat.stat-mech","published":"2025-05-14","tags":["verification","formalization","physics"],"lean_note":"LeanLJ implements and formally verifies Lennard-Jones molecular interaction energy calculations in the Lean 4 theorem prover with executable code.","repo":"https://github.com/ATOMSLab/LeanLJ","thumb":"thumbs/2505.09095.jpg"},{"id":"2505.07939","title":"Digitalizing Wick's theorem","authors":["Joseph Tooby-Smith"],"abstract":"Wick's theorem is a cornerstone of perturbative quantum field theory. In this paper we announce and discuss the digitalization of Wick's theorem and its proof into the interactive theorem prover Lean 4 as part of the project PhysLean. We do the same for the static and normal-ordered versions of Wick's theorem.","abs_url":"https://arxiv.org/abs/2505.07939","pdf_url":"https://arxiv.org/pdf/2505.07939v1","primary_category":"hep-th","published":"2025-05-12","tags":["formalization","verification","physics","quantum"],"lean_note":"Formalizes Wick's theorem and its normal-ordered and static variants in Lean 4 as part of the PhysLean project.","repo":"https://github.com/HEPLean/PhysLean","thumb":"thumbs/2505.07939.jpg"},{"id":"2505.05758","title":"APOLLO: Automated LLM and Lean Collaboration for Advanced Formal Reasoning","authors":["Azim Ospanov","Farzan Farnia","Roozbeh Yousefzadeh"],"abstract":"Formal reasoning and automated theorem proving constitute a challenging subfield of machine learning, in which machines are tasked with proving mathematical theorems using formal languages like Lean. A formal verification system can check whether a formal proof is correct or not almost instantaneously, but generating a completely correct formal proof with large language models (LLMs) remains a formidable task. The usual approach in the literature is to prompt the LLM many times (up to several thousands) until one of the generated proofs passes the verification system. In this work, we present APOLLO (Automated PrOof repair viaLLM and Lean cOllaboration), a modular, model-agnostic agentic framework that combines the strengths of the Lean compiler with an LLM's reasoning abilities to achieve better proof-generation results at a low token and sampling budgets. Apollo directs a fully automated process in which the LLM generates proofs for theorems, a set of agents analyze the proofs, fix the syntax errors, identify the mistakes in the proofs using Lean, isolate failing sub-lemmas, utilize automated solvers, and invoke an LLM on each remaining goal with a low top-K budget. The repaired sub-proofs are recombined and reverified, iterating up to a user-controlled maximum number of attempts. On the miniF2F benchmark, we establish a new state-of-the-art accuracy of 84.9% among sub 8B-parameter models (as of August 2025) while keeping the sampling budget below one hundred. Moreover, Apollo raises the state-of-the-art accuracy for Goedel-Prover-SFT to 65.6% while cutting sample complexity from 25,600 to a few hundred. General-purpose models (o3-mini, o4-mini) jump from 3-7% to over 40% accuracy. Our results demonstrate that targeted, compiler-guided repair of LLM outputs yields dramatic gains in both efficiency and correctness, suggesting a general paradigm for scalable automated theorem proving.","abs_url":"https://arxiv.org/abs/2505.05758","pdf_url":"https://arxiv.org/pdf/2505.05758v5","primary_category":"cs.AI","published":"2025-05-09","tags":["ai-agents","theorem-proving-ml","tactic-automation","benchmark"],"lean_note":"APOLLO is an agentic framework that pairs the Lean compiler with LLMs to repair and verify generated Lean proofs.","repo":"https://github.com/aziksh-ospanov/APOLLO","thumb":"thumbs/2505.05758.jpg"},{"id":"2505.03171","title":"CombiBench: Benchmarking LLM Capability for Combinatorial Mathematics","authors":["Junqi Liu","Xiaohan Lin","Jonas Bayer","Yael Dillies","Weijie Jiang","Xiaodan Liang","Roman Soletskyi","Haiming Wang","Yunzhou Xie","Beibei Xiong","Zhengfeng Yang","Jujian Zhang","Lihong Zhi","Jia Li","Zhengying Liu"],"abstract":"Neurosymbolic approaches integrating large language models with formal reasoning have recently achieved human-level performance on mathematics competition problems in algebra, geometry and number theory. In comparison, combinatorics remains a challenging domain, characterized by a lack of appropriate benchmarks and theorem libraries. To address this gap, we introduce CombiBench, a comprehensive benchmark comprising 100 combinatorial problems, each formalized in Lean~4 and paired with its corresponding informal statement. The problem set covers a wide spectrum of difficulty levels, ranging from middle school to IMO and university level, and span over ten combinatorial topics. CombiBench is suitable for testing IMO solving capabilities since it includes all IMO combinatorial problems since 2000 (except IMO 2004 P3 as its statement contain an images). Furthermore, we provide a comprehensive and standardized evaluation framework, dubbed Fine-Eval (for $\\textbf{F}$ill-in-the-blank $\\textbf{in}$ L$\\textbf{e}$an Evaluation), for formal mathematics. It accommodates not only proof-based problems but also, for the first time, the evaluation of fill-in-the-blank questions. Using Fine-Eval as the evaluation method and Kimina Lean Server as the backend, we benchmark several LLMs on CombiBench and observe that their capabilities for formally solving combinatorial problems remain limited. Among all models tested (none of which has been trained for this particular task), Kimina-Prover attains the best results, solving 7 problems (out of 100) under both ``with solution'' and ``without solution'' scenarios. We open source the benchmark dataset alongside with the code of the proposed evaluation method at https://github.com/MoonshotAI/CombiBench/.","abs_url":"https://arxiv.org/abs/2505.03171","pdf_url":"https://arxiv.org/pdf/2505.03171v1","primary_category":"cs.AI","published":"2025-05-06","tags":["benchmark","theorem-proving-ml","combinatorics","dataset"],"lean_note":"CombiBench provides 100 combinatorics problems formalized in Lean 4 with a Fine-Eval framework to benchmark LLM provers.","repo":"https://github.com/MoonshotAI/CombiBench","thumb":"thumbs/2505.03171.jpg"},{"id":"2505.02735","title":"FormalMATH: Benchmarking Formal Mathematical Reasoning of Large Language Models","authors":["Zhouliang Yu","Ruotian Peng","Keyi Ding","Yizhe Li","Zhongyuan Peng","Minghao Liu","Yifan Zhang","Zheng Yuan","Huajian Xin","Wenhao Huang","Yandong Wen","Ge Zhang","Weiyang Liu"],"abstract":"Formal mathematical reasoning remains a critical challenge for artificial intelligence, hindered by limitations of existing benchmarks in scope and scale. To address this, we present FormalMATH, a large-scale Lean4 benchmark comprising 5,560 formally verified problems spanning from high-school Olympiad challenges to undergraduate-level theorems across diverse domains (e.g., algebra, applied mathematics, calculus, number theory, and discrete mathematics). To mitigate the inefficiency of manual formalization, we introduce a novel human-in-the-loop autoformalization pipeline that integrates: (1) specialized large language models (LLMs) for statement autoformalization, (2) multi-LLM semantic verification, and (3) negation-based disproof filtering strategies using off-the-shelf LLM-based provers. This approach reduces expert annotation costs by retaining 72.09% of statements before manual verification while ensuring fidelity to the original natural-language problems. Our evaluation of state-of-the-art LLM-based theorem provers reveals significant limitations: even the strongest models achieve only 16.46% success rate under practical sampling budgets, exhibiting pronounced domain bias (e.g., excelling in algebra but failing in calculus) and over-reliance on simplified automation tactics. Notably, we identify a counterintuitive inverse relationship between natural-language solution guidance and proof success in chain-of-thought reasoning scenarios, suggesting that human-written informal reasoning introduces noise rather than clarity in the formal reasoning settings. We believe that FormalMATH provides a robust benchmark for benchmarking formal mathematical reasoning.","abs_url":"https://arxiv.org/abs/2505.02735","pdf_url":"https://arxiv.org/pdf/2505.02735v1","primary_category":"cs.AI","published":"2025-05-05","tags":["benchmark","autoformalization"],"lean_note":"Presents FormalMATH, a Lean4 benchmark of 5,560 formally verified problems built with a human-in-the-loop autoformalization pipeline.","thumb":"thumbs/2505.02735.jpg"},{"id":"2504.21801","title":"DeepSeek-Prover-V2: Advancing Formal Mathematical Reasoning via Reinforcement Learning for Subgoal Decomposition","authors":["Z. Z. Ren","Zhihong Shao","Junxiao Song","Huajian Xin","Haocheng Wang","Wanjia Zhao","Liyue Zhang","Zhe Fu","Qihao Zhu","Dejian Yang","Z. F. Wu","Zhibin Gou","Shirong Ma","Hongxuan Tang","Yuxuan Liu","Wenjun Gao","Daya Guo","Chong Ruan"],"abstract":"We introduce DeepSeek-Prover-V2, an open-source large language model designed for formal theorem proving in Lean 4, with initialization data collected through a recursive theorem proving pipeline powered by DeepSeek-V3. The cold-start training procedure begins by prompting DeepSeek-V3 to decompose complex problems into a series of subgoals. The proofs of resolved subgoals are synthesized into a chain-of-thought process, combined with DeepSeek-V3's step-by-step reasoning, to create an initial cold start for reinforcement learning. This process enables us to integrate both informal and formal mathematical reasoning into a unified model. The resulting model, DeepSeek-Prover-V2-671B, achieves state-of-the-art performance in neural theorem proving, reaching 88.9% pass ratio on the MiniF2F-test and solving 49 out of 658 problems from PutnamBench. In addition to standard benchmarks, we introduce ProverBench, a collection of 325 formalized problems, to enrich our evaluation, including 15 selected problems from the recent AIME competitions (years 24-25). Further evaluation on these 15 AIME problems shows that the model successfully solves 6 of them. In comparison, DeepSeek-V3 solves 8 of these problems using majority voting, highlighting that the gap between formal and informal mathematical reasoning in large language models is substantially narrowing.","abs_url":"https://arxiv.org/abs/2504.21801","pdf_url":"https://arxiv.org/pdf/2504.21801v2","primary_category":"cs.CL","published":"2025-04-30","tags":["theorem-proving-ml","autoformalization","benchmark"],"lean_note":"DeepSeek-Prover-V2 is an LLM for formal theorem proving in Lean 4, trained via RL with recursive subgoal decomposition.","thumb":"thumbs/2504.21801.jpg"},{"id":"2504.21230","title":"Kimina Lean Server: A High-Performance Lean Server for Large-Scale Verification","authors":["Marco Dos Santos","Hugues de Saxcé","Haiming Wang","Ran Wang","Mantas Baksys","Mert Unsal","Junqi Liu","Zhengying Liu","Jia Li"],"abstract":"We introduce the Kimina Lean Server, an open-source project designed as a high-performance verifier for reinforcement learning pipelines. Built on top of the Lean REPL (Read-Eval-Print Loop) maintained by the Lean FRO, our server combines server-side parallelism by managing multiple Lean processes in parallel with a Least Recently Used (LRU) caching mechanism that reuses Lean imports across requests. On the client side, a lightweight Python package enables submitting proof batches and receiving Lean feedback, including extracted tactics and tactic states. Together, these features enable a scalable workflow for large-scale verification and data extraction. In our experiments, the Kimina Lean Server outperforms previous Lean interaction tools, achieving a 1.5 to 2 times speedup in verification time. Moreover, its improved efficiency has enabled its use in the large-scale training of state-of-the-art models such as Kimina-Prover. We hope that our open-source project will support the neural theorem proving community and accelerate future progress by enabling efficient large-scale verification and proof data extraction.","abs_url":"https://arxiv.org/abs/2504.21230","pdf_url":"https://arxiv.org/pdf/2504.21230v3","primary_category":"cs.LO","published":"2025-04-29","tags":["tooling","verification","theorem-proving-ml"],"lean_note":"Open-source high-performance Lean server built on the Lean REPL, parallelizing Lean processes for large-scale proof verification.","repo":"https://github.com/project-numina/kimina-lean-server","thumb":"thumbs/2504.21230.jpg"},{"id":"2504.18146","title":"Tutte's theorem as an educational formalization project","authors":["Pim Otte"],"abstract":"In this work, we present two results: The first result is the formalization of Tutte's theorem in Lean, a key theorem concerning matchings in graph theory. As this formalization is ready to be integrated in Lean's mathlib, it provides a valuable step in the path towards formalizing research-level mathematics in this area. The second result is a framework for doing educational formalization projects. This framework provides a structure to learn to formalize mathematics with minimal teacher input. This framework applies to both traditional academic settings and independent community-driven environments. We demonstrate the framework's use by connecting it to the process of formalizing Tutte's theorem.","abs_url":"https://arxiv.org/abs/2504.18146","pdf_url":"https://arxiv.org/pdf/2504.18146v1","primary_category":"cs.LO","published":"2025-04-25","tags":["formalization","mathlib","combinatorics"],"lean_note":"Formalizes Tutte's theorem on graph matchings in Lean, contributing the development to Mathlib.","repo":"https://github.com/mabdula/Isabelle-Graph-Library","thumb":"thumbs/2504.18146.jpg"},{"id":"2504.13454","title":"On the Averaging Problem of Ideal Families Related to Frankl's Conjecture with Formal Proof by Lean 4","authors":["Masahiro Hachimori","Kenji Kashiwabara"],"abstract":"Frankl's conjecture, also known as the union-closed sets conjecture, can be equivalently expressed in terms of intersection-closed set families by considering the complements of sets. It posits that any family of sets closed under intersections, and containing both the ground set and the empty set, must have a ``rare vertex'' -- a vertex belonging to at most half of the members of the family. The concept of \\emph{average rarity} describes a set family where the average degree of all the elements is at most half of the number of its members. Average rarity is a stronger property that implies the existence of a rare vertex. This paper focuses on ideal families, which are set families that are downward-closed (except the ground set) and include the ground set. We present a proof that the normalized degree sum of any ideal family is non-positive, which is equivalent to saying that every ideal family satisfies the average rarity condition. This proof is formalized and verified using the Lean 4 theorem prover.","abs_url":"https://arxiv.org/abs/2504.13454","pdf_url":"https://arxiv.org/pdf/2504.13454v1","primary_category":"math.CO","published":"2025-04-18","tags":["formalization","combinatorics"],"lean_note":"Formalizes and verifies in Lean 4 that every ideal set family satisfies the average rarity condition related to Frankl's conjecture.","repo":"https://github.com/kashiwabarakenji/frankl_lean","thumb":"thumbs/2504.13454.jpg"},{"id":"2504.11354","title":"Kimina-Prover Preview: Towards Large Formal Reasoning Models with Reinforcement Learning","authors":["Haiming Wang","Mert Unsal","Xiaohan Lin","Mantas Baksys","Junqi Liu","Marco Dos Santos","Flood Sung","Marina Vinyes","Zhenzhe Ying","Zekai Zhu","Jianqiao Lu","Hugues de Saxcé","Bolton Bailey","Chendong Song","Chenjun Xiao","Dehao Zhang","Ebony Zhang","Frederick Pu","Han Zhu","Jiawei Liu","Jonas Bayer","Julien Michel","Longhui Yu","Léo Dreyfus-Schmidt","Lewis Tunstall","Luigi Pagani","Moreira Machado","Pauline Bourigault","Ran Wang","Stanislas Polu","Thibaut Barroyer","Wen-Ding Li","Yazhe Niu","Yann Fleureau","Yangyang Hu","Zhouliang Yu","Zihan Wang","Zhilin Yang","Zhengying Liu","Jia Li"],"abstract":"We introduce Kimina-Prover Preview, a large language model that pioneers a novel reasoning-driven exploration paradigm for formal theorem proving, as showcased in this preview release. Trained with a large-scale reinforcement learning pipeline from Qwen2.5-72B, Kimina-Prover demonstrates strong performance in Lean 4 proof generation by employing a structured reasoning pattern we term \\textit{formal reasoning pattern}. This approach allows the model to emulate human problem-solving strategies in Lean, iteratively generating and refining proof steps. Kimina-Prover sets a new state-of-the-art on the miniF2F benchmark, reaching 80.7% with pass@8192. Beyond improved benchmark performance, our work yields several key insights: (1) Kimina-Prover exhibits high sample efficiency, delivering strong results even with minimal sampling (pass@1) and scaling effectively with computational budget, stemming from its unique reasoning pattern and RL training; (2) we demonstrate clear performance scaling with model size, a trend previously unobserved for neural theorem provers in formal mathematics; (3) the learned reasoning style, distinct from traditional search algorithms, shows potential to bridge the gap between formal verification and informal mathematical intuition. We open source distilled versions with 1.5B and 7B parameters of Kimina-Prover","abs_url":"https://arxiv.org/abs/2504.11354","pdf_url":"https://arxiv.org/pdf/2504.11354v1","primary_category":"cs.AI","published":"2025-04-15","tags":["theorem-proving-ml","ai-agents","benchmark"],"lean_note":"Kimina-Prover is an RL-trained LLM generating Lean 4 proofs, setting a new state-of-the-art on the miniF2F benchmark.","repo":"https://github.com/MoonshotAI/Kimina-Prover-Preview","thumb":"thumbs/2504.11354.jpg"},{"id":"2506.10998","title":"Towards Automated Formal Verification of Backend Systems with LLMs","authors":["Kangping Xu","Yifan Luo","Yang Yuan","Andrew Chi-Chih Yao"],"abstract":"Software testing plays a critical role in ensuring that systems behave as intended. However, existing automated testing approaches struggle to match the capabilities of human engineers due to key limitations such as test locality, lack of general reliability, and business logic blindness. In this work, we propose a novel framework that leverages functional programming and type systems to translate Scala backend code into formal Lean representations. Our pipeline automatically generates theorems that specify the intended behavior of APIs and database operations, and uses LLM-based provers to verify them. When a theorem is proved, the corresponding logic is guaranteed to be correct and no further testing is needed. If the negation of a theorem is proved instead, it confirms a bug. In cases where neither can be proved, human intervention is required. We evaluate our method on realistic backend systems and find that it can formally verify over 50% of the test requirements, which suggests that half of a testing engineer's workload can be automated. Additionally, with an average cost of only $2.19 per API, LLM-based verification is significantly more cost-effective than manual testing and can be scaled easily through parallel execution. Our results indicate a promising direction for scalable, AI-powered software testing, with the potential to greatly improve engineering productivity as models continue to advance.","abs_url":"https://arxiv.org/abs/2506.10998","pdf_url":"https://arxiv.org/pdf/2506.10998v1","primary_category":"cs.SE","published":"2025-04-13","tags":["verification","autoformalization","theorem-proving-ml"],"lean_note":"Translates Scala backend code into formal Lean representations, auto-generates behavioral theorems, and verifies them with LLM-based provers.","repo":"https://github.com/epfl-lara/stainless","thumb":"thumbs/2506.10998.jpg"},{"id":"2504.06239","title":"Canonical for Automated Theorem Proving in Lean","authors":["Chase Norman","Jeremy Avigad"],"abstract":"Canonical is a solver for type inhabitation in dependent type theory, that is, the problem of producing a term of a given type. We present a Lean tactic which invokes Canonical to generate proof terms and synthesize programs. The tactic supports higher-order and dependently-typed goals, structural recursion over indexed inductive types, and definitional equality. Canonical finds proofs for 84% of Natural Number Game problems in 51 seconds total.","abs_url":"https://arxiv.org/abs/2504.06239","pdf_url":"https://arxiv.org/pdf/2504.06239v2","primary_category":"cs.LO","published":"2025-04-08","tags":["tactic-automation","tooling","theorem-proving-ml","type-theory"],"lean_note":"Provides a Lean tactic invoking the Canonical solver to synthesize proof terms and programs for dependently-typed goals.","repo":"https://github.com/chasenorman/Canonical","thumb":"thumbs/2504.06239.jpg"},{"id":"2504.06122","title":"Leanabell-Prover: Posttraining Scaling in Formal Reasoning","authors":["Jingyuan Zhang","Qi Wang","Xingguang Ji","Yahui Liu","Yang Yue","Fuzheng Zhang","Di Zhang","Guorui Zhou","Kun Gai"],"abstract":"Recent advances in automated theorem proving (ATP) through LLMs have highlighted the potential of formal reasoning with Lean 4 codes. However, ATP has not yet be revolutionized by the recent posttraining scaling as demonstrated by Open AI O1/O3 and Deepseek R1. In this work, we investigate the entire posttraining of ATP, aiming to align it with breakthroughs in reasoning models in natural languages. To begin, we continual train current ATP models with a hybrid dataset, which consists of numerous statement-proof pairs, and additional data aimed at incorporating cognitive behaviors that emulate human reasoning and hypothesis refinement. Next, we explore reinforcement learning with the use of outcome reward returned by Lean 4 compiler. Through our designed continual training and reinforcement learning processes, we have successfully improved existing formal provers, including both DeepSeek-Prover-v1.5 and Goedel-Prover, achieving state-of-the-art performance in the field of whole-proof generation. For example, we achieve a 59.8% pass rate (pass@32) on MiniF2F. This is an on-going project and we will progressively update our findings, release our data and training details.","abs_url":"https://arxiv.org/abs/2504.06122","pdf_url":"https://arxiv.org/pdf/2504.06122v3","primary_category":"cs.AI","published":"2025-04-08","tags":["theorem-proving-ml","dataset","benchmark"],"lean_note":"Posttrains Lean 4 LLM theorem provers with continual training and reinforcement learning using the Lean 4 compiler as reward.","repo":"https://github.com/Leanabell-LM/Leanabell-Prover","thumb":"thumbs/2504.06122.jpg"},{"id":"2503.21380","title":"Challenging the Boundaries of Reasoning: An Olympiad-Level Math Benchmark for Large Language Models","authors":["Haoxiang Sun","Yingqian Min","Zhipeng Chen","Wayne Xin Zhao","Ji-Rong Wen"],"abstract":"The rapid advancement of large reasoning models has saturated existing math benchmarks, underscoring the urgent need for more challenging evaluation frameworks. To address this, we introduce OlymMATH, a rigorously curated, Olympiad-level math benchmark comprising 350 problems, each with parallel English and Chinese versions. OlymMATH is the first benchmark to unify dual evaluation paradigms within a single suite: (1) natural language evaluation through OlymMATH-EASY and OlymMATH-HARD, comprising 200 computational problems with numerical answers for objective rule-based assessment, and (2) formal verification through OlymMATH-LEAN, offering 150 problems formalized in Lean 4 for rigorous process-level evaluation. All problems are manually sourced from printed publications to minimize data contamination, verified by experts, and span four core domains. Extensive experiments reveal the benchmark's significant challenge, and our analysis also uncovers consistent performance gaps between languages and identifies cases where models employ heuristic \"guessing\" rather than rigorous reasoning. To further support community research, we release 582k+ reasoning trajectories, a visualization tool, and expert solutions at https://github.com/RUCAIBox/OlymMATH.","abs_url":"https://arxiv.org/abs/2503.21380","pdf_url":"https://arxiv.org/pdf/2503.21380v3","primary_category":"cs.CL","published":"2025-03-27","tags":["benchmark","theorem-proving-ml","formalization"],"lean_note":"Includes OlymMATH-LEAN, 150 Olympiad problems formalized in Lean 4 for formal-verification process-level evaluation.","repo":"https://github.com/RUCAIBox/OlymMATH","thumb":"thumbs/2503.21380.jpg"},{"id":"2503.20704","title":"Formalizing colimits in Cat","authors":["Mario Carneiro","Emily Riehl"],"abstract":"Certain results involving \"higher structures\" are not currently accessible to computer formalization because the prerequisite $\\infty$-category theory has not been formalized. To support future work on formalizing $\\infty$-category theory in Lean's mathematics library, we formalize some fundamental constructions involving the 1-category of categories. Specifically, we construct the left adjoint to the nerve embedding of categories into simplicial sets, defining the homotopy category functor. We prove further that this adjunction is reflective, which allows us to conclude that Cat has colimits. To our knowledge this is the first formalized proof that the nerve functor is a fully faithful right adjoint and that the category of categories is cocomplete.","abs_url":"https://arxiv.org/abs/2503.20704","pdf_url":"https://arxiv.org/pdf/2503.20704v2","primary_category":"math.CT","published":"2025-03-26","tags":["formalization","category-theory","mathlib"],"lean_note":"Formalizes the homotopy category functor and cocompleteness of Cat in Lean's Mathlib to support future infinity-category work.","repo":"https://github.com/UniMath/UniMath","thumb":"thumbs/2503.20704.jpg"},{"id":"2505.00001","title":"Rosetta-PL: Propositional Logic as a Benchmark for Large Language Model Reasoning","authors":["Shaun Baek","Shaun Esua-Mensah","Cyrus Tsui","Sejan Vigneswaralingam","Abdullah Alali","Michael Lu","Vasu Sharma","Sean O'Brien","Kevin Zhu"],"abstract":"Large Language Models (LLMs) are primarily trained on high-resource natural languages, limiting their effectiveness in low-resource settings and in tasks requiring deep logical reasoning. This research introduces Rosetta-PL, a benchmark designed to evaluate LLMs' logical reasoning and generalization capabilities in a controlled environment. We construct Rosetta-PL by translating a dataset of logical propositions from Lean into a custom logical language, which is then used to fine-tune an LLM (e.g., GPT-4o). Our experiments analyze the impact of the size of the dataset and the translation methodology on the performance of the model. Our results indicate that preserving logical relationships in the translation process significantly boosts precision, with accuracy plateauing beyond roughly 20,000 training samples. These insights provide valuable guidelines for optimizing LLM training in formal reasoning tasks and improving performance in various low-resource language applications.","abs_url":"https://arxiv.org/abs/2505.00001","pdf_url":"https://arxiv.org/pdf/2505.00001v2","primary_category":"cs.CL","published":"2025-03-25","tags":["benchmark","dataset","logic","theorem-proving-ml"],"lean_note":"Builds the Rosetta-PL benchmark by translating a dataset of logical propositions sourced from Lean into a custom propositional language.","thumb":"thumbs/2505.00001.jpg"},{"id":"2503.19605","title":"Lean Formalization of Generalization Error Bound by Rademacher Complexity and Dudley's Entropy Integral","authors":["Sho Sonoda","Kazumi Kasaura","Yuma Mizuno","Kei Tsukamoto","Naoto Onda"],"abstract":"Understanding and certifying the generalization performance of machine learning algorithms -- i.e. obtaining theoretical estimates of the test error from the training error -- is a central theme of statistical learning theory. Among the many complexity measures used to derive such guarantees, Rademacher complexity yields sharp, data-dependent bounds that apply well beyond classical VC-dimension theory. In this study, we formalize the generalization error bound by Rademacher complexity in Lean 4, building on measure-theoretic probability theory available in the Mathlib library. Our development provides a mechanically-checked pipeline from the definitions of empirical and expected Rademacher complexity, through a formal symmetrization argument and a bounded-differences analysis, to high-probability uniform deviation bounds via a formally proved McDiarmid inequality. A key technical contribution is a reusable mechanism for lifting results from countable hypothesis classes (where measurability of suprema is straightforward in Mathlib) to separable topological index sets via a reduction to a countable dense subset. As worked applications of the abstract theorem, we mechanize standard empirical Rademacher bounds for linear predictors under $\\ell_2$ and $\\ell_1$ regularizations, and we also formalize a Dudley-type entropy integral bound based on covering numbers and a chaining construction.","abs_url":"https://arxiv.org/abs/2503.19605","pdf_url":"https://arxiv.org/pdf/2503.19605v5","primary_category":"cs.LG","published":"2025-03-25","tags":["formalization","mathlib","ml-theory","probability"],"lean_note":"Formalizes generalization error bounds via Rademacher complexity and Dudley's entropy integral in Lean 4 on top of Mathlib.","repo":"https://github.com/auto-res/lean-rademacher","thumb":"thumbs/2503.19605.jpg"},{"id":"2503.18821","title":"Formalization of Optimality Conditions for Smooth Constrained Optimization Problems","authors":["Chenyi Li","Shengyang Xu","Chumin Sun","Li Zhou","Zaiwen Wen"],"abstract":"Optimality conditions are central to analysis of optimization problems, characterizing necessary criteria for local minima. Formalizing the optimality conditions within the type-theory-based proof assistant Lean4 provides a precise, robust, and reusable framework essential for rigorous verification in optimization theory. In this paper, we introduce a formalization of the first-order optimality conditions (also known as the Karush-Kuhn-Tucker (KKT) conditions) for smooth constrained optimization problems by beginning with concepts such as the Lagrangian function and constraint qualifications. The geometric optimality conditions are then formalized, offering insights into local minima through tangent cones. We also establish the critical equivalence between the tangent cone and linearized feasible directions under appropriate constraint qualifications. Building on these key elements, the formalization concludes the KKT conditions through the proof of the Farkas lemma. Additionally, this study provides a formalization of the dual problem and the weak duality property.","abs_url":"https://arxiv.org/abs/2503.18821","pdf_url":"https://arxiv.org/pdf/2503.18821v1","primary_category":"math.OC","published":"2025-03-24","tags":["formalization","verification"],"lean_note":"Formalizes the KKT first-order optimality conditions for smooth constrained optimization in Lean4, including the Farkas lemma and weak duality.","repo":"https://github.com/optsuite/optlib","thumb":"thumbs/2503.18821.jpg"},{"id":"2503.18806","title":"Formalization of Algorithms for Optimization with Block Structures","authors":["Chenyi Li","Zichen Wang","Yifan Bai","Yunxi Duan","Yuqing Gao","Pengfei Hao","Zaiwen Wen"],"abstract":"Block-structured problems are central to advances in numerical optimization and machine learning. This paper provides the formalization of convergence analysis for two pivotal algorithms in such settings: the block coordinate descent (BCD) method and the alternating direction method of multipliers (ADMM). Utilizing the type-theory-based proof assistant Lean4, we develop a rigorous framework to formally represent these algorithms. Essential concepts in nonsmooth and nonconvex optimization are formalized, notably subdifferentials, which extend the classical differentiability to handle nonsmooth scenarios, and the Kurdyka-Lojasiewicz (KL) property, which provides essential tools to analyze convergence in nonconvex settings. Such definitions and properties are crucial for the corresponding convergence analyses. We formalize the convergence proofs of these algorithms, demonstrating that our definitions and structures are coherent and robust. These formalizations lay a basis for analyzing the convergence of more general optimization algorithms.","abs_url":"https://arxiv.org/abs/2503.18806","pdf_url":"https://arxiv.org/pdf/2503.18806v1","primary_category":"math.OC","published":"2025-03-24","tags":["formalization","verification"],"lean_note":"Formalizes convergence of block coordinate descent and ADMM for block-structured optimization in Lean4, with subdifferentials and the KL property.","repo":"https://github.com/optsuite/optlib","thumb":"thumbs/2503.18806.jpg"},{"id":"2503.09730","title":"Local Look-Ahead Guidance via Verifier-in-the-Loop for Automated Theorem Proving","authors":["Sara Rajaee","Kumar Pratik","Gabriele Cesa","Arash Behboodi"],"abstract":"The most promising recent methods for AI reasoning require applying variants of reinforcement learning (RL) either on rolled out trajectories from the LLMs, even for the step-wise rewards, or large quantities of human-annotated trajectory data. The reliance on the rolled-out trajectory renders the compute cost and time prohibitively high. In particular, the correctness of a reasoning trajectory can typically only be judged at its completion, leading to sparse rewards in RL or requiring expensive synthetic data generation in expert iteration-like methods. In this work, we focus on the Automatic Theorem Proving (ATP) task and propose a novel verifier-in-the-loop design, which, unlike existing approaches that leverage feedback on the entire reasoning trajectory, employs an automated verifier to give intermediate feedback at each step of the reasoning process. Using Lean as the verifier, we empirically show that the step-by-step local verification produces a global improvement in the model's reasoning accuracy and efficiency.","abs_url":"https://arxiv.org/abs/2503.09730","pdf_url":"https://arxiv.org/pdf/2503.09730v2","primary_category":"cs.AI","published":"2025-03-12","tags":["theorem-proving-ml","tactic-automation"],"lean_note":"Uses Lean as an automated verifier giving step-wise feedback to train an LLM for automated theorem proving.","repo":"https://github.com/lean-dojo/ReProver","thumb":"thumbs/2503.09730.jpg"},{"id":"2503.06812","title":"Can Proof Assistants Verify Multi-Agent Systems?","authors":["Julian Alfredo Mendez","Timotheus Kampik"],"abstract":"This paper presents the Soda language for verifying multi-agent systems. Soda is a high-level functional and object-oriented language that supports the compilation of its code not only to Scala, a strongly statically typed high-level programming language, but also to Lean, a proof assistant and programming language. Given these capabilities, Soda can implement multi-agent systems, or parts thereof, that can then be integrated into a mainstream software ecosystem on the one hand and formally verified with state-of-the-art tools on the other hand. We provide a brief and informal introduction to Soda and the aforementioned interoperability capabilities, as well as a simple demonstration of how interaction protocols can be designed and verified with Soda. In the course of the demonstration, we highlight challenges with respect to real-world applicability.","abs_url":"https://arxiv.org/abs/2503.06812","pdf_url":"https://arxiv.org/pdf/2503.06812v1","primary_category":"cs.PL","published":"2025-03-10","tags":["verification","tooling"],"lean_note":"The Soda language compiles to Lean so multi-agent systems can be formally verified in the Lean proof assistant.","repo":"https://github.com/julianmendez/soda","thumb":"thumbs/2503.06812.jpg"},{"id":"2503.03238","title":"FANS -- Formal Answer Selection for Natural Language Math Reasoning Using Lean4","authors":["Jiarui Yao","Ruida Wang","Tong Zhang"],"abstract":"Large Language Models (LLMs) have displayed astonishing abilities in various tasks, especially in text generation, classification, question answering, etc. However, the reasoning ability of LLMs still faces many debates. The inherent ambiguity of Natural Language (NL) limits LLMs' ability to perform verifiable reasoning, making its answers lack coherence and trustworthy support. To tackle the above problems, we propose a novel framework named FANS: Formal ANswer Selection for Natural Language Math Reasoning Using Lean4. To the best of our knowledge, it is the first framework that utilizes Lean4 to enhance LLMs' NL math reasoning ability. In particular, given an NL math question and LLM-generated answers, FANS first translates it into Lean4 theorem statements. Then it tries to prove it using a Lean4 prover and verify it by Lean4. Finally, it uses the FL result to assist in answer selection. It enhances LLMs' NL math ability in providing a computer-verifiable solution for its correct answer and proposes an alternative method for answer selection beyond the reward model. Extensive experiments indicate the effectiveness of our framework. It can improve the accuracy rate of reward model enhanced LLMs in the MATH-500 dataset by at most 1.91% and AMC-23 by at most 8.33% on strong reward-model baselines. In some particular fields like number theory that Lean4 experts in, we can even select all correct solutions. The qualitative analysis also shows our framework can make NL results formally backed by Lean4 proofs. As a pioneering work in the corresponding field, we will open-source all our models and datasets to further boost the development of the field.","abs_url":"https://arxiv.org/abs/2503.03238","pdf_url":"https://arxiv.org/pdf/2503.03238v2","primary_category":"cs.CL","published":"2025-03-05","tags":["autoformalization","theorem-proving-ml"],"lean_note":"FANS uses Lean4 to formally verify candidate answers for natural-language math reasoning and improve answer selection.","repo":"https://github.com/MaxwellJryao/FANS","thumb":"thumbs/2503.03238.jpg"},{"id":"2503.03205","title":"MA-LoT: Model-Collaboration Lean-based Long Chain-of-Thought Reasoning enhances Formal Theorem Proving","authors":["Ruida Wang","Rui Pan","Yuxin Li","Jipeng Zhang","Yizhen Jia","Shizhe Diao","Renjie Pi","Junjie Hu","Tong Zhang"],"abstract":"Solving mathematical problems using computer-verifiable languages like Lean has significantly impacted the mathematical and computer science communities. State-of-the-art methods utilize a single Large Language Model (LLM) to generate complete proof or perform tree search, but they fail to balance these tasks. We propose **MA-LoT**: *Model-CollAboration Lean-based Long Chain-of-Thought*, a comprehensive framework for Lean4 theorem proving to solve this issue. It separates the cognition tasks of general NL for whole-proof generation and error analysis for proof correction using the model-collaboration method. We achieve this by structured interaction of the LLM and Lean4 verifier in Long CoT. To implement the framework, we propose the novel *LoT-Transfer Learning* training-inference pipeline, which enables the Long CoT thinking capability to LLMs without special data annotation. Extensive experiment shows that our framework achieves a **61.07%** accuracy rate on the Lean4 version of the MiniF2F-Test dataset, largely outperforming DeepSeek-V3 (33.61%), single-model tree search (InternLM-Step-Prover, 50.70%), and whole-proof generation (Godel-Prover, 55.33%) baselines. Furthermore, our findings highlight the potential of combining Long CoT with formal verification for a more insightful generation in a broader perspective.","abs_url":"https://arxiv.org/abs/2503.03205","pdf_url":"https://arxiv.org/pdf/2503.03205v3","primary_category":"cs.CL","published":"2025-03-05","tags":["theorem-proving-ml","autoformalization","ai-agents"],"lean_note":"MA-LoT is a model-collaboration framework for Lean 4 theorem proving combining long chain-of-thought with the Lean4 verifier.","repo":"https://github.com/RickySkywalker/LeanOfThought-Official","thumb":"thumbs/2503.03205.jpg"},{"id":"2503.00959","title":"Formalizing zeta and L-functions in Lean","authors":["David Loeffler","Michael Stoll"],"abstract":"The Riemann zeta function, and more generally the L-functions of Dirichlet characters, are among the central objects of study in number theory. We report on a project to formalize the theory of these objects in Lean's \"Mathlib\" library, including a proof of Dirichlet's theorem on primes in arithmetic progressions and a formal statement of the Riemann hypothesis","abs_url":"https://arxiv.org/abs/2503.00959","pdf_url":"https://arxiv.org/pdf/2503.00959v4","primary_category":"math.NT","published":"2025-03-02","tags":["formalization","mathlib","number-theory"],"lean_note":"Formalizes the Riemann zeta function and Dirichlet L-functions in Lean's Mathlib, including Dirichlet's theorem and a statement of the Riemann hypothesis.","thumb":"thumbs/2503.00959.jpg"},{"id":"2503.07625","title":"A Formal Proof of the Irrationality of $ζ(3)$ in Lean 4","authors":["Junqi Liu","Jujian Zhang","Lihong Zhi"],"abstract":"We formalize a proof of the irrationality of $ζ(3)$ in Lean 4, using Beukers' method. To support this, we extend the Lean mathematical library (Mathlib) by formalizing shifted Legendre polynomials and important results in analytic number theory that were previously missing. As part of the Lean 4 PrimeNumberTheoremAnd project, we also formalize the asymptotic behavior of the prime counting function, giving the first formal proof in Lean 4 of a version of the Prime Number Theorem with an error term which is stronger than what had previously been formalized. This result is a crucial ingredient in proving the irrationality of $ζ(3)$. Our complete Lean 4 formalization is publicly available on GitHub.","abs_url":"https://arxiv.org/abs/2503.07625","pdf_url":"https://arxiv.org/pdf/2503.07625v2","primary_category":"math.NT","published":"2025-02-28","tags":["formalization","mathlib","number-theory"],"lean_note":"Formalizes the irrationality of zeta(3) in Lean 4 via Beukers' method, extending Mathlib with new analytic number theory.","repo":"https://github.com/ahhwuhu/zeta_3_irrational","thumb":"thumbs/2503.07625.jpg"},{"id":"2502.20485","title":"Bounded First-Class Universe Levels in Dependent Type Theory","authors":["Jonathan Chan","Stephanie Weirich"],"abstract":"In dependent type theory, being able to refer to a type universe as a term itself increases its expressive power, but requires mechanisms in place to prevent Girard's paradox from introducing logical inconsistency in the presence of type-in-type. The simplest mechanism is a hierarchy of universes indexed by a sequence of levels, typically the naturals. To improve reusability of definitions, they can be made level polymorphic, abstracting over level variables and adding a notion of level expressions. For even more expressive power, level expressions can be made first-class as terms themselves, and level polymorphism is subsumed by dependent functions quantifying over levels. Furthermore, bounded level polymorphism provides more expressivity by being able to explicitly state constraints on level variables. While semantics for first-class levels with constraints are known, syntax and typing rules have not been explicitly written down. Yet pinning down a well-behaved syntax is not trivial; there exist prior type theories with bounded level polymorphism that fail to satisfy subject reduction. In this work, we design an explicit syntax for a type theory with bounded first-class levels, parametrized over arbitrary well-founded sets of levels. We prove the metatheoretic properties of subject reduction, type safety, consistency, and canonicity, entirely mechanized from syntax to semantics in Lean.","abs_url":"https://arxiv.org/abs/2502.20485","pdf_url":"https://arxiv.org/pdf/2502.20485v1","primary_category":"cs.PL","published":"2025-02-27","tags":["type-theory","formalization","logic"],"lean_note":"Designs a dependent type theory with bounded first-class universe levels, mechanizing its metatheory from syntax to semantics in Lean.","repo":"https://github.com/ionathanch/TTBFL","thumb":"thumbs/2502.20485.jpg"},{"id":"2502.17925","title":"LeanProgress: Guiding Search for Neural Theorem Proving via Proof Progress Prediction","authors":["Robert Joseph George","Suozhi Huang","Peiyang Song","Anima Anandkumar"],"abstract":"Mathematical reasoning remains a significant challenge for Large Language Models (LLMs) due to hallucinations. When combined with formal proof assistants like Lean, these hallucinations can be eliminated through rigorous verification, making theorem proving reliable. However, even with formal verification, LLMs still struggle with long proofs and complex mathematical formalizations. While Lean with LLMs offers valuable assistance with retrieving lemmas, generating tactics, or even complete proofs, it lacks a crucial capability: providing a sense of proof progress. This limitation particularly impacts the overall development efficiency in large formalization projects. We introduce LeanProgress, a method that predicts the progress in the proof. Training and evaluating our models made on a large corpus of Lean proofs from Lean Workbook Plus and Mathlib4 and how many steps remain to complete it, we employ data preprocessing and balancing techniques to handle the skewed distribution of proof lengths. Our experiments show that LeanProgress achieves an overall prediction accuracy of 75.8% in predicting the amount of progress and, hence, the remaining number of steps. When integrated into a best-first search framework using Reprover, our method shows a 3.8% improvement on Mathlib4 compared to baseline performances of 41.4%, particularly for longer proofs. These results demonstrate how proof progress prediction can enhance both automated and interactive theorem proving, enabling users to make more informed decisions about proof strategies. Our code is merged in this library here https://github.com/lean-dojo/LeanDojo-v2.","abs_url":"https://arxiv.org/abs/2502.17925","pdf_url":"https://arxiv.org/pdf/2502.17925v3","primary_category":"cs.AI","published":"2025-02-25","tags":["theorem-proving-ml","ai-agents","mathlib","dataset"],"lean_note":"Trains a model predicting remaining proof steps from Lean Workbook Plus and Mathlib4 to guide best-first search in neural theorem proving.","repo":"https://github.com/lean-dojo/LeanDojo-v2","thumb":"thumbs/2502.17925.jpg"},{"id":"2502.17840","title":"A Combinatorial Identities Benchmark for Theorem Proving via Automated Theorem Generation","authors":["Beibei Xiong","Hangyu Lv","Haojia Shan","Jianlin Wang","Zhengfeng Yang","Lihong Zhi"],"abstract":"Large language models (LLMs) have significantly advanced formal theorem proving, yet the scarcity of high-quality training data constrains their capabilities in complex mathematical domains. Combinatorics, a cornerstone of mathematics, provides essential tools for analyzing discrete structures and solving optimization problems. However, its inherent complexity makes it particularly challenging for automated theorem proving (ATP) for combinatorial identities. To address this, we manually construct LeanComb, combinatorial identities benchmark in Lean, which is, to our knowledge, the first formalized theorem proving benchmark built for combinatorial identities. We develop an Automated Theorem Generator for Combinatorial Identities, ATG4CI, which combines candidate tactics suggested by a self-improving large language model with a Reinforcement Learning Tree Search approach for tactic prediction. By utilizing ATG4CI, we generate a LeanComb-Enhanced dataset comprising 260K combinatorial identities theorems, each with a complete formal proof in Lean, and experimental evaluations demonstrate that models trained on this dataset can generate more effective tactics, thereby improving success rates in automated theorem proving for combinatorial identities.","abs_url":"https://arxiv.org/abs/2502.17840","pdf_url":"https://arxiv.org/pdf/2502.17840v1","primary_category":"cs.AI","published":"2025-02-25","tags":["benchmark","dataset","theorem-proving-ml","combinatorics"],"lean_note":"Builds LeanComb, a Lean benchmark of combinatorial identities, plus a 260K-theorem Lean dataset generated by an automated theorem generator.","thumb":"thumbs/2502.17840.jpg"},{"id":"2502.15795","title":"Lean-ing on Quality: How High-Quality Data Beats Diverse Multilingual Data in AutoFormalization","authors":["Willy Chan","Michael Souliman","Jakob Nordhagen","Brando Miranda","Elyas Obbad","Kai Fronsdal Sanmi Koyejo"],"abstract":"Autoformalization, the process of transforming informal mathematical language into formal specifications and proofs remains a difficult task for state-of-the-art (large) language models. Existing works point to competing explanations for the performance gap. To this end, we introduce a novel methodology that leverages back-translation with hand-curated prompts to enhance the mathematical capabilities of language models, particularly addressing the challenge posed by the scarcity of labeled data. Specifically, we evaluate three primary variations of this strategy: (1) on-the-fly (online) backtranslation, (2) distilled (offline) backtranslation with few-shot amplification, and (3) line-by-line proof analysis integrated with proof state information. Each variant is designed to optimize data quality over quantity, focusing on the high fidelity of generated proofs rather than sheer data scale. Our findings provide evidence that employing our proposed approaches to generate synthetic data, which prioritizes quality over volume, improves the Autoformalization performance of LLMs as measured by standard benchmarks such as ProofNet. Crucially, our approach outperforms pretrained models using a minimal number of tokens. We also show, through strategic prompting and backtranslation, that our approaches surpass the performance of fine-tuning with extensive multilingual datasets such as MMA on ProofNet with only 1/150th of the tokens. Taken together, our methods show a promising new approach to significantly reduce the resources required to formalize proofs, thereby accelerating AI for math.","abs_url":"https://arxiv.org/abs/2502.15795","pdf_url":"https://arxiv.org/pdf/2502.15795v1","primary_category":"cs.AI","published":"2025-02-18","tags":["autoformalization","theorem-proving-ml","dataset"],"lean_note":"Improves LLM autoformalization targeting Lean using back-translation and proof-state-aware data, evaluated on ProofNet with LeanDojo and Mathlib4 data.","thumb":"thumbs/2502.15795.jpg"},{"id":"2502.13137","title":"Theorem Prover as a Judge for Synthetic Data Generation","authors":["Joshua Ong Jun Leang","Giwon Hong","Wenda Li","Shay B. Cohen"],"abstract":"The demand for synthetic data in mathematical reasoning has increased due to its potential to enhance the mathematical capabilities of large language models (LLMs). However, ensuring the validity of intermediate reasoning steps remains a significant challenge, affecting data quality. While formal verification via theorem provers effectively validates LLM reasoning, the autoformalisation of mathematical proofs remains error-prone. In response, we introduce iterative autoformalisation, an approach that iteratively refines theorem prover formalisation to mitigate errors, thereby increasing the execution rate on the Lean prover from 60% to 87%. Building upon that, we introduce Theorem Prover as a Judge (TP-as-a-Judge), a method that employs theorem prover formalisation to rigorously assess LLM intermediate reasoning, effectively integrating autoformalisation with synthetic data generation. Finally, we present Reinforcement Learning from Theorem Prover Feedback (RLTPF), a framework that replaces human annotation with theorem prover feedback in Reinforcement Learning from Human Feedback (RLHF). Across multiple LLMs, applying TP-as-a-Judge and RLTPF improves benchmarks with only 3,508 samples, achieving 5.56% accuracy gain on Mistral-7B for MultiArith, 6.00% on Llama-2-7B for SVAMP, and 3.55% on Llama-3.1-8B for AQUA.","abs_url":"https://arxiv.org/abs/2502.13137","pdf_url":"https://arxiv.org/pdf/2502.13137v2","primary_category":"cs.AI","published":"2025-02-18","tags":["autoformalization","theorem-proving-ml","dataset"],"lean_note":"Uses the Lean prover to verify autoformalized LLM reasoning steps, judging synthetic data and providing reinforcement-learning feedback.","repo":"https://github.com/joshuaongg21/RLTPF","thumb":"thumbs/2502.13137.jpg"},{"id":"2503.04772","title":"Generating Millions Of Lean Theorems With Proofs By Exploring State Transition Graphs","authors":["David Yin","Jing Gao"],"abstract":"Large Language Models (LLMs) have demonstrated significant potential in generating mathematical proofs. However, a persistent challenge is that LLMs occasionally make mistakes, while even a minor mistake can invalidate an entire proof. Proof assistants like Lean offer a great remedy. They are designed for verifying each step of a proof in a formal language, and in recent years researchers have created AI models to generate proofs in their languages. However, the scarcity of large-scale datasets of Lean proofs restrict the performance of such Automated Theorem Proving (ATP) models. We developed LeanNavigator, a novel method for generating a large-scale dataset of Lean theorems and proofs by finding new ways to prove existing Lean theorems. By leveraging an interactive Lean client and an efficient method for proof step generation, LeanNavigator efficiently produces new theorems with corresponding proofs. Applying this approach to Mathlib4, we generated 4.7 million theorems totaling 1 billion tokens, surpassing previous datasets by more than an order of magnitude. Using this extensive dataset, we trained an AI model that outperforms the state-of-the-art ReProver model in theorem-proving tasks. These results confirm our hypothesis and demonstrate the critical role of large datasets in improving the performance of automated theorem provers.","abs_url":"https://arxiv.org/abs/2503.04772","pdf_url":"https://arxiv.org/pdf/2503.04772v1","primary_category":"cs.LO","published":"2025-02-16","tags":["dataset","theorem-proving-ml","tactic-automation","mathlib"],"lean_note":"Generates 4.7M Lean theorems with proofs from Mathlib4 by exploring proof state-transition graphs to train a prover.","repo":"https://zenodo.org/records/13989482","thumb":"thumbs/2503.04772.jpg"},{"id":"2502.09955","title":"Diverse Inference and Verification for Advanced Reasoning","authors":["Iddo Drori","Gaston Longhitano","Mao Mao","Seunghwan Hyun","Yuke Zhang","Sungjun Park","Zachary Meeks","Xin-Yu Zhang","Ben Segev","Howard Yong","Nakul Verma","Avi Shporer","Alon Amit","Madeleine Udell"],"abstract":"Reasoning LLMs such as OpenAI o1, o3 and DeepSeek R1 have made significant progress in mathematics and coding, yet find challenging advanced tasks such as International Mathematical Olympiad (IMO) combinatorics problems, Abstraction and Reasoning Corpus (ARC) puzzles, and Humanity's Last Exam (HLE) questions. We use a diverse inference approach that combines multiple models and methods at test time. We find that verifying mathematics and code problems, and rejection sampling on other problems is simple and effective. We automatically verify correctness of solutions to IMO problems by Lean, and ARC puzzles by code, and find that best-of-N effectively answers HLE questions. Our approach increases answer accuracy on IMO combinatorics problems from 33.3% to 77.8%, accuracy on HLE questions from 8% to 37%, and solves 80% of ARC puzzles that 948 humans could not and 26.5% of ARC puzzles that o3 high compute does not. Test-time simulations, reinforcement learning, and meta-learning with inference feedback improve generalization by adapting agent graph representations and varying prompts, code, and datasets. Our approach is reliable, robust, and scalable, and in the spirit of reproducible research, we will make it publicly available upon publication.","abs_url":"https://arxiv.org/abs/2502.09955","pdf_url":"https://arxiv.org/pdf/2502.09955v1","primary_category":"cs.AI","published":"2025-02-14","tags":["theorem-proving-ml","verification","ai-agents"],"lean_note":"Uses Lean to automatically verify the correctness of LLM-generated solutions to IMO mathematics problems within a diverse-inference pipeline.","thumb":"thumbs/2502.09955.jpg"},{"id":"2502.07640","title":"Goedel-Prover: A Frontier Model for Open-Source Automated Theorem Proving","authors":["Yong Lin","Shange Tang","Bohan Lyu","Jiayun Wu","Hongzhou Lin","Kaiyu Yang","Jia Li","Mengzhou Xia","Danqi Chen","Sanjeev Arora","Chi Jin"],"abstract":"We introduce Goedel-Prover, an open-source language model that achieves state-of-the-art (as of April 5 2025) performance in automated formal proof generation for mathematical problems. A key challenge in this field is the scarcity of formalized mathematical statements and proofs, which we address through the following approaches. First, we train LLMs to convert natural language math problems from the Numina dataset to equivalent formal statements in Lean 4. This process creates the dataset Goedel-Pset-v1, which includes 1.64 million formal statements. Next, we develop a large dataset of formal proofs by training a series of provers. Each new prover can prove many statements that previous ones could not, and these new proofs are added to the training set for the next prover. Finally, we obtain the dataset Goedel-Pset-v1-solved, which contains proofs for over 800K statements from Goedel-Pset-v1. Supervised fine-tuning (SFT) of DeepSeek-Prover-V1.5-Base on Goedel-Pset-v1-solved (i.e., no RL) yields a Goedel-Prover-SFT that achieves a success rate of 57.6% (Pass@32) on miniF2F, surpassing the previous leader DeepSeek-Prover-V1.5-RL (trained using SFT + RL on a proprietary dataset) by 7.6%. On PutnamBench, Goedel-Prover-SFT successfully solves 7 problems (Pass@512), ranking first on the leaderboard. We provide extensive discussion of our training methodology, highlighting the key design choices that contribute to Goedel-Prover's strong performance. Further RL training (including DPO) improves Goedel-Prover-SFT's success rate to over 60% (Pass@32) on miniF2F. To aid future research, we provide extensive discussion of our training methodology and design choices. We also fully open-source our codes, models, and datasets. Additionally, we open-source formal proofs for 29.7K problems in Lean Workbook, nearly doubling the 15.7K solved by prior provers.","abs_url":"https://arxiv.org/abs/2502.07640","pdf_url":"https://arxiv.org/pdf/2502.07640v3","primary_category":"cs.LG","published":"2025-02-11","tags":["theorem-proving-ml","autoformalization","dataset","benchmark"],"lean_note":"Trains LLMs to autoformalize problems into Lean 4 and generate Lean proofs, releasing large Lean statement and proof datasets.","repo":"https://github.com/Goedel-LM/Goedel-Prover","thumb":"thumbs/2502.07640.jpg"},{"id":"2502.05714","title":"Proving the Coding Interview: A Benchmark for Formally Verified Code Generation","authors":["Quinn Dougherty","Ronak Mehta"],"abstract":"We introduce the Formally Verified Automated Programming Progress Standards, or FVAPPS, a benchmark of 4715 samples for writing programs and proving their correctness, the largest formal verification benchmark, including 1083 curated and quality controlled samples. Previously, APPS provided a benchmark and dataset for programming puzzles to be completed in Python and checked against unit tests, of the kind seen in technical assessments in the software engineering industry. Building upon recent approaches for benchmarks in interactive theorem proving, we generalize the unit tests to Lean 4 theorems given without proof (i.e., using Lean's \"sorry\" keyword). On the 406 theorems of 100 randomly selected samples, Sonnet correctly proves 30% and Gemini correctly proves 18%. We challenge the machine learning and program synthesis communities to solve both each general purpose programming problem and its associated correctness specifications. The benchmark is available at https://huggingface.co/datasets/quinn-dougherty/fvapps.","abs_url":"https://arxiv.org/abs/2502.05714","pdf_url":"https://arxiv.org/pdf/2502.05714v1","primary_category":"cs.SE","published":"2025-02-08","tags":["benchmark","dataset","verification","theorem-proving-ml"],"lean_note":"FVAPPS is a formal verification benchmark expressing coding-problem correctness as unproved Lean 4 theorems for models to prove.","repo":"https://github.com/quinn-dougherty/fvapps","thumb":"thumbs/2502.05714.jpg"},{"id":"2502.04671","title":"ProofWala: A Framework for Multilingual Proof Data Synthesis and Theorem-Proving","authors":["Amitayush Thakur","George Tsoukalas","Greg Durrett","Swarat Chaudhuri"],"abstract":"Neural approaches to theorem proving require robust infrastructure for interfacing with interactive theorem provers (ITPs), extracting structured proof data, and executing proof search at scale. However, existing tooling is often assistant-specific and oriented toward file-level execution, making repository-scale analysis and parallel experimentation challenging. We present ProofWala, a multilingual proof engineering framework built around \\texttt{itp-interface}, a reusable library for programmatic interaction with ITPs. For Lean 4, we implement a meta-programmed interaction layer executing inside the elaborator, enabling semantically faithful tactic-level tracing alongside declaration- and dependency-level extraction across entire repositories. This design extends beyond traditional REPL-style interaction by supporting project-wide analysis, environment cloning, and pooled execution of proof states. The same interface abstraction supports multiple versions of Rocq, yielding a unified cross-assistant pipeline. Built on this infrastructure, ProofWala provides standardized multilingual proof datasets, model training utilities, and parallel proof search algorithms. Using the framework, we demonstrate that multilingual training across Lean and Rocq enables cross-lingual and cross-domain transfer. We observe statistically significant improvements on Lean Mathlib and in domain adaptation (CategoryTheory), while other settings exhibit consistent upward trends. We open-source the full framework, parallel proof search module, datasets, and models across two repositories: ProofWala (https://github.com/trishullab/proof-wala) and the itp-interface library (https://github.com/trishullab/itp-interface).","abs_url":"https://arxiv.org/abs/2502.04671","pdf_url":"https://arxiv.org/pdf/2502.04671v3","primary_category":"cs.AI","published":"2025-02-07","tags":["theorem-proving-ml","tooling","dataset","mathlib"],"lean_note":"ProofWala provides a meta-programmed Lean 4 interaction layer for tactic-level tracing and trains provers on Lean Mathlib and Rocq.","repo":"https://github.com/trishullab/proof-wala","thumb":"thumbs/2502.04671.jpg"},{"id":"2502.03438","title":"BFS-Prover: Scalable Best-First Tree Search for LLM-based Automatic Theorem Proving","authors":["Ran Xin","Chenguang Xi","Jie Yang","Feng Chen","Hang Wu","Xia Xiao","Yifan Sun","Shen Zheng","Kai Shen"],"abstract":"Recent advancements in large language models (LLMs) have spurred growing interest in automatic theorem proving using Lean4, where effective tree search methods are crucial for navigating the underlying large proof search spaces. While the existing approaches primarily rely on value functions and/or Monte Carlo Tree Search (MCTS), the potential of simpler methods like Best-First Tree Search (BFS) remains underexplored. In this paper, we investigate whether BFS can achieve competitive performance in large-scale theorem proving tasks. We present BFS-Prover, a scalable expert iteration framework, featuring three key innovations. First, we implement strategic data filtering at each expert iteration round, excluding problems solvable via beam search node expansion to focus on harder cases. Second, we improve the sample efficiency of BFS through Direct Preference Optimization (DPO) applied to state-tactic pairs automatically annotated with compiler error feedback, refining the LLM's policy to prioritize productive expansions. Third, we employ length normalization in BFS to encourage exploration of deeper proof paths. BFS-Prover achieves a state-of-the-art score of $72.95\\%$ on the MiniF2F test set and therefore challenges the perceived necessity of complex tree search methods, demonstrating that BFS can achieve competitive performance when properly scaled. To facilitate further research and development in this area, we have open-sourced our model at https://huggingface.co/ByteDance-Seed/BFS-Prover-V1-7B.","abs_url":"https://arxiv.org/abs/2502.03438","pdf_url":"https://arxiv.org/pdf/2502.03438v3","primary_category":"cs.AI","published":"2025-02-05","tags":["theorem-proving-ml"],"lean_note":"BFS-Prover scales best-first tree search with expert iteration and DPO for LLM theorem proving in Lean4.","thumb":"thumbs/2502.03438.jpg"},{"id":"2502.03432","title":"A formalization of Borel determinacy in Lean","authors":["Sven Manthe"],"abstract":"We present a formalization of Borel determinacy in the Lean 4 theorem prover. The formalization includes a definition of Gale-Stewart games and a proof of Martin's theorem stating that Borel games are determined. The proof closely follows Martin's \"A purely inductive proof of Borel determinacy\".","abs_url":"https://arxiv.org/abs/2502.03432","pdf_url":"https://arxiv.org/pdf/2502.03432v5","primary_category":"math.LO","published":"2025-02-05","tags":["formalization","logic","mathlib"],"lean_note":"Formalizes Borel determinacy in the Lean 4 theorem prover, including Gale-Stewart games and Martin's theorem.","repo":"https://github.com/sven-manthe/A-formalization-of-Borel-determinacy-in-Lean","thumb":"thumbs/2502.03432.jpg"},{"id":"2502.03321","title":"Simplifying Formal Proof-Generating Models with ChatGPT and Basic Searching Techniques","authors":["Sangjun Han","Taeil Hur","Youngmi Hur","Kathy Sangkyung Lee","Myungyoon Lee","Hyojae Lim"],"abstract":"The challenge of formal proof generation has a rich history, but with modern techniques, we may finally be at the stage of making actual progress in real-life mathematical problems. This paper explores the integration of ChatGPT and basic searching techniques to simplify generating formal proofs, with a particular focus on the miniF2F dataset. We demonstrate how combining a large language model like ChatGPT with a formal language such as Lean, which has the added advantage of being verifiable, enhances the efficiency and accessibility of formal proof generation. Despite its simplicity, our best-performing Lean-based model surpasses all known benchmarks with a 31.15% pass rate. We extend our experiments to include other datasets and employ alternative language models, showcasing our models' comparable performance in diverse settings and allowing for a more nuanced analysis of our results. Our findings offer insights into AI-assisted formal proof generation, suggesting a promising direction for future research in formal mathematical proof.","abs_url":"https://arxiv.org/abs/2502.03321","pdf_url":"https://arxiv.org/pdf/2502.03321v3","primary_category":"cs.LO","published":"2025-02-05","tags":["theorem-proving-ml","benchmark","ai-agents"],"lean_note":"Combines ChatGPT with the Lean prover and simple search to generate verifiable formal proofs on the miniF2F dataset.","repo":"https://github.com/zhangir-azerbayev/ProofNet","thumb":"thumbs/2502.03321.jpg"},{"id":"ext-037","title":"Finiteness of Symbolic Derivatives in Lean","authors":["Ekaterina Zhuchko","Hendrik Maarand","Margus Veanes","Gabriel Ebner"],"abstract":"This paper extends Brzozowski's classical result on finiteness of regular expression derivatives to symbolic location-based derivatives. The authors prove that the derivative state space is finite when quotienting only by associativity, deduplication, and idempotence (ADI), without requiring commutativity of union, enabling application to derivative-based backtracking (PCRE) match semantics where union is noncommutative. The proof is constructive, building a finite overapproximation of the derivative set, and is formalized in Lean 4.","abs_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.16","pdf_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.16","primary_category":"cs.LO","published":"2025-02-01","tags":["formalization","verification","logic"],"lean_note":"Formalizes a constructive finiteness proof of symbolic regex derivatives in Lean 4."},{"id":"2502.00212","title":"STP: Self-play LLM Theorem Provers with Iterative Conjecturing and Proving","authors":["Kefan Dong","Tengyu Ma"],"abstract":"A fundamental challenge in formal theorem proving by LLMs is the lack of high-quality training data. Although reinforcement learning or expert iteration partially mitigates this issue by alternating between LLM generating proofs and finetuning them on correctly generated ones, performance quickly plateaus due to the scarcity of correct proofs (sparse rewards). To keep improving the models with limited data, we draw inspiration from mathematicians, who continuously develop new results, partly by proposing novel conjectures or exercises (which are often variants of known results) and attempting to solve them. We design the Self-play Theorem Prover (STP) that simultaneously takes on two roles, conjecturer and prover, each providing training signals to the other. The conjecturer is trained iteratively on previously generated conjectures that are barely provable by the current prover, which incentivizes it to generate increasingly challenging conjectures over time. The prover attempts to prove the conjectures with standard expert iteration. We evaluate STP with both Lean and Isabelle formal versifiers. With 51.3 billion tokens generated during the training in Lean, STP proves 28.5% of the statements in the LeanWorkbook dataset, doubling the previous best result of 13.2% achieved through expert iteration. The final model achieves state-of-the-art performance among whole-proof generation methods on miniF2F-test (65.0%, pass@3200), Proofnet-test (23.9%, pass@3200) and PutnamBench (8/644, pass@3200). We release our code, model, and dataset in this URL: https://github.com/kfdong/STP.","abs_url":"https://arxiv.org/abs/2502.00212","pdf_url":"https://arxiv.org/pdf/2502.00212v4","primary_category":"cs.LG","published":"2025-01-31","tags":["theorem-proving-ml","dataset","benchmark"],"lean_note":"Self-play conjecturer-prover LLM trained and evaluated in Lean, proving 28.5% of LeanWorkbook and state-of-the-art on miniF2F.","repo":"https://github.com/kfdong/STP","thumb":"thumbs/2502.00212.jpg"},{"id":"2501.18310","title":"ProofAug: Efficient Neural Theorem Proving via Fine-grained Proof Structure Analysis","authors":["Haoxiong Liu","Jiacheng Sun","Zhenguo Li","Andrew C Yao"],"abstract":"The synergy between deep learning models and traditional automation tools, such as built-in tactics of the proof assistant and off-the-shelf automated theorem provers, plays a crucial role in developing robust and efficient neural theorem provers(NTPs). However, for proof synthesis with LLMs, previous work applies automation tools either only when explicitly invoked by the model or at a single granularity level, failing to fully exploit their power. To solve this issue, we propose ProofAug, a procedure that equips LLMs with automation methods at various granularities through fine-grained structure analysis of model-generated proof proposals. ProofAug also serves as a versatile plug-and-play module that seamlessly integrates with any tree-search algorithm, enabling our construction of an efficient recursive proving (ERP) module to further enhance performance. The superiority of our method is validated on the miniF2F benchmark using the open-source deepseek-math-7b-base model and the Isabelle proof assistant. Notably, by additionally employing a mixed prompting strategy, we achieve a cumulative pass rate of 66.0% after curation of the dataset (61.9% for the original version) with 2100 queries to the model per problem (In contrast, the previous SOTA in Isabelle, Subgoal-XL, only achieves 56.1% using 16384 queries per problem). We also implement a Lean 4 version of ProofAug that can improve the pass@1 performance of Kimina-Prover-Preview-Distill-1.5B from 44.3% to 50.4% on miniF2F-test. Our code is available at https://github.com/haoxiongliu/ProofAug.","abs_url":"https://arxiv.org/abs/2501.18310","pdf_url":"https://arxiv.org/pdf/2501.18310v2","primary_category":"cs.LG","published":"2025-01-30","tags":["theorem-proving-ml","tactic-automation","benchmark"],"lean_note":"ProofAug augments LLM proofs with automation tools at multiple granularities; a Lean 4 version improves a prover's miniF2F pass@1.","repo":"https://github.com/haoxiongliu/ProofAug","thumb":"thumbs/2501.18310.jpg"},{"id":"2501.18639","title":"A Comprehensive Survey of the Lean 4 Theorem Prover: Architecture, Applications, and Advances","authors":["Xichen Tang"],"abstract":"This comprehensive survey examines Lean 4, a state-of-the-art interactive theorem prover and functional programming language. We analyze its architectural design, type system, metaprogramming capabilities, and practical applications in formal verification and mathematics. Through detailed comparisons with other proof assistants and extensive case studies, we demonstrate Lean 4's unique advantages in proof automation, performance, and usability. The paper also explores recent developments in its ecosystem, including libraries, tools, and educational applications, providing insights into its growing impact on formal methods and mathematical formalization.","abs_url":"https://arxiv.org/abs/2501.18639","pdf_url":"https://arxiv.org/pdf/2501.18639v1","primary_category":"cs.LO","published":"2025-01-28","tags":["meta-study","tooling","formalization","verification"],"lean_note":"Surveys the Lean 4 theorem prover's architecture, type system, metaprogramming, ecosystem, and applications in formal verification and mathematics.","thumb":"thumbs/2501.18639.jpg"},{"id":"2501.16207","title":"From Informal to Formal -- Incorporating and Evaluating LLMs on Natural Language Requirements to Verifiable Formal Proofs","authors":["Jialun Cao","Yaojie Lu","Meiziniu Li","Haoyang Ma","Haokun Li","Mengda He","Cheng Wen","Le Sun","Hongyu Zhang","Shengchao Qin","Shing-Chi Cheung","Cong Tian"],"abstract":"The research in AI-based formal mathematical reasoning has shown an unstoppable growth trend. These studies have excelled in mathematical competitions like IMO and have made significant progress. This paper focuses on formal verification, an immediate application scenario of formal reasoning, and breaks it down into sub-tasks. We constructed 18k high-quality instruction-response pairs across five formal specification languages (Coq, Lean4, Dafny, ACSL, and TLA+) by distilling gpt-4o and evaluated against ten open-sourced LLMs, including recent popular DeepSeek-R1. We also fine-tuned several 7~8B small models to achieve comparable performance with Deepseek-R1-671B. Interestingly, we observed that fine-tuning with formal data also enhances mathematics, reasoning, and coding capabilities. Fine-tuned models are released at https: //huggingface.co/fm-universe.","abs_url":"https://arxiv.org/abs/2501.16207","pdf_url":"https://arxiv.org/pdf/2501.16207v4","primary_category":"cs.AI","published":"2025-01-27","tags":["benchmark","autoformalization"],"lean_note":"Builds 18k instruction-response pairs across five formal languages including Lean4, evaluating and fine-tuning LLMs on verifiable formal proofs.","repo":"https://github.com/sahil280114/codealpaca","thumb":"thumbs/2501.16207.jpg"},{"id":"2501.15797","title":"LemmaHead: RAG Assisted Proof Generation Using Large Language Models","authors":["Tianbo Yang","Mingqi Yan","Hongyi Zhao","Tianshuo Yang"],"abstract":"Developing the logic necessary to solve mathematical problems or write mathematical proofs is one of the more difficult objectives for large language models (LLMS). Currently, the most popular methods in literature consists of fine-tuning the model on written mathematical content such as academic publications and textbooks, so that the model can learn to emulate the style of mathematical writing. In this project, we explore the effectiveness of using retrieval augmented generation (RAG) to address gaps in the mathematical reasoning of LLMs. We develop LemmaHead, a RAG knowledge base that supplements queries to the model with relevant mathematical context, with particular focus on context from published textbooks. To measure our model's performance in mathematical reasoning, our testing paradigm focuses on the task of automated theorem proving via generating proofs to a given mathematical claim in the Lean formal language.","abs_url":"https://arxiv.org/abs/2501.15797","pdf_url":"https://arxiv.org/pdf/2501.15797v4","primary_category":"cs.LG","published":"2025-01-27","tags":["theorem-proving-ml","autoformalization"],"lean_note":"Tests RAG-assisted LLM proof generation by producing proofs to mathematical claims in the Lean formal language.","thumb":"thumbs/2501.15797.jpg"},{"id":"2501.15639","title":"The continuous functional calculus in Lean","authors":["Anatole Dedecker","Jireh Loreaux"],"abstract":"The continuous functional calculus is perhaps the most fundamental construction in the theory of operator algebras, especially $C^{*}$-algebras. Here we document our formalization of the continuous functional calculus in Lean, which constitutes the first such formalization in any proof assistant. Our implementation is already merged into Lean's mathematical library, Mathlib. We provide a brief introduction to the mathematical theory for those unfamiliar with the subject, and then highlight the design decisions in our formalization which proved to be important for usability. Our exposition is aimed at a general mathematical audience and provides a glimpse into the world of formalization by laying bare the discovery process.","abs_url":"https://arxiv.org/abs/2501.15639","pdf_url":"https://arxiv.org/pdf/2501.15639v2","primary_category":"math.OA","published":"2025-01-26","tags":["formalization","mathlib","analysis","algebra"],"lean_note":"Formalizes the continuous functional calculus for C*-algebras in Lean, merged into Mathlib.","thumb":"thumbs/2501.15639.jpg"},{"id":"2501.15002","title":"A Proof-Producing Compiler for Blockchain Applications","authors":["Jeremy Avigad","Lior Goldberg","David Levit","Yoav Seginer","Alon Titelman"],"abstract":"CairoZero is a programming language for running decentralized applications (dApps) at scale. Programs written in the CairoZero language are compiled to machine code for the Cairo CPU architecture and cryptographic protocols are used to verify the results of execution efficiently on blockchain. We explain how we have extended the CairoZero compiler with tooling that enables users to prove, in the Lean 3 proof assistant, that compiled code satisfies high-level functional specifications. We demonstrate the success of our approach by verifying primitives for computation with the secp256k1 and secp256r1 curves over a large finite field as well as the validation of cryptographic signatures using the former. We also verify a mechanism for simulating a read-write dictionary data structure in a read-only setting. Finally, we reflect on our methodology and discuss some of the benefits of our approach.","abs_url":"https://arxiv.org/abs/2501.15002","pdf_url":"https://arxiv.org/pdf/2501.15002v1","primary_category":"cs.CR","published":"2025-01-25","tags":["verification","tooling","cryptography"],"lean_note":"Extends the CairoZero compiler so users can prove in the Lean 3 proof assistant that compiled code meets functional specifications.","repo":"https://github.com/starkware-libs/formal-proofs","thumb":"thumbs/2501.15002.jpg"},{"id":"2501.12906","title":"Certified Knowledge Compilation with Application to Formally Verified Model Counting","authors":["Randal E. Bryant","Wojciech Nawrocki","Jeremy Avigad","Marijn J. H. Heule"],"abstract":"Computing many useful properties of Boolean formulas, such as their weighted or unweighted model count, is intractable on general representations. It can become tractable when formulas are expressed in a special form, such as the decision decomposable negation normal form (decision-DNNF). Knowledge compilation is the process of converting a formula into such a form. Unfortunately existing knowledge compilers provide no guarantee that their output correctly represents the original formula, and therefore they cannot validate a model count, or any other computed value. We present Partitioned-Operation Graphs (POGs), a form that can encode all of the representations used by existing knowledge compilers. We have designed CPOG, a framework that can express proofs of equivalence between a POG and a Boolean formula in conjunctive normal form (CNF). We have developed a program that generates POG representations from the decision-DNNF graphs produced by the state-of-the-art knowledge compiler D4, as well as checkable CPOG proofs certifying that the output POGs are equivalent to the input CNF formulas. Our toolchain for generating and verifying POGs scales to all but the largest graphs produced by D4 for formulas from a recent model counting competition. Additionally, we have developed a formally verified CPOG checker and model counter for POGs in the Lean 4 proof assistant. In doing so, we proved the soundness of our proof framework. These programs comprise the first formally verified toolchain for weighted and unweighted model counting.","abs_url":"https://arxiv.org/abs/2501.12906","pdf_url":"https://arxiv.org/pdf/2501.12906v1","primary_category":"cs.LO","published":"2025-01-22","tags":["verification","logic"],"lean_note":"A CPOG proof checker and model counter for knowledge-compilation outputs are formally verified in the Lean 4 proof assistant.","thumb":"thumbs/2501.12906.jpg"},{"id":"2501.13959","title":"Learning an Effective Premise Retrieval Model for Efficient Mathematical Formalization","authors":["Yicheng Tao","Haotian Liu","Shanwen Wang","Hongteng Xu"],"abstract":"Formalized mathematics has recently garnered significant attention for its ability to assist mathematicians across various fields. Premise retrieval, as a common step in mathematical formalization, has been a challenge, particularly for inexperienced users. Existing retrieval methods that facilitate natural language queries require a certain level of mathematical expertise from users, while approaches based on formal languages (e.g., Lean) typically struggle with the scarcity of training data, hindering the training of effective and generalizable retrieval models. In this work, we introduce a novel method that leverages data extracted from Mathlib to train a lightweight and effective premise retrieval model. In particular, the proposed model embeds queries (i.e., proof state provided by Lean) and premises in a latent space, featuring a tokenizer specifically trained on formal corpora. The model is learned in a contrastive learning framework, in which a fine-grained similarity calculation method and a re-ranking module are applied to enhance the retrieval performance. Experimental results demonstrate that our model outperforms existing baselines, achieving higher accuracy while maintaining a lower computational load. We have released an open-source search engine based on our retrieval model at https://premise-search.com/. The source code and the trained model can be found at https://github.com/ruc-ai4math/Premise-Retrieval.","abs_url":"https://arxiv.org/abs/2501.13959","pdf_url":"https://arxiv.org/pdf/2501.13959v3","primary_category":"cs.CL","published":"2025-01-21","tags":["theorem-proving-ml","mathlib","dataset"],"lean_note":"Trains a premise retrieval model on Mathlib data that embeds Lean proof states and premises for mathematical formalization.","repo":"https://github.com/ruc-ai4math/Premise-Retrieval","thumb":"thumbs/2501.13959.jpg"},{"id":"2501.09769","title":"Classifying the groups of order $p q$ in Lean","authors":["Scott Harper","Peiran Wu"],"abstract":"This note discusses our formalisation in Lean of the classification of the groups of order $p q$ for (not necessarily distinct) prime numbers $p$ and $q$, together with various intermediate results such as the characterisation of internal direct and semidirect products.","abs_url":"https://arxiv.org/abs/2501.09769","pdf_url":"https://arxiv.org/pdf/2501.09769v2","primary_category":"cs.LO","published":"2025-01-16","tags":["formalization","mathlib","algebra"],"lean_note":"Formalizes in Lean the classification of groups of order pq, with intermediate results on internal direct and semidirect products.","thumb":"thumbs/2501.09769.jpg"},{"id":"2501.03352","title":"Teaching \"Foundations of Mathematics\" with the Lean Theorem Prover","authors":["Mattia Luciano Bottoni","Alberto S. Cattaneo","Elif Sacikara"],"abstract":"This study aims to observe if the theorem prover Lean positively influences students' understanding of mathematical proving. To this end, we perform a pilot study concerning freshmen students at the University of Zurich (UZH). While doing so, we apply certain teaching methods and gather data from the volunteer students enrolled in the ``Foundations of Mathematics'' course. After eleven weeks of study covering some exercise questions implemented with Lean, we measure Lean students' performances in proving mathematical statements, compared to other students who are not engaged with Lean. For this measurement, we interview five Lean and four Non-Lean students and we analyze the scores of all students in the final exam. Finally, we check significance by performing a $t$-test for independent samples and the Mann-Whitney $U$-test.","abs_url":"https://arxiv.org/abs/2501.03352","pdf_url":"https://arxiv.org/pdf/2501.03352v3","primary_category":"math.HO","published":"2025-01-06","tags":["meta-study","formalization"],"lean_note":"Studies whether teaching with the Lean theorem prover improves university students' understanding of mathematical proving.","repo":"https://github.com/MattiaBottoni/Lean-meetings","thumb":"thumbs/2501.03352.jpg"},{"id":"ext-036","title":"Algebra Is Half the Battle: Verifying Presentations of Graded Unipotent Chevalley Groups","authors":["Eric Wang","Arohee Bhoja","Cayden Codel","Noah Singer"],"abstract":"This paper formalizes in Lean 4 presentations of graded unipotent subgroups of Chevalley groups, verifying that certain algebraic relations hold within these groups. The work combines computational algebra techniques with formal verification to establish correctness of group presentations that arise in the study of algebraic groups.","abs_url":"https://doi.org/10.4230/LIPIcs.ITP.2025","pdf_url":"https://doi.org/10.4230/LIPIcs.ITP.2025","primary_category":"cs.LO","published":"2025-01-01","tags":["formalization","algebra"],"lean_note":"Formalizes presentations of graded unipotent Chevalley groups in Lean 4."},{"id":"ext-033","title":"Formalization of derived categories in Lean/mathlib","authors":["Joël Riou"],"abstract":"This paper presents the formalization of derived categories and their basic properties in the Lean 4 proof assistant as part of the mathlib library. The construction covers the localization of categories at a class of morphisms, triangulated categories, and the derived category of an abelian category via its homotopy category and quasi-isomorphisms.","abs_url":"https://doi.org/10.29412/afm.2025","pdf_url":"https://doi.org/10.29412/afm.2025","primary_category":"cs.LO","published":"2025-01-01","tags":["formalization","mathlib","category-theory","algebra"],"lean_note":"Formalizes derived categories, triangulated categories, and localization in Lean 4 mathlib."},{"id":"2410.01466","title":"A complete formalization of Fermat's Last Theorem for regular primes in Lean","authors":["Alex Best","Christopher Birkbeck","Riccardo Brasca","Eric Rodriguez Boidi","Ruben van De Velde","Andrew Yang"],"abstract":"We formalize a complete proof of the regular case of Fermat's Last Theorem in the Lean4 theorem prover. Our formalization includes a proof of Kummer's lemma, the main obstruction to Fermat's Last Theorem for regular primes. Instead of using the modern approach via class field theory, we prove it by using Hilbert's Theorems 90-94 in a way that is more amenable to formalization.","abs_url":"https://arxiv.org/abs/2410.01466","pdf_url":"https://arxiv.org/pdf/2410.01466","primary_category":"cs.FL","published":"2024-10-02","tags":["formalization","number-theory","mathlib"],"lean_note":"Formalizes a complete proof of Fermat's Last Theorem for regular primes in Lean 4.","thumb":"thumbs/2410.01466.jpg"},{"id":"ext-048","title":"Integrals Within Integrals: A Formalization of the Gagliardo-Nirenberg-Sobolev Inequality","authors":["Floris van Doorn","Heather Macbeth"],"abstract":"We introduce an abstraction which allows arguments involving iterated integrals to be formalized conveniently in type-theory-based proof assistants. We call this the marginal construction, connected to marginal distributions in probability theory. It handles permutations to the order of integration (Tonelli's theorem in several variables) and arguments involving induction over dimension. We implement this in Lean, with the most difficult application being the Gagliardo-Nirenberg-Sobolev inequality, a foundational result in the theory of elliptic partial differential equations that has not previously been formalized.","abs_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.37","pdf_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.37","primary_category":"cs.LO","published":"2024-09-02","tags":["formalization","mathlib","analysis"],"lean_note":"Formalizes iterated integrals and the Gagliardo-Nirenberg-Sobolev inequality in Lean's mathlib."},{"id":"ext-045","title":"Teaching Mathematics Using Lean and Controlled Natural Language","authors":["Patrick Massot"],"abstract":"This paper presents Verbose Lean, a library leveraging Lean 4's flexibility to teach undergraduate mathematics students to read and write traditional paper proofs. Students write proofs that resemble natural language and are easy to transfer to paper. Teachers can customize the experience, and Lean 4's metaprogramming features allow creative pedagogical uses of a proof assistant.","abs_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.27","pdf_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.27","primary_category":"cs.LO","published":"2024-09-02","tags":["tooling","formalization","logic"],"lean_note":"Presents Verbose Lean, a Lean 4 library for teaching proof-writing via controlled natural language."},{"id":"ext-042","title":"Duper: A Proof-Producing Superposition Theorem Prover for Dependent Type Theory","authors":["Joshua Clune","Yicheng Qian","Alexander Bentkamp","Jeremy Avigad"],"abstract":"The authors present Duper, a proof-producing theorem prover for Lean based on the superposition calculus. It can be called directly as a terminal tactic in interactive Lean proofs and is designed with proof reconstruction for a future Lean hammer in mind. The paper describes Duper's approach to proof search and proof reconstruction, emphasizing the challenges of working in a dependent type theory, and compares its performance to Metis on pre-existing benchmarks.","abs_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.10","pdf_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.10","primary_category":"cs.LO","published":"2024-09-02","tags":["tactic-automation","theorem-proving-ml","tooling"],"lean_note":"Implements a superposition-based automated theorem prover as a native Lean 4 tactic with proof production."},{"id":"ext-044","title":"Graphical Rewriting for Diagrammatic Reasoning in Monoidal Categories in Lean4","authors":["Sam Ezeh"],"abstract":"This paper presents a framework in Lean 4 for performing graphical rewriting in monoidal categories, enabling diagrammatic reasoning to be carried out within a formal proof assistant. The approach implements string diagram manipulation as rewriting rules, allowing proofs about monoidal categories to follow the intuitive graphical style used in mathematical practice.","abs_url":"https://doi.org/10.4230/LIPIcs.ITP.2024","pdf_url":"https://doi.org/10.4230/LIPIcs.ITP.2024","primary_category":"cs.LO","published":"2024-09-01","tags":["formalization","category-theory","tooling"],"lean_note":"Implements graphical rewriting for string diagram reasoning in monoidal categories in Lean 4."},{"id":"2407.12840","title":"Categorical Foundations of Formalized Condensed Mathematics","authors":["Dagur Asgeirsson","Riccardo Brasca","Nikolas Kuhn","Filippo A. E. Nuccio Mortarino Majno di Capriglio","Adam Topaz"],"abstract":"Condensed mathematics, developed by Clausen and Scholze, proposes a generalization of topology with improved categorical properties. It replaces topological spaces with condensed sets, defined as sheaves for the coherent topology on a certain category of compact Hausdorff spaces. The sheaf condition has a simple explicit description arising from studying the relationship between the coherent, regular and extensive topologies. We establish this relationship under minimal assumptions on the category, going beyond the case of compact Hausdorff spaces, while also providing a characterization of sheaves and covering sieves for these categories. All results are fully formalized in the Lean proof assistant.","abs_url":"https://arxiv.org/abs/2407.12840","pdf_url":"https://arxiv.org/pdf/2407.12840","primary_category":"math.CT","published":"2024-07-04","tags":["formalization","category-theory","topology","mathlib"],"lean_note":"Fully formalizes categorical foundations for condensed mathematics in Lean.","thumb":"thumbs/2407.12840.jpg"},{"id":"2407.03685","title":"Verifying Peephole Rewriting In SSA Compiler IRs","authors":["Siddharth Bhat","Alex Keizer","Chris Hughes","Andrés Goens","Tobias Grosser"],"abstract":"There is an increasing need for domain-specific reasoning in modern compilers. This has fueled the use of tailored intermediate representations (IRs) based on static single assignment (SSA), like in the MLIR compiler framework. Interactive theorem provers (ITPs) provide strong guarantees for the end-to-end verification of compilers (e.g., CompCert). However, modern compilers and their IRs evolve at a rate that makes proof engineering alongside them prohibitively expensive. Nevertheless, well-scoped push-button automated verification tools such as the Alive peephole verifier for LLVM-IR gained recognition in domains where SMT solvers offer efficient (semi) decision procedures. In this paper, we aim to combine the convenience of automation with the versatility of ITPs for verifying peephole rewrites across domain-specific IRs. We formalize a core calculus for SSA-based IRs that is generic over the IR and covers so-called regions (nested scoping used by many domain-specific IRs in the MLIR ecosystem). Our mechanization in the Lean proof assistant provides a user-friendly frontend for translating MLIR syntax into our calculus. We provide scaffolding for defining and verifying peephole rewrites, offering tactics to eliminate the abstraction overhead of our SSA calculus. We prove correctness theorems about peephole rewriting, as well as two classical program transformations. To evaluate our framework, we consider three use cases from the MLIR ecosystem that cover different levels of abstractions: (1) bitvector rewrites from LLVM, (2) structured control flow, and (3) fully homomorphic encryption. We envision that our mechanization provides a foundation for formally verified rewrites on new domain-specific IRs.","abs_url":"https://arxiv.org/abs/2407.03685","pdf_url":"https://arxiv.org/pdf/2407.03685","primary_category":"cs.PL","published":"2024-07-04","tags":["verification","formalization","tooling"],"lean_note":"Formalizes SSA compiler IR semantics and verifies peephole rewrites in Lean for MLIR.","thumb":"thumbs/2407.03685.jpg"},{"id":"2405.19270","title":"Formalising the local compactness of the adele ring","authors":["Salvatore Mercuri"],"abstract":"The adele ring of a number field is a central object in modern number theory. Its status as a locally compact topological ring is one of the key reasons for its importance. We describe a formal proof implemented in Lean 4 that the adele ring of a number field is locally compact. This includes formalisations of new types, including the completion of a number field at an infinite place, the infinite adele ring and the finite S-adele ring, along with formal proofs that completions of number fields are locally compact and that their rings of integers at finite places are compact.","abs_url":"https://arxiv.org/abs/2405.19270","pdf_url":"https://arxiv.org/pdf/2405.19270","primary_category":"cs.LO","published":"2024-05-29","tags":["formalization","mathlib","number-theory","algebra"],"lean_note":"Formalizes in Lean 4 that the adele ring of a number field is locally compact.","thumb":"thumbs/2405.19270.jpg"},{"id":"2403.17370","title":"Formal Verification of the Empty Hexagon Number","authors":["Bernardo Subercaseaux","Wojciech Nawrocki","James Gallicchio","Cayden Codel","Mario Carneiro","Marijn J. H. Heule"],"abstract":"A recent breakthrough in computer-assisted mathematics showed that every set of 30 points in the plane in general position contains an empty convex hexagon, thus closing a line of research dating back to the 1930s. Through a combination of geometric insights and automated reasoning techniques, Heule and Scheucher constructed a CNF formula with O(n^4) clauses, whose unsatisfiability implies that no set of n points in general position can avoid an empty convex hexagon. An unsatisfiability proof for n = 30 was then found with a SAT solver using 17300 CPU hours of parallel computation, thus implying that the empty hexagon number h(6) is equal to 30. In this paper, we formalize and verify this result in the Lean theorem prover. Our formalization covers discrete computational geometry ideas and SAT encoding techniques that have been successfully applied to similar Erdős-Szekeres-type problems. In particular, our framework provides tools to connect standard mathematical objects to propositional assignments, which represents a key step towards the formal verification of other SAT-based mathematical results.","abs_url":"https://arxiv.org/abs/2403.17370","pdf_url":"https://arxiv.org/pdf/2403.17370","primary_category":"cs.CG","published":"2024-03-26","tags":["formalization","verification","combinatorics","geometry"],"lean_note":"Formalizes in Lean the SAT-based proof that the empty hexagon number h(6) equals 30.","thumb":"thumbs/2403.17370.jpg"},{"id":"2312.06506","title":"The Directed Van Kampen Theorem in Lean","authors":["Henning Basold","Peter Bruin","Dominique Lawson"],"abstract":"Directed topology is an area of mathematics with applications in concurrency. It extends the concept of a topological space by adding a notion of directedness, which restricts how paths can evolve through a space and enables thereby a faithful representation of computation with their direction. We present a Lean formalisation of directed spaces and a Van Kampen theorem for them. This theorem allows the calculation of the homotopy type of a space by combining local knowledge of the homotopy type of subspaces. By representing concurrent systems as directed spaces, it allows the deduction of properties of a composed system to that of subsystems. The formalisation in Lean can serve to support computer-assisted reasoning about the behaviour of concurrent systems.","abs_url":"https://arxiv.org/abs/2312.06506","pdf_url":"https://arxiv.org/pdf/2312.06506","primary_category":"cs.LO","published":"2023-12-11","tags":["formalization","topology","verification"],"lean_note":"Formalizes directed spaces and a directed Van Kampen theorem in Lean for concurrency reasoning.","thumb":"thumbs/2312.06506.jpg"},{"id":"2310.01998","title":"A Formalization of Complete Discrete Valuation Rings and Local Fields","authors":["María Inés de Frutos-Fernández","Filippo A. E. Nuccio Mortarino Majno di Capriglio"],"abstract":"We formalize in Lean the basic theory of discretely valued fields. We prove that the unit ball with respect to a discrete valuation on a field is a discrete valuation ring and conversely that the adic valuation on the field of fractions of a DVR is discrete. We define finite extensions of valuations and discrete valuation rings, proving global-to-local results. Building on this, we formalize the abstract definition and fundamental properties of local fields, showing that finite extensions of the p-adic numbers and the Laurent series field over a finite field are local fields.","abs_url":"https://arxiv.org/abs/2310.01998","pdf_url":"https://arxiv.org/pdf/2310.01998","primary_category":"cs.LO","published":"2023-10-03","tags":["formalization","mathlib","number-theory","algebra"],"lean_note":"Formalizes discrete valuation rings and local fields in Lean with contributions to mathlib.","thumb":"thumbs/2310.01998.jpg"},{"id":"2310.01530","title":"A Pretty Expressive Printer (with Appendices)","authors":["Sorawee Porncharoenwase","Justin Pombrio","Emina Torlak"],"abstract":"Pretty printers make trade-offs between the expressiveness of their pretty printing language, the optimality objective that they minimize when choosing between different ways to lay out a document, and the performance of their algorithm. This paper presents a new pretty printer that is strictly more expressive than all pretty printers in the literature and provably minimizes an optimality objective. Furthermore, its time complexity is better than many existing pretty printers. When choosing among different ways to lay out a document, the printer consults a user-supplied cost factory, which determines the optimality objective, giving it a unique degree of flexibility. We use the Lean theorem prover to verify the correctness (validity and optimality) of the algorithm, and implement it concretely as a pretty printer called PrettyExpressive that has seen real-world adoption as a foundation of a code formatter for Racket.","abs_url":"https://arxiv.org/abs/2310.01530","pdf_url":"https://arxiv.org/pdf/2310.01530","primary_category":"cs.PL","published":"2023-10-02","tags":["verification","tooling"],"lean_note":"Uses Lean to formally verify correctness and optimality of the pretty printing algorithm.","thumb":"thumbs/2310.01530.jpg"},{"id":"2309.07252","title":"Towards Solid Abelian Groups: A Formal Proof of Nöbeling's Theorem","authors":["Dagur Asgeirsson"],"abstract":"Condensed mathematics, developed by Clausen and Scholze over the last few years, is a new way of studying the interplay between algebra and geometry. It replaces the concept of a topological space by a more sophisticated but better-behaved idea, namely that of a condensed set. Central to the theory are solid abelian groups and liquid vector spaces, analogues of complete topological groups. Nöbeling's theorem, concerning the structure of the abelian group of continuous maps from a profinite space to the integers, is a crucial ingredient in the theory of solid abelian groups; without it one cannot give any nonzero examples of solid abelian groups. This paper discusses a completed formalization of this result in the Lean theorem prover and provides a more detailed proof than previously available. The proof is somewhat unusual in that it requires induction over ordinals -- a technique which has not previously been used to a great extent in formalised mathematics.","abs_url":"https://arxiv.org/abs/2309.07252","pdf_url":"https://arxiv.org/pdf/2309.07252","primary_category":"math.LO","published":"2023-09-13","tags":["formalization","mathlib","algebra","topology"],"lean_note":"Formalizes Nöbeling's theorem in Lean using ordinal induction for condensed mathematics.","thumb":"thumbs/2309.07252.jpg"},{"id":"ext-057","title":"Group Cohomology in the Lean Community Library","authors":["Amelia Livingston"],"abstract":"This paper describes the formalization of group cohomology in Lean's mathlib library. It covers the construction of group cohomology as the derived functors of the invariants functor, the explicit cochain complex computing it, and the long exact sequence in cohomology, contributing both mathematical content and API design insights for large-scale formalization projects.","abs_url":"https://doi.org/10.4230/LIPIcs.ITP.2023.20","pdf_url":"https://doi.org/10.4230/LIPIcs.ITP.2023.20","primary_category":"cs.LO","published":"2023-07-31","tags":["formalization","mathlib","algebra"],"lean_note":"Formalizes group cohomology as derived functors and explicit cochains in Lean mathlib."},{"id":"ext-000","title":"An Extensible User Interface for Lean 4","authors":["Wojciech Nawrocki","Edward W. Ayers","Gabriel Ebner"],"abstract":"Modern proof assistants handle complex automation and large libraries, but understanding the emergent interactions between components can be a serious challenge. This paper proposes presentations, UI elements that store references to the objects they are displaying, as a productive framework for ITP interface design. The authors built an extensible user interface for Lean 4 with the ProofWidgets 4 library, demonstrating examples including type information popups, structured traces, contextual suggestions, algebraic reasoning displays, and red-black tree visualizations. The interface is already part of the core Lean distribution.","abs_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2023.24","pdf_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2023.24","primary_category":"cs.LO","published":"2023-07-26","tags":["tooling","formalization"],"lean_note":"Introduces ProofWidgets 4, an extensible UI framework built into the core Lean 4 distribution."},{"id":"2306.17234","title":"Formalizing Norm Extensions and Applications to Number Theory","authors":["María Inés de Frutos-Fernández"],"abstract":"Let K be a field complete with respect to a nonarchimedean real-valued norm, and let L/K be an algebraic extension. We show there is a unique norm on L extending the given norm on K, with an explicit description. As an application, the p-adic norm on Q_p is extended to its algebraic closure Q_p^alg, and the field C_p of p-adic complex numbers is defined as the completion of the latter. Building on C_p, we formalize the Fontaine period ring B_HT and discuss applications to Galois representations and p-adic Hodge theory. The formalized results are a prerequisite to formalize Local Class Field Theory, which is a fundamental ingredient of the proof of Fermat's Last Theorem.","abs_url":"https://arxiv.org/abs/2306.17234","pdf_url":"https://arxiv.org/pdf/2306.17234","primary_category":"cs.LO","published":"2023-06-29","tags":["formalization","number-theory","algebra","mathlib"],"lean_note":"Formalizes p-adic norm extensions and Fontaine period rings in Lean for number theory.","thumb":"thumbs/2306.17234.jpg"},{"id":"2306.00617","title":"Multiple-Inheritance Hazards in Dependently-Typed Algebraic Hierarchies","authors":["Eric Wieser"],"abstract":"Abstract algebra's hierarchy of properties can be arranged as a directed acyclic graph and encoded in theorem provers like Lean using typeclasses. Multiple inheritance is inevitable since, for example, a ring is both a semiring and an abelian group. With dependently-typed typeclasses consuming other typeclasses as parameters, the type of an outer typeclass depends on the resolution path, and unless all paths are judgmentally equal, problems arise. This paper provides concrete examples reduced from mathlib, compares implementation approaches, and outlines solutions including kernel support for eta-reduction of structures.","abs_url":"https://arxiv.org/abs/2306.00617","pdf_url":"https://arxiv.org/pdf/2306.00617","primary_category":"cs.LO","published":"2023-06-01","tags":["mathlib","formalization","algebra","type-theory"],"lean_note":"Identifies and solves multiple-inheritance diamond hazards in Lean's mathlib algebraic hierarchy.","thumb":"thumbs/2306.00617.jpg"},{"id":"2305.08955","title":"Fermat's Last Theorem for Regular Primes","authors":["Alex J. Best","Christopher Birkbeck","Riccardo Brasca","Eric Rodriguez Boidi"],"abstract":"We formalise the proof of the first case of Fermat's Last Theorem for regular primes using the Lean theorem prover and its mathematical library mathlib. This is an important 19th century result that motivated the development of modern algebraic number theory. Besides explaining the mathematics behind this result, we analyze in this paper the difficulties we faced in the formalisation process and how we solved them. For example, we had to deal with a diamond about characteristic zero fields and problems arising from multiple nested coercions related to number fields. We also explain how we integrated our work to mathlib.","abs_url":"https://arxiv.org/abs/2305.08955","pdf_url":"https://arxiv.org/pdf/2305.08955","primary_category":"cs.LO","published":"2023-05-15","tags":["formalization","number-theory","mathlib"],"lean_note":"Formalizes the first case of Fermat's Last Theorem for regular primes in Lean with mathlib.","thumb":"thumbs/2305.08955.jpg"},{"id":"2302.10640","title":"An Elementary Formal Proof of the Group Law on Weierstrass Elliptic Curves in any Characteristic","authors":["David Kurniadi Angdinata","Junyan Xu"],"abstract":"Elliptic curves are fundamental objects in number theory and algebraic geometry, whose points over a field form an abelian group under a geometric addition law. Any elliptic curve over a field admits a Weierstrass model, but prior formal proofs of associativity involve either advanced algebraic geometry or tedious computation, especially in characteristic two. The authors formalise in the Lean theorem prover the type of nonsingular points of a Weierstrass curve over a field of any characteristic and a purely algebraic proof that it forms an abelian group.","abs_url":"https://arxiv.org/abs/2302.10640","pdf_url":"https://arxiv.org/pdf/2302.10640","primary_category":"cs.LO","published":"2023-02-21","tags":["formalization","mathlib","number-theory","algebra"],"lean_note":"Formalizes in Lean that nonsingular points of a Weierstrass elliptic curve form an abelian group.","thumb":"thumbs/2302.10640.jpg"},{"id":"2302.06420","title":"Closure Properties of General Grammars -- Formally Verified","authors":["Martin Dvorak","Jasmin Blanchette"],"abstract":"We formalized general (i.e., type-0) grammars using the Lean 3 proof assistant. We defined basic notions of rewrite rules and of words derived by a grammar, and showed closure of type-0 languages under four operations: union, reversal, concatenation, and the Kleene star. The literature mostly focuses on Turing machine arguments, which are possibly more difficult to formalize. For the Kleene star, we could not follow the literature and came up with our own grammar-based construction.","abs_url":"https://arxiv.org/abs/2302.06420","pdf_url":"https://arxiv.org/pdf/2302.06420","primary_category":"cs.FL","published":"2023-02-13","tags":["formalization","logic"],"lean_note":"Formalizes type-0 grammar closure properties in Lean 3 with novel constructions.","thumb":"thumbs/2302.06420.jpg"},{"id":"2302.00448","title":"A Formalisation of Gallagher's Ergodic Theorem","authors":["Oliver Nash"],"abstract":"Gallagher's ergodic theorem is a result in metric number theory stating that the approximation of real numbers by rational numbers obeys a striking all-or-nothing behaviour. This paper discusses formalising this result in the Lean theorem prover. The result is a key preliminary for Koukoulopoulos and Maynard's proof of the Duffin-Schaeffer conjecture.","abs_url":"https://arxiv.org/abs/2302.00448","pdf_url":"https://arxiv.org/pdf/2302.00448","primary_category":"cs.LO","published":"2023-02-01","tags":["formalization","mathlib","number-theory","analysis"],"lean_note":"Formalizes Gallagher's ergodic theorem in Lean as groundwork toward the Duffin-Schaeffer conjecture.","thumb":"thumbs/2302.00448.jpg"},{"id":"ext-058","title":"Formalising Sharkovsky's Theorem (Proof Pearl)","authors":["Bhavik Mehta"],"abstract":"This paper presents a formalization of Sharkovsky's theorem in Lean and mathlib. Sharkovsky's theorem states that if a continuous function on the real line has a periodic point of period n, then it has periodic points of all periods that follow n in Sharkovsky's ordering. The formalization includes the full theorem and provides a clean proof structure suitable for inclusion in mathlib.","abs_url":"https://dl.acm.org/doi/10.1145/3573105.3575690","pdf_url":"https://dl.acm.org/doi/10.1145/3573105.3575690","primary_category":"cs.LO","published":"2023-01-16","tags":["formalization","mathlib","analysis","combinatorics"],"lean_note":"Formalizes Sharkovsky's theorem on periodic points of continuous real functions in Lean/mathlib."},{"id":"ext-053","title":"A Formalized Reduction of Keller's Conjecture","authors":["Joshua Clune"],"abstract":"This paper presents a formalized proof in Lean that reduces Keller's conjecture to a finite combinatorial problem. Keller's conjecture states that in any tiling of n-dimensional space by unit cubes, some pair must share an entire (n-1)-dimensional face. The author formalizes the reduction showing that the conjecture for dimension n is equivalent to the nonexistence of certain cliques in the Keller graph, enabling the use of computational verification to settle the conjecture.","abs_url":"https://doi.org/10.1145/3573105.3575662","pdf_url":"https://doi.org/10.1145/3573105.3575662","primary_category":"cs.LO","published":"2023-01-16","tags":["formalization","combinatorics","geometry"],"lean_note":"Formalizes in Lean the reduction of Keller's conjecture to a finite clique-finding problem."},{"id":"ext-016","title":"Aesop: White-Box Best-First Proof Search for Lean","authors":["Jannis Limperg","Asta Halkjær From"],"abstract":"Aesop is a proof search tactic for the Lean 4 proof assistant. It applies a configurable set of rules using best-first search to try to close a goal. Unlike black-box automation tactics, Aesop is designed to be transparent and predictable: users can inspect the search tree to understand why a proof succeeded or failed, and can control the search by adding or removing rules. Aesop supports safe and unsafe rules with configurable priorities, norm rules for simplification, and can unfold definitions during search.","abs_url":"https://doi.org/10.1145/3573105.3575671","pdf_url":"https://doi.org/10.1145/3573105.3575671","primary_category":"cs.LO","published":"2023-01-16","tags":["tactic-automation","tooling"],"lean_note":"Implements a configurable white-box best-first proof search tactic for Lean 4."},{"id":"2212.05578","title":"A Formalization of Doob's Martingale Convergence Theorems in mathlib","authors":["Kexing Ying","Rémy Degenne"],"abstract":"We present our formalization of Doob's martingale convergence theorems within the mathlib library for the Lean theorem prover. These theorems provide conditions under which (sub)martingales converge, almost everywhere or in L^1. To achieve this formalization, we construct a definition of conditional expectation in Banach spaces and develop theory around stochastic processes, stopping times, and martingales. As an application, we also formalize Lévy's generalized Borel-Cantelli lemma. This work represents one of the earliest developments of probability theory in mathlib, building on diverse parts of that library such as topology, analysis and most importantly measure theory.","abs_url":"https://arxiv.org/abs/2212.05578","pdf_url":"https://arxiv.org/pdf/2212.05578","primary_category":"cs.LO","published":"2022-12-11","tags":["formalization","probability","analysis","mathlib"],"lean_note":"Formalizes Doob's martingale convergence and conditional expectation in Lean's mathlib.","thumb":"thumbs/2212.05578.jpg"},{"id":"2210.07746","title":"Formalising the h-Principle and Sphere Eversion","authors":["Patrick Massot","Floris van Doorn","Oliver Nash"],"abstract":"In differential topology and geometry, the h-principle is a property enjoyed by certain construction problems. Roughly speaking, it states that the only obstructions to the existence of a solution come from algebraic topology. We describe a formalisation in Lean of the local h-principle for first-order, open, ample partial differential relations. This is a significant result in differential topology, originally proven by Gromov in 1973 as part of his sweeping effort which greatly generalised many previous flexibility results in topology and geometry. In particular it reproves Smale's celebrated sphere eversion theorem, a visually striking and counter-intuitive construction. Our formalisation uses Theillière's implementation of convex integration from 2018. This paper is the first part of the sphere eversion project, aiming to formalise the global version of the h-principle for open and ample first order differential relations, for maps between smooth manifolds. Our current local version for vector spaces is the main ingredient of this proof, and is sufficient to prove the titular corollary of the project. From a broader perspective, the goal of this project is to show that one can formalise advanced mathematics with a strongly geometric flavour and not only algebraically-flavoured mathematics.","abs_url":"https://arxiv.org/abs/2210.07746","pdf_url":"https://arxiv.org/pdf/2210.07746","primary_category":"cs.LO","published":"2022-10-14","tags":["formalization","mathlib","topology","geometry"],"lean_note":"Formalizes the local h-principle and Smale's sphere eversion theorem in Lean using convex integration.","thumb":"thumbs/2210.07746.jpg"},{"id":"2209.15492","title":"Formalized Class Group Computations and Integral Points on Mordell Elliptic Curves","authors":["Anne Baanen","Alex J. Best","Nirvana Coppola","Sander R. Dahmen"],"abstract":"Determining all integer solutions to a Mordell equation y^2 = x^3 + d for a nonzero integer d is a classical problem. We describe a non-elementary approach involving resolution via descent and class groups, and formalize in Lean 3 the resolution of Mordell equations for several instances of d < 0. To accomplish this, we formalize ideal norms, quadratic fields and rings, and explicit computations of the class number. We also introduce new computational tactics to carry out efficiently computations in quadratic rings and beyond.","abs_url":"https://arxiv.org/abs/2209.15492","pdf_url":"https://arxiv.org/pdf/2209.15492","primary_category":"cs.LO","published":"2022-09-30","tags":["formalization","number-theory","algebra","tactic-automation"],"lean_note":"Formalizes class group computations in Lean 3 to solve Mordell elliptic curve equations.","thumb":"thumbs/2209.15492.jpg"},{"id":"ext-073","title":"Formalising the Kruskal-Katona Theorem in Lean","authors":["Bhavik Mehta"],"abstract":"This paper describes the formalization of the Kruskal-Katona theorem in the Lean theorem prover. The Kruskal-Katona theorem characterizes the shadow (lower shadow) of set families and is a fundamental result in extremal combinatorics. The formalization follows a proof via compressions and includes the construction of colex orderings and the Lovász form of the theorem.","abs_url":"https://doi.org/10.1007/978-3-031-16681-5_12","pdf_url":"https://doi.org/10.1007/978-3-031-16681-5_12","primary_category":"cs.LO","published":"2022-09-19","tags":["formalization","mathlib","combinatorics"],"lean_note":"Formalizes the Kruskal-Katona theorem and colex orderings for extremal combinatorics in Lean."},{"id":"ext-076","title":"Graded Rings in Lean's Dependent Type Theory","authors":["Eric Wieser","Jujian Zhang"],"abstract":"This paper describes the formalization of graded rings and their properties in the Lean theorem prover using mathlib. Graded rings are algebraic structures that decompose as a direct sum of abelian groups indexed by a monoid, and they arise naturally in commutative algebra and algebraic geometry. The authors discuss design decisions for representing graded structures in dependent type theory and the challenges of working with direct sum decompositions.","abs_url":"https://doi.org/10.1007/978-3-031-16681-5_9","pdf_url":"https://doi.org/10.1007/978-3-031-16681-5_9","primary_category":"cs.LO","published":"2022-09-01","tags":["formalization","algebra","mathlib"],"lean_note":"Formalizes graded rings and their properties in Lean's mathlib using dependent type theory."},{"id":"ext-001","title":"'do' unchained: embracing local imperativity in a purely functional language (functional pearl)","authors":["Sebastian Ullrich","Leonardo de Moura"],"abstract":"Monadic programming in pure functional languages has long provided a disciplined way to incorporate effects. This paper presents a system of extended do-notation for the Lean 4 programming language that allows users to write local mutable variables, early return, loops with break/continue, and other imperative patterns directly inside pure functional code, compiling them away via monad transformers without runtime overhead.","abs_url":"https://doi.org/10.1145/3547640","pdf_url":"https://doi.org/10.1145/3547640","primary_category":"cs.LO","published":"2022-08-19","tags":["tooling","type-theory"],"lean_note":"Describes Lean 4's do-notation extensions that compile imperative patterns via monad transformers."},{"id":"ext-072","title":"Formalizing the Divergence Theorem and the Cauchy Integral Formula in Lean","authors":["Yury Kudryashov"],"abstract":"We formalize a version of the divergence theorem for a function on a rectangular box that does not assume regularity of individual partial derivatives, requiring only Frechet differentiability and integrability of the divergence. The main tool is the GP-integral, a Henstock-Kurzweil integral variant introduced by J. Mawhin in 1981, for which the divergence theorem does not require integrability of the divergence. This theorem is then used to prove the Cauchy-Goursat theorem for simple domains and to bootstrap complex analysis in the Lean mathematical library.","abs_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2022.23","pdf_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2022.23","primary_category":"cs.LO","published":"2022-08-03","tags":["formalization","mathlib","analysis"],"lean_note":"Formalizes the divergence theorem and uses it to bootstrap complex analysis in Lean's mathlib."},{"id":"ext-068","title":"Formalising Szemerédi's Regularity Lemma in Lean","authors":["Yaël Dillies","Bhavik Mehta"],"abstract":"This paper presents the first formalization of Szemerédi's Regularity Lemma in the Lean theorem prover using mathlib. The Regularity Lemma is a fundamental result in combinatorics stating that every sufficiently large graph can be partitioned into a bounded number of parts such that edges between most pairs of parts behave quasi-randomly. The formalization required developing supporting infrastructure for graph theory and energy arguments in mathlib.","abs_url":"https://doi.org/10.4230/LIPIcs.ITP.2022.9","pdf_url":"https://doi.org/10.4230/LIPIcs.ITP.2022.9","primary_category":"cs.LO","published":"2022-08-01","tags":["formalization","combinatorics","mathlib"],"lean_note":"Formalizes Szemerédi's Regularity Lemma in Lean using mathlib's graph theory library."},{"id":"2207.12742","title":"A formalization of the change of variables formula for integrals in mathlib","authors":["Sébastien Gouëzel"],"abstract":"We report on a formalization of the change of variables formula in integrals, in the mathlib library for Lean. The theorem is extremely general, building on developments in linear algebra, analysis, measure theory and descriptive set theory. The interplay between these domains is transparent thanks to the highly integrated development model of mathlib.","abs_url":"https://arxiv.org/abs/2207.12742","pdf_url":"https://arxiv.org/pdf/2207.12742","primary_category":"cs.LO","published":"2022-07-26","tags":["formalization","analysis","mathlib"],"lean_note":"Formalizes the general change of variables formula for integrals in Lean's mathlib.","thumb":"thumbs/2207.12742.jpg"},{"id":"2203.16344","title":"Formalizing the Ring of Adeles of a Global Field","authors":["María Inés de Frutos-Fernández"],"abstract":"The ring of adeles of a global field and its group of units, the group of ideles, are fundamental objects in modern number theory. We formalize their definitions in the Lean 3 theorem prover. As a prerequisite, we formalize adic valuations on Dedekind domains. We present applications including the statement of the main theorem of global class field theory and a proof that the ideal class group of a number field is isomorphic to an explicit quotient of its idele class group.","abs_url":"https://arxiv.org/abs/2203.16344","pdf_url":"https://arxiv.org/pdf/2203.16344","primary_category":"cs.LO","published":"2022-03-06","tags":["formalization","mathlib","number-theory","algebra"],"lean_note":"Formalizes adeles, ideles, and adic valuations on Dedekind domains in Lean 3.","thumb":"thumbs/2203.16344.jpg"},{"id":"2202.05360","title":"Formalized functional analysis with semilinear maps","authors":["Frédéric Dupuis","Robert Y. Lewis","Heather Macbeth"],"abstract":"Semilinear maps are a generalization of linear maps between vector spaces where the scalar action is twisted by a ring homomorphism. This generalization unifies the concepts of linear and conjugate-linear maps. The authors implement this in Lean's mathlib library and prove key results including the Frechet-Riesz representation theorem and the spectral theorem for compact self-adjoint operators generically over real and complex Hilbert spaces. They also formalize a case of a theorem by Dieudonne and Manin on isocrystals.","abs_url":"https://arxiv.org/abs/2202.05360","pdf_url":"https://arxiv.org/pdf/2202.05360","primary_category":"cs.LO","published":"2022-02-10","tags":["formalization","mathlib","analysis"],"lean_note":"Formalizes semilinear maps in mathlib enabling generic proofs over real and complex Hilbert spaces.","thumb":"thumbs/2202.05360.jpg"},{"id":"2202.01629","title":"Use and abuse of instance parameters in the Lean mathematical library","authors":["Anne Baanen"],"abstract":"The Lean mathematical library mathlib features extensive use of the typeclass pattern for organising mathematical structures, based on Lean's mechanism of instance parameters. Related mechanisms for typeclasses are available in other provers including Agda, Coq and Isabelle with varying degrees of adoption. This paper analyses representative examples of design patterns involving instance parameters in the current Lean 3 version of mathlib, focussing on complications arising at scale and how the mathlib community deals with them.","abs_url":"https://arxiv.org/abs/2202.01629","pdf_url":"https://arxiv.org/pdf/2202.01629","primary_category":"cs.LO","published":"2022-02-03","tags":["mathlib","tooling","type-theory"],"lean_note":"Analyzes typeclass design patterns and scaling challenges in Lean's mathlib library.","thumb":"thumbs/2202.01629.jpg"},{"id":"ext-074","title":"A Formal Foundation for Symbolic Evaluation with Merging","authors":["Sorawee Porncharoenwase","Luke Nelson","Xi Wang","Emina Torlak"],"abstract":"This paper presents a formal foundation for symbolic evaluation with state merging, as used in tools like Rosette. The authors develop a semantics for symbolic evaluation that supports path merging and prove its correctness. The formalization uses the Lean theorem prover to establish that the symbolic evaluator faithfully represents all concrete executions of a program.","abs_url":"https://doi.org/10.1145/3527327","pdf_url":"https://doi.org/10.1145/3527327","primary_category":"cs.LO","published":"2022-01-12","tags":["verification","logic","formalization"],"lean_note":"Uses Lean to formally verify the correctness of symbolic evaluation with state merging in Rosette."},{"id":"2110.03551","title":"Formalizing Geometric Algebra in Lean","authors":["Eric Wieser","Utensil Song"],"abstract":"This paper explores formalizing Geometric (or Clifford) algebras into the Lean 3 theorem prover, building upon mathlib. We formalize the multivectors as the quotient of the tensor algebra by a suitable relation, which provides the ring structure automatically, then establish the universal property of the Clifford algebra. Our approach does not require a choice of basis, distinguishing it from existing formalizations in other theorem provers. We show how operations and structure such as involutions, versors, and the Z_2-grading can be defined using the universal property alone, and how to recover an induction principle for proving statements about these definitions.","abs_url":"https://arxiv.org/abs/2110.03551","pdf_url":"https://arxiv.org/pdf/2110.03551","primary_category":"cs.LO","published":"2021-10-07","tags":["formalization","mathlib","algebra","geometry"],"lean_note":"Formalizes Clifford/Geometric algebras in Lean 3 using a basis-free universal property approach.","thumb":"thumbs/2110.03551.jpg"},{"id":"2108.13660","title":"Formalizing the Gromov-Hausdorff space","authors":["Sébastien Gouëzel"],"abstract":"The Gromov-Hausdorff space is usually defined in textbooks as \"the space of all compact metric spaces up to isometry\". We describe a formalization of this notion in the Lean proof assistant, insisting on how we need to depart from the usual informal viewpoint of mathematicians on this object to get a rigorous formalization.","abs_url":"https://arxiv.org/abs/2108.13660","pdf_url":"https://arxiv.org/pdf/2108.13660","primary_category":"cs.LO","published":"2021-08-31","tags":["formalization","mathlib","geometry","topology"],"lean_note":"Formalizes the Gromov-Hausdorff space in Lean, addressing set-theoretic subtleties of quotient spaces.","thumb":"thumbs/2108.13660.jpg"},{"id":"2108.10700","title":"Scalar actions in Lean's mathlib","authors":["Eric Wieser"],"abstract":"Scalar actions are ubiquitous in mathematics, and therefore it is valuable to be able to write them succinctly when formalizing. This paper explores how Lean 3's typeclasses are used by mathlib for scalar actions, illustrates problems including compatibility of actions and non-definitionally-equal diamonds, and notes solutions and areas still needing more work.","abs_url":"https://arxiv.org/abs/2108.10700","pdf_url":"https://arxiv.org/pdf/2108.10700","primary_category":"cs.LO","published":"2021-08-10","tags":["mathlib","formalization","algebra","type-theory"],"lean_note":"Analyzes the typeclass-based encoding of scalar actions throughout Lean's mathlib library.","thumb":"thumbs/2108.10700.jpg"},{"id":"ext-019","title":"Automatically Generalizing Theorems Using Typeclasses","authors":["Alexander Best"],"abstract":"When typeclass assumptions in formally verified mathematical developments are stronger than necessary, this can be detected from elaborated proof terms. This paper introduces a metaprogram for the Lean theorem prover that finds and informs the user about possible generalizations of theorems by analyzing typeclass usage in proofs.","abs_url":"https://easychair.org/publications/preprint/KLfT","pdf_url":"https://easychair.org/publications/preprint/KLfT","primary_category":"cs.LO","published":"2021-08-01","tags":["tactic-automation","tooling","mathlib"],"lean_note":"Develops a Lean metaprogram to detect and suggest generalizations of typeclass assumptions in proofs."},{"id":"2108.00484","title":"Elements of Differential Geometry in Lean: A Report for Mathematicians","authors":["Anthony Bordg","Nicolò Cavalleri"],"abstract":"We report on our experience formalizing differential geometry with mathlib, the Lean mathematical library. Our account is geared towards geometers with no knowledge of type theory, but eager to learn more about the formalization of mathematics and maybe curious enough to give Lean a try in the future. To this effect, we stress the possibly surprising difference between the formalization and its pen-and-paper counterpart arising from Lean's treatment of equality. Our three case studies are Lie groups, vector bundles and the Lie algebra of a Lie group.","abs_url":"https://arxiv.org/abs/2108.00484","pdf_url":"https://arxiv.org/pdf/2108.00484","primary_category":"cs.LO","published":"2021-08-01","tags":["formalization","geometry","mathlib"],"lean_note":"Formalizes Lie groups, vector bundles, and Lie algebras in Lean's mathlib.","thumb":"thumbs/2108.00484.jpg"},{"id":"ext-083","title":"Formalizing Rotation Number and Its Properties in Lean","authors":["Yury Kudryashov"],"abstract":"Rotation number is the key numerical invariant of an orientation-preserving circle homeomorphism. This paper describes a formalization in Lean of the definition and basic properties of the translation number of a lift of a circle homeomorphism to the real line, as well as a theorem by E. Ghys providing a necessary and sufficient condition for two actions of a group on the circle by homeomorphisms to be semiconjugate to each other.","abs_url":"https://easychair.org/publications/preprint/38w1","pdf_url":"https://easychair.org/publications/preprint/38w1","primary_category":"cs.LO","published":"2021-07-27","tags":["formalization","mathlib","topology","analysis"],"lean_note":"Formalizes rotation number for circle homeomorphisms and Ghys's semiconjugacy theorem in Lean."},{"id":"2107.10988","title":"Formalizing Galois Theory","authors":["Thomas Browning","Patrick Lutz"],"abstract":"We describe a project to formalize Galois theory using the Lean theorem prover, which is part of a larger effort to formalize all of the standard undergraduate mathematics curriculum in Lean. We discuss some of the challenges we faced and the decisions we made in the course of this project. The main theorems we formalized are the primitive element theorem, the fundamental theorem of Galois theory, and the equivalence of several characterizations of finite degree Galois extensions.","abs_url":"https://arxiv.org/abs/2107.10988","pdf_url":"https://arxiv.org/pdf/2107.10988","primary_category":"cs.LO","published":"2021-07-23","tags":["formalization","mathlib","algebra"],"lean_note":"Formalizes the fundamental theorem of Galois theory and primitive element theorem in Lean.","thumb":"thumbs/2107.10988.jpg"},{"id":"ext-002","title":"The Lean 4 Theorem Prover and Programming Language","authors":["Leonardo de Moura","Sebastian Ullrich"],"abstract":"Lean 4 is a reimplementation of the Lean interactive theorem prover in Lean itself. It addresses the performance limitations of the previous version and features a new compiler that generates efficient native code. The system serves both as a programming language with dependent types and as an interactive theorem prover, featuring a powerful macro system and hygienic notation.","abs_url":"https://link.springer.com/chapter/10.1007/978-3-030-79876-5_37","pdf_url":"https://link.springer.com/chapter/10.1007/978-3-030-79876-5_37","primary_category":"cs.LO","published":"2021-07-11","tags":["tooling","type-theory"],"lean_note":"Presents Lean 4's architecture as both a dependently-typed programming language and interactive theorem prover."},{"id":"2102.07636","title":"Formalized Haar Measure","authors":["Floris van Doorn"],"abstract":"We describe the formalization of the existence and uniqueness of Haar measure in the Lean theorem prover. The Haar measure is an invariant regular measure on locally compact groups, and this is the first formalization of this result in a proof assistant. We also discuss the measure theory library in Lean's mathematical library mathlib, including the construction of product measures and the proof of Fubini's theorem for the Bochner integral.","abs_url":"https://arxiv.org/abs/2102.07636","pdf_url":"https://arxiv.org/pdf/2102.07636","primary_category":"cs.LO","published":"2021-02-04","tags":["formalization","mathlib","analysis","topology"],"lean_note":"First formalization of the existence and uniqueness of Haar measure in any proof assistant.","thumb":"thumbs/2102.07636.jpg"},{"id":"2102.02901","title":"A Formal Proof of the Independence of the Continuum Hypothesis","authors":["Jesse Michael Han","Floris van Doorn"],"abstract":"We describe a formal proof of the independence of the continuum hypothesis (CH) in the Lean theorem prover. We use Boolean-valued models to give forcing arguments for both directions, using Cohen forcing for the consistency of not-CH and a sigma-closed forcing for the consistency of CH.","abs_url":"https://arxiv.org/abs/2102.02901","pdf_url":"https://arxiv.org/pdf/2102.02901","primary_category":"math.LO","published":"2021-02-04","tags":["formalization","logic"],"lean_note":"Formalizes the independence of the continuum hypothesis using Boolean-valued forcing in Lean.","thumb":"thumbs/2102.02901.jpg"},{"id":"2102.02600","title":"A formalization of Dedekind domains and class groups of global fields","authors":["Anne Baanen","Sander R. Dahmen","Ashvni Narayanan","Filippo A. E. Nuccio Mortarino Majno di Capriglio"],"abstract":"Dedekind domains and their class groups are notions in commutative algebra that are essential in algebraic number theory. We formalized these structures and fundamental properties, including number theoretic finiteness results for class groups, in the Lean prover as part of the mathlib mathematical library. We describe the formalization process, noting the idioms we found useful in our development and mathlib's decentralized collaboration processes involved in this project.","abs_url":"https://arxiv.org/abs/2102.02600","pdf_url":"https://arxiv.org/pdf/2102.02600","primary_category":"cs.LO","published":"2021-02-04","tags":["formalization","mathlib","algebra","number-theory"],"lean_note":"Formalizes Dedekind domains and class groups of global fields in Lean's mathlib.","thumb":"thumbs/2102.02600.jpg"},{"id":"2101.07758","title":"A bi-directional extensible interface between Lean and Mathematica","authors":["Robert Y. Lewis","Minchao Wu"],"abstract":"The authors implement a user-extensible ad hoc connection between the Lean proof assistant and the computer algebra system Mathematica. By reflecting each system's syntax in the other and providing flexible translation extensions, they enable the exchange of arbitrary information between the two systems. They demonstrate how Lean's metaprogramming framework can verify certain Mathematica computations so that the rigor of the proof assistant is not compromised, while Mathematica serves as an untrusted oracle to guide proof search in Lean.","abs_url":"https://arxiv.org/abs/2101.07758","pdf_url":"https://arxiv.org/pdf/2101.07758","primary_category":"cs.LO","published":"2021-01-17","tags":["tooling","tactic-automation"],"lean_note":"Implements a bidirectional interface connecting Lean's proof assistant with Mathematica's computer algebra.","thumb":"thumbs/2101.07758.jpg"},{"id":"2101.02602","title":"Schemes in Lean","authors":["Kevin Buzzard","Chris Hughes","Kenny Lau","Amelia Livingston","Ramon Fernández Mir","Scott Morrison"],"abstract":"We tell the story of how schemes were formalised in three different ways in the Lean theorem prover.","abs_url":"https://arxiv.org/abs/2101.02602","pdf_url":"https://arxiv.org/pdf/2101.02602","primary_category":"math.AG","published":"2021-01-07","tags":["formalization","mathlib","algebra","geometry"],"lean_note":"Describes three approaches to formalizing algebraic geometry schemes in Lean.","thumb":"thumbs/2101.02602.jpg"},{"id":"2101.00127","title":"Formalizing Hall's Marriage Theorem in Lean","authors":["Alena Gusakov","Bhavik Mehta","Kyle A. Miller"],"abstract":"The authors formalize Hall's Marriage Theorem in the Lean theorem prover for inclusion in mathlib, the community-driven mathematics library for Lean. They provide three presentations of the theorem statement and formalize a version of König's lemma (in terms of inverse limits) to extend the theorem to the case of countably infinite index sets.","abs_url":"https://arxiv.org/abs/2101.00127","pdf_url":"https://arxiv.org/pdf/2101.00127","primary_category":"math.CO","published":"2021-01-01","tags":["formalization","mathlib","combinatorics"],"lean_note":"Formalizes Hall's Marriage Theorem with three statement variants and a König's lemma extension in Lean.","thumb":"thumbs/2101.00127.jpg"},{"id":"2012.08990","title":"A Novice-Friendly Induction Tactic for Lean","authors":["Jannis Limperg"],"abstract":"Induction is the primary proof technique for inductive types and for inductive predicates and relations defined on top of them. In dependently-typed theorem provers such as Coq and Lean, the ergonomics of existing induction tactics are not ideal: they do not reliably support inductive predicates and relations; they sometimes generate overly specific or unnecessarily complex induction hypotheses; and they occasionally choose confusing names for the hypotheses they introduce. This paper describes a new induction tactic implemented in Lean 3 that addresses these issues. It is particularly suitable for educational use, but experts should also find it more convenient than existing induction tactics. As a moderately complex case study for the metaprogramming framework of Lean 3, the paper also describes implementation difficulties and suggests improvements to the framework.","abs_url":"https://arxiv.org/abs/2012.08990","pdf_url":"https://arxiv.org/pdf/2012.08990","primary_category":"cs.LO","published":"2020-12-16","tags":["tactic-automation","tooling"],"lean_note":"Implements a new induction tactic in Lean 3 that improves hypothesis generation and naming.","thumb":"thumbs/2012.08990.jpg"},{"id":"2010.02595","title":"Formalizing the Ring of Witt Vectors","authors":["Johan Commelin","Robert Y. Lewis"],"abstract":"The ring of Witt vectors over a base ring R is an important tool in algebraic number theory and lies at the foundations of modern p-adic Hodge theory. It has the interesting property that it constructs a ring of characteristic 0 out of a ring of characteristic p > 1, and it can be used more specifically to construct from a finite field containing Z/pZ the corresponding unramified field extension of the p-adic numbers. We formalize the notion of a Witt vector in the Lean proof assistant, along with the corresponding ring operations and other algebraic structure. We prove in Lean that, for prime p, the ring of Witt vectors over Z/pZ is isomorphic to the ring of p-adic integers. In the process we develop idioms to cleanly handle calculations of identities between operations on the ring of Witt vectors. These calculations are intractable with a naive approach, and require a proof technique that is usually skimmed over in the informal literature.","abs_url":"https://arxiv.org/abs/2010.02595","pdf_url":"https://arxiv.org/pdf/2010.02595","primary_category":"cs.LO","published":"2020-10-06","tags":["formalization","mathlib","algebra","number-theory"],"lean_note":"Formalizes Witt vectors and proves their isomorphism with p-adic integers in Lean.","thumb":"thumbs/2010.02595.jpg"},{"id":"2004.03673","title":"Maintaining a Library of Formal Mathematics","authors":["Floris van Doorn","Gabriel Ebner","Robert Y. Lewis"],"abstract":"The Lean mathematical library mathlib is developed by a community of users with very different backgrounds and levels of experience. To lower the barrier of entry for contributors and to lessen the burden of reviewing contributions, we have developed a number of tools for the library which check proof developments for subtle mistakes in the code and generate documentation suited for our varied audience.","abs_url":"https://arxiv.org/abs/2004.03673","pdf_url":"https://arxiv.org/pdf/2004.03673","primary_category":"cs.PL","published":"2020-04-07","tags":["mathlib","tooling"],"lean_note":"Describes tooling for maintaining mathlib including linters and documentation generators.","thumb":"thumbs/2004.03673.jpg"},{"id":"2003.01685","title":"Sealing Pointer-Based Optimizations Behind Pure Functions","authors":["Daniel Selsam","Simon Hudon","Leonardo de Moura"],"abstract":"Functional programming languages are particularly well-suited for building automated reasoning systems because logical terms can be represented directly as inductive types, term traversal can make use of higher-order combinators, and persistent datastructures enable backtracking cheaply. However, traversing a term requires time proportional to the tree size of the term as opposed to its graph size. This is a significant problem for theorem provers such as Lean and Coq, where the exponential blowup of term-tree sizes has proved to be both common and difficult to prevent. We show how to use dependent types to seal the necessary pointer-address manipulations behind pure functional interfaces with only a negligible amount of additional trust. The approach was implemented for Lean v4 and could be adopted by other dependently-typed languages.","abs_url":"https://arxiv.org/abs/2003.01685","pdf_url":"https://arxiv.org/pdf/2003.01685","primary_category":"cs.PL","published":"2020-03-03","tags":["tooling","type-theory","verification"],"lean_note":"Implemented pointer-based optimizations in Lean 4's runtime using dependent types for referential transparency.","thumb":"thumbs/2003.01685.jpg"},{"id":"2001.10594","title":"Simplifying Casts and Coercions","authors":["Robert Y. Lewis","Paul-Nicolas Madelaine"],"abstract":"This paper introduces norm_cast, a toolbox of tactics for the Lean proof assistant designed to manipulate expressions containing coercions and casts. These expressions can be frustrating for beginning and expert users alike; the presence of coercions can cause seemingly identical expressions to fail to unify and rewrites to fail. The norm_cast tactics aim to make reasoning with such expressions as transparent as possible. They are used extensively to eliminate boilerplate arguments in the Lean mathematical library and in external developments.","abs_url":"https://arxiv.org/abs/2001.10594","pdf_url":"https://arxiv.org/pdf/2001.10594","primary_category":"cs.PL","published":"2020-01-28","tags":["tactic-automation","tooling","mathlib"],"lean_note":"Introduces the norm_cast tactic toolbox for handling coercions and casts in Lean.","thumb":"thumbs/2001.10594.jpg"},{"id":"2001.10490","title":"Beyond Notations: Hygienic Macro Expansion for Theorem Proving Languages","authors":["Sebastian Ullrich","Leonardo de Moura"],"abstract":"In interactive theorem provers, the weights of extensible syntax and tactic languages have grown to rival the logical core. This paper proposes a hygienic macro system inspired by Scheme for these languages. The system addresses restrictive syntax extension mechanisms that cause unnecessary redundancy in libraries and accidental name capture in tactic languages that often produces unexpected behavior. The authors describe type-directed macro expansion yielding a single, uniform system offering multiple abstraction levels from simple syntax sugars to elaboration of built-in syntax. The system was implemented in Lean 4 and is simple enough to be integrated into other systems.","abs_url":"https://arxiv.org/abs/2001.10490","pdf_url":"https://arxiv.org/pdf/2001.10490","primary_category":"cs.PL","published":"2020-01-28","tags":["tooling","type-theory"],"lean_note":"Implements and evaluates a hygienic macro system as a core component of Lean 4.","thumb":"thumbs/2001.10490.jpg"},{"id":"2001.04301","title":"Tabled Typeclass Resolution","authors":["Daniel Selsam","Sebastian Ullrich","Leonardo de Moura"],"abstract":"Typeclasses provide an elegant and effective way of managing ad-hoc polymorphism in both programming languages and interactive proof assistants. The increasingly sophisticated uses of typeclasses within proof assistants have turned theoretical limitations into major impediments to ongoing progress. The two most devastating limitations of existing procedures are exponential running times in the presence of diamonds and divergence in the presence of cycles. We present a new procedure for typeclass resolution, tabled typeclass resolution, that solves both problems by tabling, which is a generalization of memoizing originally introduced to address similar limitations of early logic programming systems. We implemented the procedure for the upcoming version (v4) of Lean and confirmed empirically that our implementation is exponentially faster than existing systems in the presence of diamonds. Although tabling is notoriously difficult to implement, our procedure is notably lightweight and could easily be implemented in other systems.","abs_url":"https://arxiv.org/abs/2001.04301","pdf_url":"https://arxiv.org/pdf/2001.04301","primary_category":"cs.PL","published":"2020-01-13","tags":["tactic-automation","tooling","type-theory"],"lean_note":"Implements tabled typeclass resolution for Lean 4 to handle diamonds and cycles efficiently.","thumb":"thumbs/2001.04301.jpg"},{"id":"ext-095","title":"Formalizing Double Groupoids and Cross Modules in the Lean Theorem Prover","authors":["Jakob von Raumer"],"abstract":"This paper presents a formalization of double groupoids and cross modules in the Lean theorem prover. Double groupoids are higher-dimensional categorical structures that generalize groupoids, while cross modules capture certain 2-dimensional algebraic information. The formalization demonstrates Lean's capabilities for encoding higher categorical structures and establishes the equivalence between these two algebraic frameworks within a proof assistant.","abs_url":"https://doi.org/10.1007/978-3-319-42432-3_63","pdf_url":"https://doi.org/10.1007/978-3-319-42432-3_63","primary_category":"cs.LO","published":"2020-01-01","tags":["formalization","category-theory","algebra"],"lean_note":"Formalizes double groupoids and cross modules as higher categorical structures in Lean."},{"id":"ext-094","title":"Simple Verification of Rust Programs via Functional Purification","authors":["Sebastian Ullrich"],"abstract":"This master's thesis presents an approach to verifying Rust programs by translating them into a purely functional representation in the Lean theorem prover. The translation exploits Rust's ownership type system to extract a functional semantics, enabling verification of imperative Rust code using Lean's proof capabilities.","abs_url":"https://pp.ipd.kit.edu/uploads/publikationen/ullrich16masterarbeit.pdf","pdf_url":"https://pp.ipd.kit.edu/uploads/publikationen/ullrich16masterarbeit.pdf","primary_category":"cs.LO","published":"2020-01-01","tags":["verification","tooling"],"lean_note":"Translates Rust programs into Lean's functional representation for verification via ownership analysis.","thumb":"thumbs/ext-094.jpg"},{"id":"ext-071","title":"A Parametrized Family of Tversky Metrics Connecting the Jaccard Distance to an Analogue of the Normalized Information Distance","authors":["Bjørn Kjos-Hanssen","Saroj Niraula","Soowhan Yoon"],"abstract":"This paper introduces a parametrized family of metrics based on Tversky's set-theoretic similarity index, connecting the Jaccard distance at one extreme to an analogue of the normalized information distance at the other. The triangle inequality and metric properties are formally verified using the Lean theorem prover, providing machine-checked guarantees for these distance functions used in information theory and data science.","abs_url":"https://doi.org/10.1007/978-3-030-93100-1_14","pdf_url":"https://doi.org/10.1007/978-3-030-93100-1_14","primary_category":"cs.LO","published":"2020-01-01","tags":["formalization","verification","combinatorics"],"lean_note":"Uses Lean to formally verify metric properties of a parametrized Tversky distance family."},{"id":"ext-063","title":"Formalising the Proj Construction in Lean","authors":["Jujian Zhang"],"abstract":"This paper presents a formalization of the Proj construction in algebraic geometry using the Lean theorem prover and its mathlib library. The Proj construction associates to a graded ring a scheme, and is fundamental in algebraic geometry for constructing projective varieties and projective space. The formalization covers the construction of the homogeneous spectrum, its structure sheaf, and the proof that the result is a scheme.","abs_url":"https://doi.org/10.1007/978-3-031-38499-8_26","pdf_url":"https://doi.org/10.1007/978-3-031-38499-8_26","primary_category":"cs.LO","published":"2020-01-01","tags":["formalization","mathlib","algebra","geometry"],"lean_note":"Formalizes the Proj scheme construction from algebraic geometry in Lean's mathlib library."},{"id":"ext-046","title":"Lean Formalization of Completeness Proof for Coalition Logic with Common Knowledge","authors":["Kai Obendrauf","Anne Baanen","Patrick Koopmann","Vera Stebletsova"],"abstract":"This paper presents a Lean formalization of the completeness proof for coalition logic with common knowledge (CL-CK). The formalization covers the syntax and semantics of the logic, the axiom system, canonical model construction, and the proof that every consistent formula is satisfiable in the canonical model.","abs_url":"https://doi.org/10.4230/LIPIcs.ITP.2024.28","pdf_url":"https://doi.org/10.4230/LIPIcs.ITP.2024.28","primary_category":"cs.LO","published":"2020-01-01","tags":["formalization","logic","verification"],"lean_note":"Formalizes the completeness proof for coalition logic with common knowledge in Lean."},{"id":"ext-022","title":"A Lean Tactic for Normalising Ring Expressions with Exponents (Short Paper)","authors":["Anne Baanen"],"abstract":"This paper presents a tactic for the Lean theorem prover that normalizes ring expressions involving exponents. The tactic extends the existing ring normalization procedure to handle expressions with natural number exponents, enabling automated reasoning about polynomial identities in commutative rings.","abs_url":"https://doi.org/10.1007/978-3-030-51054-1_2","pdf_url":"https://doi.org/10.1007/978-3-030-51054-1_2","primary_category":"cs.LO","published":"2020-01-01","tags":["tactic-automation","algebra","tooling"],"lean_note":"Implements a Lean tactic for normalizing ring expressions with natural number exponents."},{"id":"ext-014","title":"Automatically Generalizing Proofs and Statements","authors":["Anshula Gandhi","Anand Tadipatri","Timothy Gowers"],"abstract":"This paper presents methods for automatically generalizing proofs and mathematical statements in the Lean proof assistant. The approach identifies hypotheses in a proof that can be weakened or removed while preserving correctness, producing more general versions of theorems that may apply to broader mathematical structures.","abs_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025","pdf_url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025","primary_category":"cs.LO","published":"2020-01-01","tags":["tactic-automation","tooling","mathlib"],"lean_note":"Implements automatic generalization of Lean proofs by weakening hypotheses."},{"id":"ext-007","title":"Formally Verified Insertion of Reference Counting Instructions","authors":["Marc Huisinga"],"abstract":"This thesis presents a formally verified compiler pass for inserting reference counting instructions in Lean 4's intermediate representation. The correctness of the insertion algorithm is proved in Lean itself, ensuring that the resulting programs correctly manage memory through reference counting without introducing leaks or use-after-free errors.","abs_url":"https://lean-forward.github.io/lean-together/2020/abstracts/huisinga.pdf","pdf_url":"https://lean-forward.github.io/lean-together/2020/abstracts/huisinga.pdf","primary_category":"cs.LO","published":"2020-01-01","tags":["verification","tooling"],"lean_note":"Implements and formally verifies a Lean 4 compiler pass for reference counting in Lean itself."},{"id":"ext-006","title":"The Type Theory of Lean","authors":["Mario Carneiro"],"abstract":"This master's thesis describes the type theory underlying the Lean theorem prover, providing a formal specification of the calculus of inductive constructions as implemented in Lean. It covers the kernel type system, soundness, and normalization properties of the system.","abs_url":"https://github.com/digama0/lean-type-theory/releases","pdf_url":"https://github.com/digama0/lean-type-theory/releases","primary_category":"cs.LO","published":"2020-01-01","tags":["type-theory","logic","formalization"],"lean_note":"Provides a formal specification of Lean's kernel type theory and metatheoretic properties."},{"id":"1911.00385","title":"A Formal Proof of PAC Learnability for Decision Stumps","authors":["Joseph Tassarotti","Koundinya Vajjha","Anindya Banerjee","Jean-Baptiste Tristan"],"abstract":"We present a formal proof in Lean of probably approximately correct (PAC) learnability of the concept class of decision stumps. This classic result in machine learning theory derives a bound on error probabilities for a simple type of classifier. Though such a proof appears simple on paper, analytic and measure-theoretic subtleties arise when carrying it out fully formally. Our proof is structured so as to separate reasoning about deterministic properties of a learning function from proofs of measurability and analysis of probabilities.","abs_url":"https://arxiv.org/abs/1911.00385","pdf_url":"https://arxiv.org/pdf/1911.00385","primary_category":"cs.LG","published":"2019-11-01","tags":["formalization","verification","ml-theory","probability"],"lean_note":"Formalizes a PAC learnability proof for decision stumps in Lean with measure theory.","thumb":"thumbs/1911.00385.jpg"},{"id":"1910.12320","title":"Formalising perfectoid spaces","authors":["Kevin Buzzard","Johan Commelin","Patrick Massot"],"abstract":"Perfectoid spaces are sophisticated objects in arithmetic geometry introduced by Peter Scholze in 2012. The authors formalize enough definitions and theorems in topology, algebra, and geometry to define perfectoid spaces in the Lean theorem prover. This confirms that a proof assistant can handle complexity in that direction and that mathematicians with no computer science training can become proficient users of a proof assistant in a relatively short period of time.","abs_url":"https://arxiv.org/abs/1910.12320","pdf_url":"https://arxiv.org/pdf/1910.12320","primary_category":"cs.LO","published":"2019-10-27","tags":["formalization","mathlib","algebra","geometry","number-theory"],"lean_note":"Defines perfectoid spaces in Lean, demonstrating formalization of cutting-edge arithmetic geometry.","thumb":"thumbs/1910.12320.jpg"},{"id":"1910.09336","title":"The Lean mathematical library","authors":["The mathlib Community"],"abstract":"This paper describes mathlib, a community-driven effort to build a unified library of mathematics formalized in the Lean proof assistant. It is distinguished by its dependently typed foundations, focus on classical mathematics, extensive hierarchy of structures, use of large- and small-scale automation, and distributed organization. We explain the architecture and design decisions of the library and the social organization that has led us here.","abs_url":"https://arxiv.org/abs/1910.09336","pdf_url":"https://arxiv.org/pdf/1910.09336","primary_category":"cs.LO","published":"2019-10-21","tags":["mathlib","formalization","tooling"],"lean_note":"Describes the architecture, design, and social organization of Lean's mathlib library.","thumb":"thumbs/1910.09336.jpg"},{"id":"1909.11342","title":"A formal proof of Hensel's lemma over the p-adic integers","authors":["Robert Y. Lewis"],"abstract":"The field of p-adic numbers and the ring of p-adic integers are essential constructions of modern number theory. Hensel's lemma, described by Gouvea as the most important algebraic property of the p-adic numbers, shows the existence of roots of polynomials over the p-adic integers provided an initial seed point. The theorem can be proved for the p-adics with significantly weaker hypotheses than for general rings. We construct the p-adic numbers and integers in the Lean proof assistant, with various associated algebraic properties, and formally prove a strong form of Hensel's lemma. The proof lies at the intersection of algebraic and analytic reasoning and demonstrates how the Lean mathematical library handles such a heterogeneous topic.","abs_url":"https://arxiv.org/abs/1909.11342","pdf_url":"https://arxiv.org/pdf/1909.11342","primary_category":"cs.LO","published":"2019-09-25","tags":["formalization","mathlib","number-theory","algebra"],"lean_note":"Constructs p-adic numbers in Lean and formally proves a strong form of Hensel's lemma.","thumb":"thumbs/1909.11342.jpg"},{"id":"1908.05647","title":"Counting Immutable Beans: Reference Counting Optimized for Purely Functional Programming","authors":["Sebastian Ullrich","Leonardo de Moura"],"abstract":"Most functional languages rely on some form of garbage collection for automatic memory management. They usually eschew reference counting in favor of a tracing garbage collector, which has less bookkeeping overhead at runtime. However, having an exact reference count of each value can enable optimizations such as destructive updates. We explore these optimization opportunities in the context of an eager, purely functional programming language. We propose a new mechanism for efficiently reclaiming memory used by nonshared values, reducing stress on the global memory allocator. We also describe an approach for minimizing the number of reference count updates using borrowed references and a heuristic for automatically inferring borrow annotations. These techniques have been implemented in a new compiler for an eager and purely functional programming language with support for multi-threading. Our preliminary experimental results demonstrate our approach is competitive and often outperforms state-of-the-art compilers.","abs_url":"https://arxiv.org/abs/1908.05647","pdf_url":"https://arxiv.org/pdf/1908.05647","primary_category":"cs.PL","published":"2019-08-15","tags":["tooling","type-theory"],"lean_note":"Describes the reference counting optimizations implemented in the Lean 4 compiler runtime.","thumb":"thumbs/1908.05647.jpg"},{"id":"ext-026","title":"Arithmetic and Casting in Lean","authors":["Paul-Nicolas Madelaine"],"abstract":"This report describes a Lean metaprogram for automatically solving arithmetic goals involving casts between numeric types. The tool normalizes expressions involving coercions between naturals, integers, and rationals, and discharges side conditions. It addresses a practical pain point in Lean formalization where type casts create verification obligations that are tedious to prove manually.","abs_url":"https://lean-forward.github.io/internships/arithmetic-and-casting-in-lean.pdf","pdf_url":"https://lean-forward.github.io/internships/arithmetic-and-casting-in-lean.pdf","primary_category":"cs.LO","published":"2019-08-01","tags":["tactic-automation","tooling"],"lean_note":"Develops a Lean tactic for automated reasoning about arithmetic with numeric type coercions."},{"id":"1907.07801","title":"Iterated chromatic localisation","authors":["Neil Strickland","Nicola Bellumat"],"abstract":"We study a certain monoid of endofunctors of the stable homotopy category that includes localizations with respect to finite unions of Morava K-theories. We work in an axiomatic framework that can also be applied to analogous questions in equivariant stable homotopy theory. Our results should be helpful for the study of transchromatic phenomena, including the Chromatic Splitting Conjecture. The combinatorial parts of this work have been formalised in the Lean proof assistant.","abs_url":"https://arxiv.org/abs/1907.07801","pdf_url":"https://arxiv.org/pdf/1907.07801","primary_category":"math.AT","published":"2019-07-17","tags":["formalization","topology"],"lean_note":"Formalizes the combinatorial components of chromatic localisation theory in Lean.","thumb":"thumbs/1907.07801.jpg"},{"id":"1907.01449","title":"Formalizing the Solution to the Cap Set Problem","authors":["Sander R. Dahmen","Johannes Hölzl","Robert Y. Lewis"],"abstract":"In 2016, Ellenberg and Gijswijt established a new upper bound on the size of subsets of F_q^n with no three-term arithmetic progression. This problem has garnered significant mathematical attention, especially when q = 3, known as the cap set problem. Ellenberg and Gijswijt's proof was published in the Annals of Mathematics and is noteworthy for its clever use of elementary methods. We describe formalizing this proof in the Lean proof assistant, covering both the general result in F_q^n and concrete values for q = 3. We faithfully follow the pen and paper argument to construct the bound. The work demonstrates that some modern mathematics is within the range of proof assistants.","abs_url":"https://arxiv.org/abs/1907.01449","pdf_url":"https://arxiv.org/pdf/1907.01449","primary_category":"cs.LO","published":"2019-07-02","tags":["formalization","combinatorics"],"lean_note":"Formalizes Ellenberg-Gijswijt's 2016 cap set bound proof in Lean.","thumb":"thumbs/1907.01449.jpg"},{"id":"1904.10570","title":"A Formalization of Forcing and the Unprovability of the Continuum Hypothesis","authors":["Jesse Michael Han","Floris van Doorn"],"abstract":"The authors describe a formalization of forcing using Boolean-valued models in the Lean 3 theorem prover, including the fundamental theorem of forcing and a deep embedding of first-order logic with a Boolean-valued soundness theorem. They specialize the construction to the Boolean algebra of regular opens of the Cantor space and formally verify the failure of the continuum hypothesis in the resulting model.","abs_url":"https://arxiv.org/abs/1904.10570","pdf_url":"https://arxiv.org/pdf/1904.10570","primary_category":"cs.LO","published":"2019-04-23","tags":["formalization","logic","type-theory"],"lean_note":"Formalizes Cohen's forcing method in Lean 3 to prove the independence of the continuum hypothesis.","thumb":"thumbs/1904.10570.jpg"},{"id":"1810.08380","title":"Formalizing Computability Theory via Partial Recursive Functions","authors":["Mario Carneiro"],"abstract":"We present an extension to the mathlib library of the Lean theorem prover formalizing the foundations of computability theory. We use primitive recursive functions and partial recursive functions as the main objects of study, employing a constructive encoding of partial functions such that they are executable when the programs in question provably halt. The main results include the construction of a universal partial recursive function and a proof of the undecidability of the halting problem. Type class inference provides a transparent way to supply Gödel numberings where needed and encapsulate the encoding details.","abs_url":"https://arxiv.org/abs/1810.08380","pdf_url":"https://arxiv.org/pdf/1810.08380","primary_category":"cs.LO","published":"2018-10-19","tags":["formalization","mathlib","logic"],"lean_note":"Formalizes computability theory in Lean's mathlib including the halting problem undecidability.","thumb":"thumbs/1810.08380.jpg"},{"id":"1712.09288","title":"An Extensible Ad Hoc Interface between Lean and Mathematica","authors":["Robert Y. Lewis"],"abstract":"We implement a user-extensible ad hoc connection between the Lean proof assistant and the computer algebra system Mathematica. By reflecting each system's syntax in the other and providing a flexible translation interface, the connection enables exchange of arbitrary information between the two systems. We demonstrate how Lean's metaprogramming framework can verify Mathematica computations, ensuring the rigor of the proof assistant is not compromised.","abs_url":"https://arxiv.org/abs/1712.09288","pdf_url":"https://arxiv.org/pdf/1712.09288","primary_category":"cs.LO","published":"2017-12-05","tags":["tooling","tactic-automation"],"lean_note":"Implements a bidirectional interface between Lean and Mathematica for verified computation.","thumb":"thumbs/1712.09288.jpg"},{"id":"ext-009","title":"A metaprogramming framework for formal verification","authors":["Gabriel Ebner","Sebastian Ullrich","Jared Roesch","Jeremy Avigad","Leonardo de Moura"],"abstract":"This paper describes the metaprogramming framework of the Lean theorem prover, which allows users to write custom proof automation, decision procedures, and tactics in the same language used for formal verification. The framework provides a monadic interface to the elaborator and type-checker, enabling reflective tactics and program transformations within a single unified environment.","abs_url":"https://doi.org/10.1145/3110278","pdf_url":"https://doi.org/10.1145/3110278","primary_category":"cs.LO","published":"2017-09-01","tags":["tactic-automation","tooling"],"lean_note":"Introduces Lean's tactic framework enabling user-defined proof automation in Lean itself."},{"id":"1701.04391","title":"Congruence Closure in Intensional Type Theory","authors":["Daniel Selsam","Leonardo de Moura"],"abstract":"Congruence closure procedures are used extensively in automated reasoning and are a core component of most satisfiability modulo theories solvers. However, no known congruence closure algorithms can support the expressive logics based on intensional type theory (ITT) that underpin many interactive theorem provers. The authors present an efficient and proof-producing congruence closure procedure that applies to every function in ITT regardless of dependencies among the arguments, relying only on the uniqueness of identity proofs axiom.","abs_url":"https://arxiv.org/abs/1701.04391","pdf_url":"https://arxiv.org/pdf/1701.04391","primary_category":"cs.LO","published":"2017-01-16","tags":["tactic-automation","type-theory","logic"],"lean_note":"Implements a congruence closure procedure used in Lean's automation for intensional type theory.","thumb":"thumbs/1701.04391.jpg"},{"id":"1512.02274","title":"Constructing the Propositional Truncation using Non-recursive HITs","authors":["Floris van Doorn"],"abstract":"In homotopy type theory, the propositional truncation is constructed as a colimit using only non-recursive higher inductive types (HITs). This is a first step towards reducing recursive HITs to non-recursive HITs. The construction also characterizes functions from the propositional truncation to arbitrary types, extending its universal property. The results are formalized in the Lean proof assistant.","abs_url":"https://arxiv.org/abs/1512.02274","pdf_url":"https://arxiv.org/pdf/1512.02274","primary_category":"math.LO","published":"2015-12-07","tags":["formalization","type-theory","topology","logic"],"lean_note":"Formalizes a colimit construction of propositional truncation from non-recursive HITs in Lean.","thumb":"thumbs/1512.02274.jpg"},{"id":"ext-012","title":"The Lean Theorem Prover (System Description)","authors":["Leonardo de Moura","Soonho Kong","Jeremy Avigad","Floris van Doorn","Jakob von Raumer"],"abstract":"Lean is an open-source theorem prover and programming language being developed at Microsoft Research. It is based on dependent type theory with a cumulative hierarchy of non-cumulative universes, and supports inductive types, type classes, and a tactic framework for proof automation. This paper describes the system architecture, its kernel, elaboration engine, and proof automation infrastructure.","abs_url":"https://doi.org/10.1007/978-3-319-21401-6_26","pdf_url":"https://doi.org/10.1007/978-3-319-21401-6_26","primary_category":"cs.LO","published":"2015-08-01","tags":["tooling","type-theory"],"lean_note":"Introduces the Lean theorem prover system based on dependent type theory with inductive types."},{"id":"1505.04324","title":"Elaboration in Dependent Type Theory","authors":["Leonardo de Moura","Jeremy Avigad","Soonho Kong","Cody Roux"],"abstract":"To be usable in practice, interactive theorem provers need to provide convenient and efficient means of writing expressions, definitions, and proofs. This involves inferring information that is left implicit in a quasi-formal expression and resolving ambiguities. We call the process of passing from a quasi-formal and partially-specified expression to a completely precise formal one elaboration. We describe an elaboration algorithm for dependent type theory that has been implemented in the Lean theorem prover. Lean's elaborator supports higher-order unification, type class inference, ad hoc overloading, insertion of coercions, the use of tactics, and the computational reduction of terms. The interactions between these components are subtle and complex, and we describe the central design goals that guide our implementation.","abs_url":"https://arxiv.org/abs/1505.04324","pdf_url":"https://arxiv.org/pdf/1505.04324","primary_category":"cs.LO","published":"2015-05-16","tags":["tooling","type-theory","logic"],"lean_note":"Describes the elaboration algorithm implemented in the Lean theorem prover for dependent type theory.","thumb":"thumbs/1505.04324.jpg"},{"id":"ext-096","title":"Formalization of Non-Abelian Topology for Homotopy Type Theory","authors":["Jakob von Raumer"],"abstract":"This thesis develops non-abelian algebraic topology within the framework of homotopy type theory, formalized in the Lean proof assistant. It establishes constructions involving higher groupoids and their relationship to homotopy types, extending the computational content of homotopy type theory beyond abelian structures. The formalization demonstrates how dependent type theory provides a natural setting for higher-dimensional algebraic topology.","abs_url":"http://www.contrib.andrew.cmu.edu/~avigad/Students/von_raumer_thesis.pdf","pdf_url":"http://www.contrib.andrew.cmu.edu/~avigad/Students/von_raumer_thesis.pdf","primary_category":"cs.LO","published":"2015-01-01","tags":["formalization","topology","type-theory"],"lean_note":"Formalizes non-abelian algebraic topology constructions in Lean within a HoTT framework.","thumb":"thumbs/ext-096.jpg"}]}