Extremal descendant integrals on moduli spaces of curves: An inequality discovered and proved in collaboration with AI
Johannes Schmitt
math.AG
Dec 16, 2025 · v1
TL;DR
A combinatorial optimization theorem underlying the extremal-descendant result was formalized and machine-checked in Lean 4.
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).
Problem
For pure psi-class intersection numbers on the moduli space of stable curves, it was unknown for which exponent vectors the descendant integral becomes extremal (minimal or maximal).
Approach
The authors determine that the intersection number is minimal for powers of a single psi-class and maximal for balanced vectors. The proof uses nefness of psi-classes combined with Khovanskii-Teissier log-concavity. The result was discovered and the proof was found by AI models (GPT-5 and Gemini 3 Pro), with large parts of the paper drafted by Claude Opus 4.5. A portion of the argument was formalized in Lean with the help of Claude Code and GPT-5.2.
Results
The extremal values are completely characterized: minimal for concentrated vectors (all exponents but one vanish) and maximal for balanced vectors (|e_i - e_j| <= 1 for all i,j). The paper demonstrates a workflow combining human mathematicians and AI models for discovery, writing, and formal verification.
