Binary Belief Propagation Fixed Points: Finiteness, Rigidity, and Topological Complexity
Yusuke Watanabe
cs.IT
Nov 9, 2026 · v1
math.AG math.CO
TL;DR
The three main theorems on belief propagation fixed points, including the logarithmic size bounds, are formally verified in Lean 4 with Mathlib and the Tau Ceti differential topology library.
Abstract
A fixed finite binary Ising model has only finitely many admissible complex belief propagation fixed points, even at degenerate parameter values. Families of positive fixed points obtained by varying external fields can nevertheless have nontrivial topology. We study this distinction through the external-field projection. For a fixed field, we prove finiteness by bounding the pole orders of the magnetizations on a hypothetical complex solution curve. The generic degree on the support graph then bounds the total local multiplicity. A graph has a single smooth real line as its full positive response for every choice of positive couplings, base field, and strictly positive sweep direction exactly when it contains no two vertex-disjoint connected subgraphs of cycle rank at least two. Two such subgraphs suffice to create a circular component. For a plane obtained by varying two vertex fields, connected simple bipartite cubic graphs with one uniform coupling and a universal bound on the fixed fields realize every genus. The minimum number of vertices grows logarithmically in the genus, by a matching algebraic upper bound on Betti numbers. Finally, we certify the complete field discriminant of an eighteen-vertex product model and recover its torus component's genus from the counts of distinct roots on all field cells. The three main theorems, including the logarithmic size bounds, are formally verified in Lean 4 with Mathlib.
Problem
The paper studies the fixed points of binary belief propagation on Ising models. It asks how many complex fixed points a single model has, and when the positive responses along one-field or two-field families have nontrivial topology.
Approach
Finiteness at a fixed field is proved by bounding pole orders of the magnetizations on a hypothetical complex solution curve. One-parameter rigidity is characterized graph-theoretically using sign-domain and Hessian eigenvalue arguments. Every genus is realized by iterated fold doubling in connected simple bipartite cubic graphs with a uniform coupling. The main theorems are formalized in Lean 4 with Mathlib, and interval-arithmetic certificates cover an 18-vertex example.
Results
Theorem A: every fixed model has finitely many admissible complex fixed points. Theorem B: a graph's response along every positive sweep is a single smooth line exactly when it lacks two vertex-disjoint connected subgraphs of cycle rank at least two. Theorem C: every genus is realized, and the minimum number of vertices grows logarithmically in the genus; all three theorems are machine-checked in Lean.
| Result | Declaration | Module |
|---|
| Theorem A | finite_admissibleSolutions_ising | UniversalFiniteness |
| Theorem B | theoremB_smooth | TheoremB |
| Theorem C, realization | theoremC_genus | TheoremC |
| Theorem C, size bounds | theoremC_minVertices | TheoremC |
Final Lean declarations for the main theorems
