Random Signs into Matchings: A Godsil-Gutman Identity, Formalized in Lean 4 (Part I)
Description
A machine-checked formalization, in Lean 4 / Mathlib, of the Godsil–Gutman identity: the average characteristic polynomial of a uniformly random ±1 signing of a graph is its matching polynomial. The development also contributes the matching polynomial and its deletion recurrence, the ℤ/2 sign-averaging engine, and the Bilu–Linial 2-lift spectral decomposition. The headline theorems are sorry-free (axioms: propext, Classical.choice, Quot.sound). Heilmann–Lieb real-rootedness and interlacing families are mapped, not formalized — a path-tree route is proposed. The paper is included in English and Spanish editions, with figures and exact numerical cross-checks. Lean sources: github.com/karlesmarin/godsil-gutman-lean.
Notes
The sequel proposed at the end of this paper is now written. Part II, "Unfolding a Graph into a Tree: A Machine-Checked Proof of the Heilmann–Lieb Theorem in Lean 4" (doi 10.5281/zenodo.20561832), carries out the path-tree route proposed here as the "next stone": it builds Godsil's path tree, proves the divisibility μ_G ∣ μ_{T(G,u)} and the forest identity, and proves the Ramanujan bound, giving the first machine-checked Heilmann–Lieb theorem (both halves, sorry-free). Part II thus continues and completes Part I.
Files
godsil-gutman-lean.pdf
Additional details
Related works
- Is referenced by
- 10.5281/zenodo.20600326 (DOI)
- Is supplemented by
- Software: https://github.com/karlesmarin/godsil-gutman-lean (URL)