An exponential lower bound for the bit pigeonhole principle in resolution over parities

Abstract

Resolution over parities, $\mathrm{Res}(\oplus)$, is the characteristic-two version of resolution over linear equations: clauses are disjunctions of affine equations over $\mathbb F_2$. Superpolynomial size lower bounds were previously known only for restricted refutations: tree-like, regular, or of bounded depth. We prove that every DAG-like $\mathrm{Res} (\oplus)$ refutation of the bit pigeonhole principle with $n+1$ pigeons and $n=2^\ell$ holes has more than $\exp(n/(32768\ell^2))=2^{Ω(n/\log^2 n)}$ clauses, for every $\ell\ge32$, with no restriction on regularity or depth. The proof translates an arbitrary refutation with $S$ clauses into a polynomial calculus refutation of degree $O(\log n)$ over $O(S+n^2)$ groups of extension variables in the style of Buss, Impagliazzo, Krajicek, Pudlak, Razborov, and Sgall. One substitution then removes all extension variables at once and leaves a nonzero low-degree polynomial derived from the pigeonhole axioms alone at degree at most $n/2$; a degree lower bound in the style of Razborov, proved through the homology of chessboard complexes, shows that no such derivation exists. The argument also yields a general sufficient condition for $\mathrm{Res}(\oplus)$ size lower bounds. The main theorem, this condition, and all their dependencies are formalized in Lean 4, and every statement links to its formal proof. The proof was developed with substantial AI assistance within an open research framework described in the final section.