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

Claimed: for every $l >= 32$ , any dag-like $\mathrm{Res}(\oplus)$ refutation of the bit pigeonhole principle with $2^l + 1$ pigeons and $n = 2^l$ holes has more than $exp(n/(32768 l^2)) = 2^{\Omega(n/log^2 n)}$ nodes; exponential in the number of holes up to a $log^2$ factor, for a proof system with no regularity or depth restriction. The whitepaper's route is a transfer to degree- $O(log n)$ polynomial calculus. Candidate: kernel-checked, read by no specialist.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 15, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-verified. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: An exponential lower bound for the bit pigeonhole principle in resolution over parities · Bit PHP in dag-like Res(⊕)

Confidence: Not scored

Registry verification: lean verified · announcement · candidate

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Attribution

VibeMathed
registry · event recorded by

GPT-6 Astra
model · ai model contributor · maker attribution ambiguous: OpenAI, Anthropic

Claude Sonnet 5
model · ai model contributor · maker attribution ambiguous: OpenAI, Anthropic

Claude Fable 5.1
model · ai model contributor · maker attribution ambiguous: OpenAI, Anthropic

Claude Opus 5
model · ai model contributor · maker attribution ambiguous: OpenAI, Anthropic

Kamil Braun
human · human collaborator

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

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

specialist registry · passed

Held on 15 September 2026 for one reason: nothing outside the author's machine had compiled the proof. Lifted on 22 September. GitHub Actions run 34953139115 (commit 3979e0c, fresh ubuntu-24.04 runner, mathlib cache for dependencies only, .lake/build deleted) ran `lake --wfail build claims.BitPHPSuperpolynomial` & replayed the module in a fresh leanchecker kernel. Run 35282748811 (17 September UTC, commit e4c7723, same setup) did the same for `claims.BitPHPPreprintRevision1`, which adds the revised version's MathResearch.bitPHP_exponential: for $l >= 32$ , more than $exp(2^l/(32768 l^2))$ nodes. Both theorems use only propext, Classical.choice and Quot.sound. Audit at 54f0937, byte-identical to 3979e0c: the superpolynomial theorem quantifies over every AffineDAG over the standard bit-PHP initial clauses with a node deriving the empty clause; AffineDAGStep allows initial clauses, semantic weakening, resolution on a parity literal against its complement & any sound two-premise rule, nodes reference any earlier node, no regularity or depth constraint, a superset of dag-like $\mathrm{Res}(\oplus)$ ; usualCNFClause is the standard bit-PHP axiom over all pairs & all $2^l$ labels; the chessboard homology is proved, not assumed. e4c7723 only adds files, & the exponential theorem uses the same definitions. No sorry, custom axiom or native_decide at either commit. Kernel-checked on a machine other than the author's, statement audited: Lean-verified. No specialist has read the argument.

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Final Lean theorem

lean artifact · passed

lean-proof link supplied by the VibeMathed registry.

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

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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GitHub Actions build on a fresh runner: axioms and kernel replay

independent reproduction · pending

independent link supplied by the VibeMathed registry.

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

paper · pending

paper link supplied by the VibeMathed registry.

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Artifacts and verifiers

Final Lean theorem

lean artifact · passed

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

This event attributed to Claude Fable 5.1

VibeMathed record: An exponential lower bound for the bit pigeonhole principle in resolution over parities evidence for this event

This event attributed to Claude Opus 5

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

Final Lean theorem evidence for this event

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

GitHub Actions build on a fresh runner: axioms and kernel replay evidence for this event

An exponential lower bound for the bit pigeonhole principle in resolution over parities parent of this event

This event attributed to Claude Sonnet 5

This event attributed to Kamil Braun

This event attributed to GPT-6 Astra

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