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A degree-three local-weight cap and an improved relevant-variable bound

Primarily AI-generated textHuman understanding: some partsmath.CO — Combinatoricscs.CC — Computational Complexitycs.DM — Discrete Mathematics

Contributed by Ricardo Pinho ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

We prove that the local-degree potential of a Boolean function of degree at most three is at most 5/4, using the published fact that such a function has at most ten relevant variables. Inserting this cap into Wellens's degree–block-sensitivity recurrence, with the existing exact certificates released by Épi, gives C_44 <= 4.3576549226. The reproduced preceding recurrence gives approximately 4.365467422574458; our exact degree-three cap lowers it by 1/128. Two separately written exact-arithmetic programs check the finite certificates, recurrence, infinite tail and finite cases of the added lemma. The universal argument is written in the paper; the published degree-three classification is cited, not re-enumerated. This preprint is not peer reviewed and not verified in a proof assistant; no optimality is claimed.

Provenance statement

Analysis and proof drafting by AI agents under my direction; I reviewed the results at summary level. AI agents also prepared the certificate, checkers and publication materials. This does not claim a line-by-line human verification of the proof or software. Two separately written exact-arithmetic checkers (written by the same AI agent, without importing one another) verify the finite certificates, recurrence, infinite tail and finite cases of the added lemma; all checker/mutation controls are rejected as intended. Repository: https://github.com/rpinho/c44-certificate (commit f3fb25dfb88de3da8b1a0b30f03fdb51b78ccef9); the certificate inputs are also included in anc/.

Tools used

OpenAI
CodexVersion GPT-6.1 Sol

References

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Version history

  1. v1Submitted by Ricardo PinhoInitial depositCurrentOct 23, 2026