The ceiling of the single-letter entropy method for the union-closed sets conjecture
Every lower bound on the union-closed constant since Gilmer (2022) — including , Sawin–Yu–Cambie's and Liu's — is certified by the same single-letter inequality: a memoryless protocol coupling two conditional bits, together with a class of admissible joint laws of the two prefix-conditional probabilities. It was not known how far this framework can go: whether it could be pushed to and settle Frankl's conjecture, or whether it has a hard ceiling strictly below it.
- Result
- Proved(see note)
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Extremal set theory / entropy method
- Posed by
- Raised in the entropy-method literature after Gilmer (2022): Cambie (2022) studies the approach's boundaries, Liu (2023) asks whether other couplings can improve it
- Year posed
- 2022
- Years open
- 4y
- Solved
- 2026-09-08
- Model
- Fable 5.1
- Vendor
- —
- Collaborators
- Andrew Moffat
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Significance
- 18 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Two unconditional ceilings. Every single-letter certificate whose classes contain product laws certifies at most . Every certificate that uses the i.i.d. protocol and whose other classes admit component hiding — a property of every class in the literature — certifies at most . So the entropy method, in every form used since 2022, cannot reach : Frankl's conjecture is out of its range, and only of headroom remained above the previous record. The paper also constructs a protocol attaining (computer-assisted, conditional on two numerically verified hypotheses). Frankl's conjecture itself remains wide open.
What the AI did
A team of Claude agents working under the author's direction. Claude Fable 5.1 selected the problem, found both ceiling theorems — including the component-hiding adversary, which is the new idea — and designed the optimal-diagonal protocol and its certification. Claude Opus 5 agents refereed the manuscript over four rounds (catching an error in the lead agent's own lemma), proved the small-entropy lemma, and independently re-implemented the numerical certificate from the written statement alone. Claude Sonnet agents transcribed the cited literature. The author set the goal and the standard, checked the claims, and decided what to publish. Referee reports and logs are in the repository.
Verification
Filed at Lean-checked rather than Lean-verified, and the distinction is the one that tier exists for. The two unconditional ceiling theorems are machine-checked: the lean/ project builds with no sorry and only propext, Classical.choice and Quot.sound, check.sh enforces both, and the lean and verify workflows were green at HEAD on 10 September. What Lean proves, though, is a theorem about Certifies, the author's own finite-model definition of a single-letter certificate. Whether every certificate in the literature is an instance is Lemma 3.3, whose maximal-correlation case is paper-only, and Gilmer's reduction from union-closed families to the certificate is a hypothesis in the development rather than a theorem. So the artifact compiles and proves what the paper's framework says; the framework's fidelity to the informal claim that the entropy method cannot reach was not audited here. The computer-assisted constant is conditional on two numerically verified hypotheses of the same kind as those behind Liu's record, and is not in the formalisation. Not peer reviewed and not checked by a named expert; four rounds of referee reports were produced by the author's own model agents and are in the repository.
Sources
Submitted by CobaltPanther851 on