Source authenticated

The ceiling of the single-letter entropy method for the union-closed sets conjecture

Two unconditional ceilings. Every single-letter certificate whose classes contain product laws certifies at most $1-h(1/\sqrt2)/\sqrt2 = 0.383099\ldots$ . 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 $c^{**} = 0.382885260\ldots$ . So the entropy method, in every form used since 2022, cannot reach $1/2$ : Frankl's conjecture is out of its range, and only $1.8\cdot10^{-4}$ of headroom remained above the previous record. The paper also constructs a protocol attaining $0.38284$ (computer-assisted, conditional on two numerically verified hypotheses). Frankl's conjecture itself remains wide open.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 8, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. 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: The ceiling of the single-letter entropy method for the union-closed sets conjecture · Entropy-method ceiling for Frankl

Confidence: Not scored

Registry verification: lean checked · announcement · resolved

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Attribution

VibeMathed
registry · event recorded by

Fable 5.1
model · ai model contributor

Andrew Moffat
human · human collaborator

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: The ceiling of the single-letter entropy method for the union-closed sets conjecture

specialist registry · passed

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 $1/2$ was not audited here. The computer-assisted constant $0.38284$ 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.

Source ↗

The ceiling of the single-letter entropy method for the union-closed sets conjecture

witness · pending

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

Source ↗

The paper, 31 pages with the Lean appendices, at v1.2

paper · pending

paper link supplied by the VibeMathed registry.

Source ↗

lean/: Theorems 3.1 and 3.4 machine-checked, with check.sh

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Cambie 2022, which studies the entropy approach's boundaries

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

Source ↗

Artifacts and verifiers

lean/: Theorems 3.1 and 3.4 machine-checked, with check.sh

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

Cambie 2022, which studies the entropy approach's boundaries evidence for this event

The paper, 31 pages with the Lean appendices, at v1.2 evidence for this event

This event attributed to Fable 5.1

VibeMathed record: The ceiling of the single-letter entropy method for the union-closed sets conjecture evidence for this event

The ceiling of the single-letter entropy method for the union-closed sets conjecture parent of this event

lean/: Theorems 3.1 and 3.4 machine-checked, with check.sh evidence for this event

The ceiling of the single-letter entropy method for the union-closed sets conjecture evidence for this event

This event attributed to Andrew Moffat

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