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The ceiling of the single-letter entropy method for the union-closed sets conjecture, and a protocol that reaches it

Andrew Moffat

Moffat’s Theorem 3.1 claims to bound any single-letter entropy certificate whose classes contain product laws by c<=1-h(1/sqrt(2))/sqrt(2)=0.383099... . Theorem 3.4 gives a refined c<=0.382885260... ceiling when the iid protocol has a singleton class and other classes admit component hiding. These are limits of specified certificate methods, not bounds on the true union-closed constant. The paper's c>=0.38284 construction remains conditional on numerical hypotheses and is separate.

Sources for this record

What was already known

Gilmer's entropy method and later coupling refinements led to Liu's conditional computer-assisted 0.382709 lower bound; none established this method-level upper ceiling. Frankl's 1/2 conjecture remains open. The paper identifies the product-law and hiding assumptions needed for its barrier.

AI involvement AI-led

Moffat credits Claude Fable 5.1 with both ceiling theorems and the component-hiding argument. He set the goal, checked claims and selected the release; other AI agents refereed and formalized parts of it.

The lead model is expressly credited with both central theorems and the new adversary while the human author directed the campaign.

Importance Notable

Identifies a limitation of specified single-letter entropy certificates.

Limitations

These are ceilings for certificate classes satisfying the stated product-law and component-hiding assumptions, not bounds on the true union-closed constant. Lean covers finite-model versions and a c≤0.3829 corollary; the broad bridge and exact 0.382885260 value remain outside that check. The separate c≥0.38284 construction is conditional.

Technical verification Partial

Lean checks finite-model versions of the unconditional ceiling arguments with standard axioms only, including c<=0.3829 for the refined case. The paper-to-Lean certificate bridge, the maximal-correlation case and the exact 0.382885260 fixed point are outside Lean; the author reports checking claims but not a completed human proof review of the full paper theorem.

Human understanding Few

The available process account says the model found the central arguments and the author checked claims, but does not document a human explanation of the component-hiding mechanism. Few is an evidence-gap default.

Correction and publication history