Panel review: A uniform influence bound for hypergraph properties

Independent model assessments of the frozen manuscript. Model conclusions are not human proof grades. Each model received only the supplied manuscript; external-reference inspection was not enabled. Baseline-v2 and source-first-v2 share the typed reference schema and differ in experimental review strategy.

Returned findings

gpt-6.1-sol / baseline-v2

All five stated mathematical elements have complete, valid proofs on the supplied text; no demonstrated error or substantive logical gap was found. The argument is self-contained and handles nonmonotonicity, arbitrary bias, real degree cutoffs, and small vertex counts explicitly. The historical attribution, comparisons with prior bounds, and identification of the threshold result with the cited conjecture were not independently checked against the external references.

gpt-6.1-sol / source-first-v2

All five theorem-like statements have complete, valid proofs in the supplied manuscript. The central restriction-to-capture transfer and the uniform treatment of extreme biases withstand direct checking, and the proof does not depend on uninspected external theorems. The historical comparisons and the precise identification of the cited conjecture remain outside this manuscript-only audit.

claude-opus-5-5 / baseline-v2

The paper is short and self-contained, and every step I checked holds. This covers the biased (2,4) inequality, the KKL-type low-degree bound, the Sym(B)-restriction averaging, the minimum-degree capture lemma, the bias bookkeeping and the Margulis–Russo integration. I found no mathematical error or logical gap. The proof uses no external theorem as a black box; citations serve only as attribution and context.

claude-opus-5-5 / source-first-v2

Every step of the proof was recomputed and checks out under the stated hypotheses: the hypercontractive constants, the orbit-size argument, the Fourier averaging identity, the degree-counting capture lemma, the bias cancellation, and the small-$n$ case. The argument is self-contained; the cited companion and classical works serve as background, not imported proof inputs. No demonstrated error was found. Remaining caveats are attributional (the exact form of the Friedgut--Kalai conjecture, not checked against FK96) and the non-explicit constant.

gemini-4-argon / baseline-v2

This is an exceptionally clear, self-contained, and rigorous paper that proves a uniform influence bound with exponent r/(r-1) for symmetric Boolean functions on r-uniform hypergraphs and resolves the Friedgut-Kalai threshold-width conjecture. All proofs are complete, elementary, and verified without any gaps.

gemini-4-argon / source-first-v2

This is an exceptionally clear, self-contained, and mathematically rigorous paper. Every lemma, theorem, and corollary is proved from first principles with no logical gaps, hidden assumptions, or unverified external dependencies. The combination of biased 2-to-4 hypercontractivity, vertex-block restrictions, and hypergraph degree bounds cleanly resolves the Friedgut-Kalai threshold-width conjecture for r-uniform hypergraphs.

Remaining assessment

Human substantiation, localization, evidence, uncertainty and coverage grades remain pending in the separate anonymous comparison packet. Finite output caps and failed responses are resource outcomes, not mathematical verdicts. External publications and claimed formalizations have not been independently audited.