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Claim. Theorem 1 of Chvátal's conjecture: a proof from The Book states that a finite downset has an intersecting family of maximum size that is a star: the corrected Statement of Problem 701, which has a finite ground set. The proof is spectral. For an intersecting $\mathcal{I}\subseteq\mathcal{D}$, let $h$ be $+1$ on the upset generated by $\mathcal{I}$, $-1$ on the complements of its members and $0$ elsewhere, and let $H$ be the Fourier multiplier by $h$, so that $H(A,B)=\hat h(A\oplus B)$. Restricted to $\mathcal{D}$, $H$ has at least $|\mathcal{I}|$ eigenvalues $+1$ and $|\mathcal{I}|$ eigenvalues $-1$, so $2|\mathcal{I}|\le\sum_{A,B\in\mathcal{D}}H(A,B)^2$, while Parseval and a two-case count give $\sum_{A,B\in\mathcal{D}}H(A,B)^2\le\max_{T\ne\emptyset}|\mathcal{D}\cap(\mathcal{D}\oplus T)|\le2\max_i|\mathcal{S}_i|$. Version 2 of the note (5 October 2026) adds a spectral proof of Kleitman's conjecture along the same lines (Theorem 2); the note also proves a strengthening bounding the projection packing number and proposes two spectral conjectures supported by numerical experiments. The authors describe their argument as using ideas from the preprint of Chang, Liu and Liu (claim page), which they credit with the first proof, and record that Keevash (claim page) proved Kahn's conjecture in complete generality shortly after.

Claimant. David Ellis, Yuval Filmus and Ehud Friedgut. Their AI disclosure says that they had sought proofs of two spectral conjectures of their own with ChatGPT-6 Astra and Claude Fable 5.1 without success; that after the preprint of Chang, Liu and Liu appeared they gave it to ChatGPT-6 Astra, which produced a proof that the projection packing number is tight for every downset and, when asked to strip it down, the spectral proof the note presents; and that they prepared the final paper themselves, using ChatGPT only for proofreading. The note has no proof-claims entry of its own on the erdosproblems.com forum; the entry of 30 September 2026 for the first proof names it as related work.

Acceptance. None: the note is not refereed, no outside reviewer has endorsed it, and the site labels the problem OPEN (2026-10-07).