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Declan Gessel, Vanishing minimal lemniscate area at critical transfinite diameter, a manuscript dated 6 September 2026 and published as a gist, with the byline "Research prepared with GPT-6 Astra in Codex, directed by Declan Gessel", claims the second question. The statement, as the manuscript and the formal statement give it, is that for every closed infinite $F\subseteq\mathbb C$ of transfinite diameter at least $1$, the infimum of the areas of $\{z:|p(z)|<1\}$ over monic polynomials $p$ of positive degree with all roots in $F$, repeated roots allowed, is $0$; the critical case of transfinite diameter exactly $1$ needs no smoothness or connectedness. The argument works with a compact $K$ of transfinite diameter at least $1$ and never identifies that diameter with logarithmic capacity. Section 2 takes maximizing (Fekete) configurations: monotonicity of the normalized Vandermonde maxima gives $V_n\ge1$ for every $n$, and along a subsequence where $\log V_n$ drops by at most $1$ per step, Lagrange interpolation bounds the Fekete polynomial from below outside $n$ disks of radius $1/n$; a weak limit $\mu$ of the empirical measures is a probability measure on $K$ whose potential is nonnegative almost everywhere and, by the minimum principle, positive off the filled hull of its support. Section 1 proves the continuity of the logarithmic kernel in planar $L^2$, which turns weak convergence of measures into norm convergence of their potentials and lets any density potential be approximated by an equal-weight mean over points of $K$. Sections 3 and 4 show that the potentials of bounded signed densities against $\mu$ approximate, in $L^2$ of the hull, every real harmonic polynomial annihilated by one normalized linear functional, and build such a polynomial through an explicit polynomial separator so that it is at least $1$ outside a strip of arbitrarily small area. Section 5 adds a large multiple of $\mu$ to make the density positive, normalizes to a probability measure whose potential has a positive margin outside a set of small area, and discretizes it to equal weights on $K$; the resulting monic polynomial has all roots in $K$ and a unit sublevel set inside a fixed disk of arbitrarily small area. Section 6 treats an unbounded $F$ by two roots far apart, whose quadratic has sublevel set inside two small disks. The manuscript says that the capacity-above-one result and the smooth capacity-one result were already known, that the proposed new contribution is arbitrary compact sets of capacity one, and that the first question, already refuted, is not claimed. The proof is not checked here.

Submission note. Posted to erdosproblems.com as a proof claim by Declan Gessel (account declangessel) on 6 September 2026, giving "GPT-6 Astra (Codex)" as the AI used:

We prove the remaining implication in part (ii): for every closed infinite set F in the complex plane with transfinite diameter at least 1, the infimum of the areas of {|p(z)|<1}, over positive-degree monic polynomials with all roots in F, is zero. Repeated roots are allowed. The critical diameter-one case requires no smoothness or connectedness. Fekete configurations supply a probability measure with nonnegative logarithmic potential. Planar L² duality and a polynomial separator yield a signed potential positive outside a small exceptional set. A positive tilt and equal-weight discretization then produce root polynomials with arbitrarily small sublevel area. Unbounded sets are handled by two widely separated roots. The complete statement is checked in Lean. Part (i) was already refuted by Aletheia; we claim no new result for that part. Notes: Extends the special cases of Krishnapur–Lundberg–Ramachandran and Ghosh–Ramachandran; references are in the manuscript. Jig's exact-statement and standard-axiom checks passed: https://jig.so/p/277 .

Covers. The second question: $\mu(F)=0$ for every closed infinite set of transfinite diameter at least $1$. The first question is not claimed; its negative answer is recorded on Aletheia's page.

Standing. The result was filed on the site's proof-claims tab on 6 September 2026; the tab names GPT-6 Astra (Codex). The Lean 4 proof is kept in the Jig verifier's submissions repository, and the Jig page the notes cite reports that the submission, filed by two contributors, passed the verifier's build, axiom (propext, Classical.choice and Quot.sound only), manifest, static-policy and anti-restatement checks under Lean v4.33.0 with a pinned Mathlib; the Jig problem, posed on 25 August 2026 and closed on 6 September 2026, is closed with the resolution "Independent rediscovery", whose prior-art entry is the preprint of Krishnapur, Lundberg and Ramachandran of 24 March 2025, described there as covering every compact set of unit capacity although that paper's Theorem 6 assumes a $C^2$ boundary; the statement carries a separate priority correction saying that shlummi's earlier public manuscript and Lean source cover the same full assertion, that the contributors have not rebuilt that project, and that they withdraw any claim of first public resolution and credit the earlier work while holding their own proof and its kernel verification valid. The manuscript says the same: historical priority and independent expert review remain separate from kernel verification, and the document supersedes an earlier equilibrium and polar-boundary manuscript. A pull request to formal-conjectures opened the same day proposes a statement file for the problem marked research solved and pointing to this proof; it was open on 2026-10-07, and a statement file is in any case not a formalization link. The Jig verdict is an outside kernel check, not a review by a named mathematician, and the corpus has built and audited nothing, so the claim lists no evidence and stays claimed; the site labels the problem OPEN. The two other claims of the same statement are Tzachristas's page and shlummi's page; an independent Lean development of the same day, for a local definition of the transfinite diameter, is Kitamura's page.