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The author publishing under the pen name shlummi, Small-area polynomial lemniscates with prescribed roots, public review manuscript version 1 of 6 September 2026, claims the second question. The full theorem states that if a closed infinite $F\subseteq\mathbb C$ has geometric transfinite diameter $\rho(F)\ge1$, defined as the limit of the normalized maximal Vandermonde products $\delta_N(F)$, then for every $\varepsilon>0$ there is a monic polynomial $p$ of positive degree with every root in $F$ whose strict sublevel set $\{z:|p(z)|<1\}$ has area below $\varepsilon$; a corollary gives the same conclusion for every compact set of logarithmic capacity at least $1$, including arbitrary compact sets of capacity exactly $1$, with no boundary regularity. The argument constructs a finite-energy minimizing probability measure whose potential is nonnegative everywhere and positive outside a compact filled hull, approximates mean-zero harmonic polynomials by density potentials in planar $L^2$, shifts a density to a positive probability with a positive potential margin on almost all of the observation area, discretizes it to actual roots in the set, and handles unbounded sets separately. An appendix formalizes the known negative answer to the first question, which the manuscript does not claim as new. The manuscript credits Ghosh and Ramachandran with the compact case of capacity above $1$ and Krishnapur, Lundberg and Ramachandran with the regular capacity-one case and the potential-approximation strategy. The statements are those of the manuscript at the pinned revision; the proofs are not checked here.

Submission note. Posted to erdosproblems.com as a proof claim by shlummi (account shlummi) on 6 September 2026, giving "OpenAI Codex (proof development and Lean formalization); DeepSeek and Claude/Opus (additional AI review)" as the AI used:

Notes: Publishing name: shlummi (pen name). I coordinated AI-assisted proof development, formalization and checking. This submission and its documentation were prepared with AI assistance. Independent human peer review is not claimed. The repository links the paper and the complete Lean source ZIP, pins dependency revisions, and provides reproduction instructions and verification records. The recorded checks cover 274 tracked declarations. Their same-host verification and dependency-cache reuse are disclosed in VERIFICATION.md. The optional mathematical summary is left blank because this form requests a human-written summary. The detailed argument and exact theorem statements are in the linked artifacts. The paper distinguishes its threshold assertion from the already-known negative answer to the separate capacity-only dependence question; the included counterexample is not claimed as new.

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 with no summary; the tab's tools line names OpenAI Codex for proof development and Lean formalization and DeepSeek and Claude/Opus for additional AI review, and the manuscript says that no comprehensive independent human review is claimed and that it claims neither journal acceptance nor acceptance by the site. The thread carries no comment on the claim, the site labels the problem OPEN, no reviewer is named and nothing is refereed, so the claim stays claimed. The two other claims of the same statement are Tzachristas's page and Gessel's page.

Formalization. The repository holds the Lean 4 sources (54 files, 274 tracked declarations: 241 theorems and 33 definitions, Lean v4.33.1 with a pinned Mathlib revision) and a verification record describing same-host checks and dependency-cache reuse. The manuscript itself says that the checking records concern the exact Lean statements and that correspondence with the informal problem is a separate assessment. The corpus has built and audited nothing, so the claim lists no formalized evidence.