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Claim. Let $f$ be a nonzero entire function that is not a monomial, and write

$$f(z)=z^m\bigl(c_0+c_kz^k+O(z^{k+1})\bigr),\qquad m\ge0,\ k\ge1,\ c_0c_k\ne0,$$

so that $k$ is the gap between the exponents of the first two nonzero terms. Theorem 1.1 of the manuscript A negative answer to a question of Erdős on maximum modulus points asserts that there is a countable set $E\subset(0,\infty)$ with $\nu(r)\le2k$ for every $r\notin E$, where $\nu(r)$ counts the points of $|z|=r$ at which $|f|$ attains its maximum modulus as on Problem 1117. Hence $\liminf_{r\to\infty}\nu(r)\le2k<\infty$: no non-monomial entire function has $\nu(r)\to\infty$, which is a negative answer to the problem's second question. The manuscript says the factor $2$ is attained by an explicit example (its Remark 5.1). The route: at radii where $\log M(r,f)$ is differentiable in $\log r$, every maximum modulus point has the same real value of $A=zf'/f$, so the points lie in one fiber of the correspondence $A(z)=\overline{A(\overline w)}$ with a common value of $zw$; every irreducible component of that correspondence is shown to contain points with $|zw|$ arbitrarily small, by continuation of implicit functions (the Iversen property, cited to Stoïlow) and growth in a direct tract (a lemma of Fuchs, cited through Bergweiler, Rippon and Stallard); the map $(z,w)\mapsto(zw,A(z))$ is then finite and proper, and since $A-m$ has local degree $k$ at the origin, a degree count near the coordinate axes bounds the generic fibers by $2k$. This page states the manuscript's theorem and the route its introduction describes; the proof is not verified in this corpus.

Submission note. Posted to erdosproblems.com as a proof claim by Qiyuan Gu (account fireflysentinel) on 5 September 2026, giving "GPT 6 Astra, GPT 5.6 Sol and Claude Opus 5" as the AI used:

The proof studies the logarithmic derivative A(z) = zf′(z)/f(z), giving a negative answer to Erdős's second question on maximum modulus points. At radii where log M(r,f) is differentiable, all maximum-modulus points have the same real value of A, so they lie in one fibre of the correspondence A(z) = overline{A(overline{w})}. The key step is to show that every irreducible component of this correspondence contains points with |zw| arbitrarily small, using analytic continuation of implicit functions and growth in a direct tract. The map (z,w) ↦ (zw, A(z)) is then finite and proper, and a local degree-k count near the coordinate axes gives the bound 2k. Notes: GPT-6 Astra was used to generate the mathematical proofs and draft the manuscript. GPT-5.6 Sol and Claude Opus 5 was used only for editorial review of the exposition and did not contribute to the mathematical arguments. The author reviewed the final manuscript and takes full responsibility for its content.

Covers. The second question, whether $\liminf\nu(r)=\infty$ is possible: the theorem answers it no. The first question, whether $\limsup\nu(r)=\infty$ is possible, is outside this claim; its affirmative answer by Herzog and Piranian (1968), which the manuscript cites, is on their page.

Standing. The claim was posted to the site's proof-claims tab on 5 September 2026 on the author's behalf, crediting Qiyuan Gu, with a Zenodo record as the write-up. The version of the record that the tab links was removed on 23 September 2026, its tombstone says; the record's surviving version (v6, 6 September 2026), reached through the concept DOI in the link, lists its author as anonymous, carries the title A bound for the number of maximum modulus points, and holds the PDF (titled as above) and a Lean archive. The attribution to Qiyuan Gu is the site's. The thread has no comments, the site labels the problem OPEN (page last edited 29 December 2025), no reviewer independent of the author has endorsed the manuscript, and it has no refereed publication. The claim therefore stays claimed.

AI systems. The claim's notes on the site say that GPT-6 Astra generated the mathematical proofs and drafted the manuscript, that GPT-5.6 Sol and Claude Opus 5 were used only for editorial review of the exposition, and that the author reviewed the final manuscript and takes responsibility for it; the Lean archive's README adds that its formalization was generated with OpenAI Codex (GPT-6).

Formalization. The Lean archive in the record (toolchain leanprover/lean4:v4.33.0) describes itself as a Lean 4 formalization of the manuscript. Its README says that the theorem is derived, as theorem_1_1_of_entire_component_data, from two component properties of the correspondence that remain hypotheses of the manuscript's analytic argument, so the analytic core of the proof is not formalized; a bridge file states the negative answer from those two named hypotheses. This corpus has not built or kernel-checked the development, and the archive awards nothing here.

Depends on. Nothing on the wiki: the theorem rests on the cited manuscript alone.