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Claim. The first two questions of Problem 514 have the answer yes. P. Chojecki, A note on an Erdős path problem for transcendental entire functions, a seven-page note dated 20 April 2026 and posted in the problem's discussion thread the same day, proves in its Theorem 1 that every transcendental entire function $f$ has a path to infinity $\gamma$ with

$$\frac{\log\lvert f(\gamma(t))\rvert}{\log\lvert\gamma(t)\rvert}\to\infty \quad(t\to\infty), \qquad\text{so}\qquad \Bigl\lvert\frac{f(\gamma(t))}{\gamma(t)^n}\Bigr\rvert\to\infty \text{ for every fixed }n,$$

and that the same path satisfies $\ell_\gamma(R)=O(M(R,f)^\varepsilon)$ as $R\to\infty$ for every $\varepsilon>0$, where $\ell_\gamma(R)$ is the length of the initial segment of $\gamma$ up to its first exit from the disk $\lvert z\rvert<R$ and $M(R,f)$ the maximum modulus. The proof applies Theorem B of Wu's 1985 paper (quoted as the note's Theorem 4) to $u=\max\{\log\lvert f\rvert,-1\}$, whose hypothesis $\max_{\lvert z\rvert=r}u/\log r\to\infty$ follows from Cauchy's estimate for a transcendental $f$ (Lemma 3); the theorem gives the path and the convergence of $\int_\gamma e^{-\delta u}\lvert\mathrm{d}z\rvert$ for every $\delta>0$, and the length bound follows by writing $1=e^{\varepsilon u}e^{-\varepsilon u}$ along the segment and bounding $u$ by $\log M(R,f)$ there. The note's Theorem 2 constructs, from Theorem 1.4 of Langley's 2022 paper, one transcendental entire $f=e^G$ such that for every unbounded connected set $E$ and every $\varepsilon>0$ there are $w_k\in E$ with $\lvert w_k\rvert\to\infty$ and $\lvert f(w_k)\rvert\le M(\lvert w_k\rvert,f)^\varepsilon$; so no fixed power of $M(r,f)$ is a lower bound along a path for every $f$, which answers the power example of the third question no. By the note's own Remark 7 this does not settle the third question as the problem page reads it, with an arbitrary fixed comparison function; that reading is answered on Oriike's page. The thread post announcing the note says that GPT-5.4 Pro produced the proof. The statement follows the note's print.

Submission note. Posted to the site's forum by Przemek Chojecki on 20 April 2026:

GPT-5.4 Pro claims to have a proof of the first question (+ second) and a negative answer to the third. Here are the notes. Mostly used is a pretty recent result of Langley, and also Wu.

Covers. The first question (the part path), answered yes, and the second (the part length), answered yes with the bound $\ell_\gamma(R)=O(M(R,f)^\varepsilon)$ for every $\varepsilon>0$. Not covered: the third question (the part growth); Theorem 2 refutes only its power example.

Depends on. [[problems/analysis/E0514/claims/1984_12_01_lewis_rossi_weitsman|Lewis, Rossi and Weitsman 1984]], the path theorem the note applies in Wu's restatement, an accepted page; the length estimate and Theorem 2 are the note's own, with Langley's theorem as outside input.

Standing. The note was posted in the site's discussion thread on 20 April 2026 and not on the proof-claims tab; the site's label is OPEN and its page was last edited on 18 January 2026, before the note, so its commentary does not mention it. In the thread, Sothanaphan reported on 20 April 2026 that a check found no issue and that the note settles the first two questions and only the power version of the third, and on 10 May 2026 that they had worked through the proof of the first two questions and could vouch for it, describing the result as a short derivation from Wu's Theorem B applied to $\max\{\log\lvert f\rvert,0\}$, and posted a streamlined version of the argument produced with GPT-5.5 Thinking; those are thread posts and are not listed as reviewed evidence. On 5 June 2026 Chojecki announced in the thread a joint paper with Oriike collecting the arguments for the whole problem; no such preprint is recorded. The note is not refereed and not formalized, and the claim stays claimed.