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Claim. The third question of [[problems/analysis/E0514/_index|Problem 514]], read with the comparison function fixed before the entire function is chosen, has the answer no. Yuta Oriike, A negative answer to the universal-function version of Erdős's third question in Problem 514, a note dated 28 April 2026 and revised on 10 May 2026, proves in its Theorem 1 that for every nondecreasing $\Phi:[T_0,\infty)\to(0,\infty)$ with $\Phi(T)\to\infty$ there is a transcendental entire function $f$ such that every path to infinity $\gamma$ satisfies

$$\liminf_{t\to\infty} \frac{\lvert f(\gamma(t))\rvert}{\Phi(M_f(\lvert\gamma(t)\rvert))}=0, \qquad M_f(r)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert.$$

Its Corollary 1 draws the consequence: no function $\Psi(T)\to\infty$ chosen independently of $f$ makes $\lvert f(z)\rvert/\Psi(M_f(\lvert z\rvert))\to\infty$ along some path for every transcendental entire $f$, and in particular no power $\Psi(T)=T^\varepsilon$ does. The revision derives Theorem 1 from Hayman's theorem on the growth of entire functions along asymptotic paths (W. K. Hayman, On the growth of integral functions on asymptotic paths, J. Indian Math. Soc. (N.S.) 24 (1960), 251--264, Theorem 2), which gives an entire function of infinite lower order with $\log M(r,f)>e^{\lambda(r)}$ for a prescribed increasing $\lambda$ with $\log\lambda(r)/\log r\to\infty$ while $\log\log\lvert f\rvert=O(\log r)$ along every asymptotic path, after a selection lemma (its Lemma 1) that chooses $\lambda$ so that $\Phi(e^{e^{\lambda(r)}})\ge e^{r^k}$ for every $k$; the revision says that Hayman's theorem alone refutes the power example and keeps the note's original direct construction, an explicit lacunary series with alternating signs (its Lemma 2), as a second, self-contained proof of the same theorem. The thread post announcing the note says that GPT-5.5 Pro produced the proof and its Lean formalization. The statement follows the two versions' print. The first two questions are not addressed in the note; they are answered on [[problems/analysis/E0514/claims/1984_12_01_lewis_rossi_weitsman|Lewis, Rossi and Weitsman 1984]] and Chojecki's page, whose Theorem 2 refutes the power example independently.

Covers. The third question (the part growth) in the reading the problem page adopts, a comparison function fixed independently of $f$, answered no; the power example $M(r)^\varepsilon$ is the case $\Psi(T)=T^\varepsilon$. Not covered: the existence and length of a path on which $f$ outgrows every power of $z$.

Depends on. Nothing in this wiki: the first proof rests on Hayman's published theorem and the second is the note's own construction.

Standing. The note and the Lean file were posted in the site's discussion thread on 28 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. The note prints no author's name; it was posted by the forum user YutaOriike and is held in the GitHub repository of the user yuta0x89. In the thread, Sothanaphan reported on 29 April 2026 that a check found no issue and that the Lean file corresponds to the stated result; that is a thread post and is not listed as reviewed evidence. The revision of 10 May 2026 followed a thread comment of 1 May 2026 that the result was close to Hayman's theorem; it changes attribution and presentation, not the theorem. The Lean file Erdos514.lean, linked above at its revision of 28 April 2026, declares in its header that it formalizes the original note, builds on Lean 4.28.0 with Mathlib, and proves the theorem in an equivalent epsilon form (comparison_negative_sphere); it has not been built or audited in this repository, so it is a link and not formalized evidence. The note is not refereed, and the claim stays claimed.