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Claim. Let $O_n$ count the real zeros of $f_n(x)=\sum_{k\le n}\epsilon_kx^k$ outside $[-1,1]$ for independent uniform signs. Then $\mathbb{P}(O_n=0\text{ for infinitely many }n)=1$; with Do's strong law for the zeros in $[-1,1]$ this gives

$$\liminf_{n\to\infty}\frac{R_n}{\log n}=\frac{1}{\pi},\qquad \limsup_{n\to\infty}\frac{R_n}{\log n}\ge\frac{2}{\pi}$$

almost surely, so $R_n/\log n$ does not converge almost surely and Problem 521 has a negative answer for $\{-1,1\}$ coefficients. The note A cone-event obstruction to almost-sure convergence in Erdős Problem #521 by Yongchan Kwon and James Zou strengthens the elementary cone event (two simultaneous positivity conditions on random walks built from the reversed coefficients, which force every real zero into $[-1,1]$) from positive probability to an almost-sure statement by a Lévy zero–one argument applied to a quantitative quadrant-survival estimate, and obtains the upper limit from reversal symmetry and a deterministic subsequence. The repository's README says the two notes it holds reach their conclusions independently of the April 2026 thread postings and by different routes, and Kwon's comment on the site's thread for Problem 522 (2026-04-28) says the work used ChatGPT 5.5 Pro; that comment, which announced the first version of the Problem 522 note, is the only posting by either author on the site's forums and the discussion link this page carries. The repository carries a statement on the use of AI. The note was added to the repository on 2026-05-07, the date this page carries.

Depends on. No page of this wiki.

Standing. Claimed. No proof claim was registered on the site's proof-claims tab, the thread records no check of this note, and no refereed or arXiv version was found. The same mechanism, with a different zero–one step, is the Sneiderman 2026 claim; the first posting of the cone-record route is the note on Kovač 2026. The site labels the problem OPEN (page last edited 19 October 2025).