---

Claim. For independent uniform signs $\epsilon_k\in\{-1,1\}$, with $R_n$ the number of real roots of $f_n(x)=\sum_{k\le n}\epsilon_kx^k$,

$$\mathbb{P}\Bigl(\liminf_{n\to\infty}\frac{R_n}{\log n}\le\frac{1}{\pi}\Bigr)>0,$$

so $R_n/\log n\to2/\pi$ cannot hold almost surely, which answers Problem 521 in the negative for the $\{-1,1\}$ reading. The note On Erdős problem #521 for Kac polynomials, dated 2026-04-29, names ChatGPT 5.5 Pro as its author; Vjekoslav Kovač posted it on the site's discussion thread on 2026-04-30 as a raw, unpolished and undigested write-up generated by that system, and the corpus records Kovač as the claimant because Kovač published it. Its general route, for coefficient laws satisfying Do's moment hypotheses, reduces the question to two persistence inputs (a one-time bulk gap estimate of Gaussian order and a two-time correlation estimate), which it verifies for standard Gaussian coefficients. For the problem's coefficients it uses instead a cone-record criterion: when the planar walk built from the reversed coefficients has a divergent cone-survival series, the Kochen–Stone lemma gives infinitely many $n$ at which every real root lies in $[-1,1]$, and Do's strong law $R_n[-1,1]/\log n\to1/\pi$ then bounds the lower limit by $1/\pi$ on an event of positive probability. The note credits the thread's ideas: Letwin's Gaussian-process limit of the rescaled polynomial near the endpoints, Tao's buckets, and Kovač's suggestion to look for degrees with no root outside $[-1,1]$. An earlier note by the same route, posted by Kovač on 2026-04-25 (Erdos_521_Gaussian.pdf), treats standard Gaussian coefficients only, which answers a question of Pritsker rather than this problem; Kovač reported that the system produced that note after Kovač's guidance and much proofreading and correction, and Nat Sothanaphan's AI check of it found no issues.

Depends on. No page of this wiki.

Standing. Claimed. Kovač posted the note as an unpolished draft that Kovač had not digested, reserved no credit, and wrote that Letwin, whose ideas were the most developed, should take the project forward; no proof claim was registered on the site's proof-claims tab, no check of this note is posted on the thread, and no refereed or arXiv version exists. Later notes build on it: the Kwon–Zou note on Kwon–Zou 2026 and Sneiderman's note on Sneiderman 2026, which cites it as a public working note without assigning human authorship; the Lean development on Alexeev 2026 lists Section 7 of this note (cone records) among its informal sources. The site labels the problem OPEN (page last edited 19 October 2025).