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Claim. For independent uniform signs, almost surely there are infinitely many even $n$ for which every real zero of $f_n(x)=\sum_{k\le n}\epsilon_kx^k$ lies in $[-1,1]$. With Do's almost-sure asymptotic for the zeros in $[-1,1]$ this gives $\liminf_{n\to\infty}R_n/\log n=1/\pi$ almost surely, so $R_n/\log n$ cannot converge almost surely to $2/\pi$ and Problem 521 has a negative answer for the $\{-1,1\}$ reading. The note A subsequence with no exterior real roots for random Littlewood polynomials by XianJun An and Vincent Lin was registered on the site's proof-claims tab on 2026-08-01 by the user nujan, with GPT 5.5 Pro named as the system used; the preprint lists Lin before An, and this page's name follows the order on the tab. The route is a deterministic Abel summation lemma, which shows that a simultaneous coordinatewise record of a two-dimensional random walk built from the coefficients (with $\epsilon_n=-1$) leaves no real root outside $[-1,1]$, and a recurrence argument for a two-dimensional reflected simple random walk giving infinitely many such record times, with a predictable-sampling independence argument for the coefficient unused at each record time.

Submission note. Posted to erdosproblems.com as a proof claim by XianJun An, Vincent Lin (account nujan) on 1 August 2026, giving "GPT 5.5 Pro" as the AI used:

The argument disproves the conjectured almost sure limit

$$\frac{R_n}{\log n} > \longrightarrow \frac{2}{\pi}$$

by combining Do's theorem,

$$> \frac{R_n([-1,1])}{\log n} \longrightarrow \frac{1}{\pi},$$

with the fact that

infinitely many random polynomials have no real roots outside $[-1,1]$. This comes via deterministic Abel summation lemma and another probabilistic lemma establishing infinitely many simultaneous coordinatewise record times of a two-dimensional random walk with $\varepsilon_n=-1$. So,

$$R_n = R_n([-1,1]) >$$

infinitely often, which implies

$$\liminf_{n\to\infty}\frac{R_n}{\log > n}=\frac{1}{\pi}.$$

Since

$$\frac{1}{\pi}<\frac{2}{\pi},$$

this contradicts

the conjectured almost sure convergence

$$\frac{R_n}{\log > n}\longrightarrow\frac{2}{\pi}.$$

Depends on. No page of this wiki.

Standing. Claimed. The claim has no comments on the tab, the site labels the problem OPEN (page last edited 19 October 2025), and no refereed or arXiv version exists. The same conclusion, by the cone-record route, is claimed on Kovač 2026, Kwon–Zou 2026 and Sneiderman 2026; Sneiderman's note cites an earlier reflected-walk proposal for the exact lower limit by the user vvncent, a Mathematics Stack Exchange question (5140475) of 2026-06-13, closed with no answers, whose title describes the same simultaneous-record event of a reflected two-dimensional walk. The Lean development on Alexeev 2026 lists this note, as the selected oscillation claim, among its informal sources.