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Claim. For every $\varepsilon\in(0,1)$ there is $N_0$ such that every length $N\ge N_0$ admits signs $\varepsilon_0,\dots,\varepsilon_{N-1}\in\{-1,1\}$ with

$$(1-\varepsilon)\sqrt N\le\Bigl\lvert\sum_{k=0}^{N-1}\varepsilon_k z^k\Bigr\rvert \le(1+\varepsilon)\sqrt N \qquad\text{for every }\lvert z\rvert=1,$$

the real points $z=\pm1$ included (the main theorem of the release's manuscript Ultraflat real Littlewood polynomials, 2026-10-05, carded as [[../library/polynomials/openai_2026_ultraflat_real_littlewood_polynomials/_index|Ultraflat real Littlewood polynomials]]). A companion manuscript of the same date, Nearly minimal maxima and positive minima of Littlewood polynomials, carded as [[../library/polynomials/openai_2026_nearly_minimal_maxima_positive_minima_littlewood_polynomials/_index|Nearly minimal maxima and positive minima of Littlewood polynomials]], claims the weaker two-sided bound $\sqrt N/16\le\lvert P(z)\rvert\le(1+\eta)\sqrt N$ for every $\eta>0$ and every large $N$. The release names OpenAI as the author of both manuscripts; its README says that its manuscripts and proof artifacts were produced by an internal OpenAI model and that the collection includes results at different stages of verification. Either statement answers Problem 228 yes for every large degree and is strictly stronger than the accepted answer of [[problems/polynomials/E0228/claims/2019_07_22_balister_bollobas_morris_sahasrabudhe_tiba|Balister et al. 2020]], whose constants are fixed but far from $1$ (its Theorem 2.1 gives $2^{-160}\sqrt n\le\lvert P(z)\rvert\le2^{12}\sqrt n$ on the circle for centered polynomials $\sum_{k=-2n}^{2n}\varepsilon_kz^k$, of degree $4n$): the ultraflat theorem forces the ratio $\lvert P(z)\rvert/\sqrt N$ to $1$ uniformly on the circle, and the companion improves the constants to $1/16$ below and $1+\eta$ above. The same family bears on Problem 1150, which asks whether the maximum modulus of every such polynomial exceeds $(1+c)\sqrt n$ for one fixed $c>0$; that page carries its own account. The release describes its route as relaxed coefficients in $[-1,1]$ with small defect, a coefficient cap built from quadratic-phase waves sampled off an auxiliary torus polynomial with Pippenger–Spencer packing, and a defect-sensitive rounding to signs by the Spencer and Lovett–Meka partial colorings.

Depends on. No page of this wiki.

Standing. The claim is a manuscript statement and stays claimed. The release's lean/ folder (the formalization link, at the pinned revision) proves two statements from the earlier manuscript Asymptotically minimal maxima of real Littlewood polynomials (2026-09-23, carded as [[../library/polynomials/openai_2026_asymptotically_minimal_maxima_real_littlewood_polynomials/_index|Asymptotically minimal maxima of real Littlewood polynomials]]): OAI.AsymptoticallyMinimalLittlewood.main, that for every $\eta>0$ and every large $N$ some real signing has $\max_{\lvert z\rvert=1}\lvert P(z)\rvert\le(1+\eta)\sqrt N$, and OAI.AsymptoticallyMinimalLittlewoodFiniteFlatness.main, that one all-length family of real signings has the $L^p$ mean of $\bigl\lvert\lvert P_N(z)\rvert/\sqrt N-1\bigr\rvert$ over the circle tending to $0$ for every finite $p>0$; the comparator challenges AsymptoticallyMinimalLittlewood.lean and LittlewoodFiniteFlatness.lean of the same folder pin the two statements. Neither declaration states a lower bound for $\lvert P(z)\rvert$ on the circle, which is the half of the question that the accepted proof made hard and that the $L^p$ statement leaves open (an $L^p$ mean near $0$ allows zeros on the circle); the two October 5 manuscripts that carry the lower bound have no Lean in the release. The release's family document for that formalization says its results are existential, with no effective rate and no signing algorithm. Neither declaration bounds $\lvert P(z)\rvert$ from below, so the formalization settles nothing this problem asks. An upper bound of order $\sqrt N$ is classical (Rudin–Shapiro); the release's sharper $(1+\eta)\sqrt N$ bound answers [[problems/polynomials/E1150/_index|Problem 1150]] but does not touch the lower half of the question. The formalization link is therefore recorded for the stronger claim's provenance and contributes no formalized evidence. No outside review, referee report or acceptance by the site is recorded.