---

Cristian Budala, Erdős Problem 1038: A Proposed Clean-Room Proof with Quantitative Near-Minimizers, dated 24 August 2026 and published in the author's repository (release v1.0.0), claims both extremal values. For $\mathcal F$ the nonconstant monic real polynomials with all roots in $[-1,1]$ and $\mu(f)=|\{x\in\mathbb R:|f(x)|<1\}|$, its Target Theorem 1.1 states that $\inf_{\mathcal F}\mu(f)=D=1.834430475762661711090753635124786143\ldots$, where $D$ is defined exactly by a finite transcendental system, that every $f$ has $\mu(f)>D$, that $\sup_{\mathcal F}\mu(f)=2\sqrt2$ with equality exactly for $(x^2-1)^m$, $m\ge1$, and that an explicit monic sequence $f_n$ of degree $n$ has $\mu(f_n)=D+O(\log n/\sqrt n)$. The argument replaces the roots in each component of the sublevel set by their barycenter, normalizes at the endpoint, compares the configuration with an explicit limiting measure whose potential is constant on its continuous support, reduces the comparison by a quantile-adjoint formula to a one-variable sign inequality checked on a frozen interval ledger of 512 cells, and discretizes the limiting measure by biased quantiles for the near-minimizers. Section 1.3 records that the sharp upper value had been proved earlier by Elbert (Elbert's page); the paper proves it again by its own argument. No independent check of the proofs is recorded.

Submission note. Posted to erdosproblems.com as a proof claim by Cristian Budala (account cristianbudala) on 24 August 2026, giving "GPT 5.6 Sol" as the AI used:

For monic real polynomials $f$ with all roots in $[-1,1]$, I propose that

$$> D = 1.83443047576266171109\ldots, >$$

and that this infimum is never attained. The proof first replaces the roots in each sublevel component by their barycenter. After normalization, the resulting configuration is compared with an explicit limiting measure whose logarithmic potential is constant on its continuous support. A quantile-adjoint formula reduces the comparison to a one-variable sign inequality. Discretizing the limiting measure by biased quantiles then gives finite polynomials converging to $D$. I also obtain the supremum $2\sqrt2$, attained exactly by $f(x)=(x^2-1)^m$ for $m\ge1$. Notes: Altough two separate full claims were made, I believe this version is still useful as an independent route: it uses a different argument, provides quantitative near-minimizers, and includes reproducible exact certificates. No priority claim is made. The research was done with extensive AI assistance. Paper | Source, certificates, and verification guide

Standing. The tab names GPT 5.6 Sol, and the paper's Section 1.4 says that AI systems were used extensively for exploration, drafting, code generation, adversarial checking and editorial organization, that every machine-certified assertion is decided by exact rational arithmetic or outward-rounded interval arithmetic, and that the disclosure is not a claim of peer review. The claim's notes present the work as an independent route, with no priority claim, after the two earlier full claims; the paper calls itself logically independent of their packages. The claim carries no comment, the site's label is unchanged (OPEN), no reviewer is named and nothing is refereed, so the claim stays claimed. The earlier claims of the same infimum are Wang's page and [[problems/analysis/E1038/claims/2026_07_15_darvas_peng_tao|the page of Darvas, Peng and Tao]].