Wang 2026 proposed complete solution erdos problem 1038

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Shouqiao Wang, A Proposed Complete Solution to Erdős Problem 1038. preprint (GitHub repository github.com/ShouqiaoW/erdos) (2026). The file prints no copyright or license line of its own; the repository holding it carries a LICENSE file that GitHub shows as "MIT license", the MIT License for the repository as a whole, a software license applied here to a paper (https://github.com/ShouqiaoW/erdos, read 2026-10-02).

This is a claimed solution rather than a verified one. For f ranging over nonconstant monic real polynomials with all zeros in [-1,1] and E_f = {x : |f(x)| < 1}, Tao had determined sup |E_f| = 2 sqrt 2; the preprint claims to determine the infimum, stating (Theorem 1.1, Main theorem) that an explicit one-variable function Lambda has a unique minimizer q_* on (0, q_s] and that inf |E_f| = L = Lambda(q_) = 1.834430475762661..., with certified outward enclosures 1.834430475762661 < L < 1.834430475762662 and 0.025715536866527 < q_ < 0.025715536866528. The infimum is claimed not to be attained, every finite polynomial satisfying the inequality strictly, while the supremum 2 sqrt 2 is attained by (x^2-1)^m for all m >= 1. For the lower bound, the roots are collapsed to one atom per component of the sublevel set, a direct energy estimate handles small endpoint-to-residual mass ratios, a convex comparison with a constant-platform reference measure and an endpoint-corrected adjoint handle the main component, and a circle rearrangement inequality (Theorem 5.1, circle block inequality) reduces each residual quantile block to one-variable inequalities. The scalar and parameter-uniform signs left over are certified by directed outward interval arithmetic in a companion Python file, and together these give Theorem 8.1 (uniform lower bound); sharpness is claimed through empirical measures with a positive platform approaching the one-cut extremal from above. The starting point is Tao's updated note, whose structural reductions for the lower problem (the componentwise barycentric reduction among them) partly adapt an argument of Erdős, Herzog and Piranian. The document states that the proposed solution was found by GPT-5.6. Its bearing on #1038 is the claimed computer-assisted determination of L = 1.8344...; for the open #114, it is a technique candidate from the same Erdős–Herzog–Piranian family.

Source: https://github.com/ShouqiaoW/erdos/tree/main/1038.

Bears on. #114, #1038

Results to transcribe.

• Theorem 1.1 (Main theorem): Lambda has a unique minimizer q_* on (0, q_s]; inf |E_f| = L = Lambda(q_*) with 1.834430475762661 < L < 1.834430475762662, not attained, while sup |E_f| = 2 sqrt 2 is attained by (x^2-1)^m for every m >= 1.
• Theorem 5.1 (Circle block inequality): A circle rearrangement inequality stated for every nonempty angular interval, which reduces each residual quantile block of the lower-bound comparison to explicit one-variable inequalities.
• Theorem 8.1 (Uniform lower bound): Every f in the class satisfies |E_f| > L; its proof uses Lemma 2.1, Proposition 3.1, Certificate 7.1, Theorem 5.1 and Proposition 4.4, with signs certified by directed outward interval arithmetic in a companion Python file.
• Theorem 10.1 (Sharp upper bound; Tao): Every f in the class satisfies |E_f| <= 2 sqrt 2, with equality exactly for f = (x^2-1)^m, m >= 1; deduced from Tao's theorem for probability measures with its equality characterization.
• Certificate 6.1 (One-cut global minimization): Lambda has exactly one stationary point q_* on (0, q_s], with the enclosures of Theorem 1.1 and 0.123630684649383 < q_s < 0.123630684649384; L is obtained by certified one-dimensional root isolation from F(q,u) = A(q) log((u-q)/|1-qu|) - log u and W(u) = u + 1/u, not by grid minimization.