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Claim. Let $x_{1,n}<\cdots<x_{n,n}$ be any triangular array of distinct nodes in $[-1,1]$, with rows that need not be nested, and let $\lambda_n(x)=\sum_k\lvert\ell_{k,n}(x)\rvert$ be its Lebesgue function. The $L_n$ of Problem 1132 is the Lebesgue function of the nested rows $x_1,\ldots,x_n$ of one infinite sequence, a special case of such an array. The write-up Sharp pointwise bounds for Lebesgue functions (Zenodo; the link above is version 9 of 2026-09-07, under the concept record 10.5281/zenodo.22322281, which lists its creator as anonymous) states as Theorem 1 that, for every such array, (i) the set of $x\in(-1,1)$ for which some finite constant $C(x)$ gives

$$\lambda_n(x)>\frac2\pi\log n-C(x)$$

for infinitely many $n$ is dense in $(-1,1)$, and (ii) for Lebesgue almost every $x\in(-1,1)$,

$$\limsup_{n\to\infty}\frac{\lambda_n(x)}{\log n}\ge\frac2\pi .$$

Both parts apply to the Statement's setting, since a sequence is a special array. Part (ii) is the second question of the problem as asked. Part (i) answers the first question in the reading where the $O(1)$ term may depend on the point $x$ and on the array; the write-up says so in its introduction, and the notes of the forum submission say the same. The same record carries a companion note by the claimant, Nonuniform additive constants for Lebesgue functions, whose Theorem 1 concerns triangular arrays whose rows are not nested, Erdős's question for every point group in [Er67]: for every $M>0$ there is such an array with $\lambda_n(x)\le\frac2\pi\log n-M$ for all large $n$ at every fixed $x\in(-1,1)$, the threshold depending on $x$, and the array can be chosen so that for every finite $C$ the set of points where the bound with $C$ holds infinitely often is not dense. Its Section 4 builds row $n$ from the $n$ roots of a polynomial whose parameter changes along a sequence of thresholds, so consecutive rows share no nodes and the construction is not a single sequence. So, as claimed, the first question has the answer yes with a point-dependent constant; for non-nested arrays no constant independent of the array can serve; and for a single sequence, as the Statement poses the question, the uniform reading is not settled by either document. The Statement does not fix the dependence, a gap Tao [Ta26b] had already pointed out, and a comment on the forum claim of 2026-09-05 objects that the claim answers only the weaker, point-dependent variant and that the stronger one remains open. The claimant posted the claim as a full one, treating the point-dependent reading as the question. In the reading the problem page's Formulation records, it would settle the second question only, so this page records it as partial. The proof of (ii) is described as self-contained, through cancellation in positive Cauchy transforms and a harmonic-measure identity; the proof of (i) applies the local estimates of Tao's local Bernstein theory ([Ta26b], card tao_2026_local_bernstein_theory_lower_bounds_lebesgue) with Riesz's differentiation formula, a Riesz energy bound and Baire's theorem. The write-up places itself after Erdős's dense-set statement in his Mathematica (Cluj) paper (the site's [Er67], p. 68; the site credits the density to Bernstein [Be31]), Erdős's $O(1)$ bound for the maximum of $L_n$ [Er61c], the almost-everywhere positive coefficient of Erdős and Vértesi, and Tao's Corollary 1.11, which gives a dense set with any loss $\omega(n)\to\infty$ in place of a constant.

Submission note. Posted to erdosproblems.com as a proof claim by Qiyuan Gu (account fireflysentinel) on 5 September 2026, giving "GPT-6 Astra, GPT-5.6 Sol, Claude Opus 5" as the AI used:

For an arbitrary triangular array of distinct interpolation nodes in [−1, 1], let λₙ be the corresponding Lebesgue function. The note proves that there is a fixed x ∈ (−1, 1) and C such that λₙ(x) > (2/π) log n − C for infinitely many n, and that lim supₙ→∞ λₙ(x) / log n ≥ 2/π for almost every x. The first result combines Tao’s local Bernstein theory with an energy estimate for derivative jumps and a second-moment argument. The almost-everywhere result uses positive Cauchy transforms, harmonic measure, and a symmetric approximate-identity argument. Notes: For the first question, the (O(1)) constant is allowed to depend on the node array and on the point (x), as noted explicitly in the writeup. GPT-6 Astra was used to generate the mathematical proofs and draft the manuscript. GPT-5.6 Sol and Claude Opus 5 were used for editorial review of the exposition. The author reviewed the final manuscript and takes full responsibility for its content.

Covers. The second question: $\limsup_n L_n(x)/\log n\ge2/\pi$ for almost every $x\in(-1,1)$. It does not cover the first question as the problem page's Formulation reads it, with one absolute constant; Theorem 1(i) gives only the variant whose constant depends on the point and the sequence.

Depends on. No page of this wiki: the write-up's inputs are Tao's local estimates and classical potential theory, cited from the literature.

Standing. A manuscript claim, claimed. The claimant is Qiyuan Gu, named on the site's proof-claims tab, where the claim was submitted on 2026-09-05 by another account on Gu's behalf with the disclosure that GPT-6 Astra generated the proofs and drafted the manuscript, GPT-5.6 Sol and Claude Opus 5 reviewed the exposition, and the author reviewed the final manuscript and takes responsibility for it; the Zenodo record lists its creator as anonymous. Version 9 of the record carries a Lean 4 archive said to formalize both theorems and the companion note; no build or audit of the archive is recorded, so the archive is a link and contributes no formalized evidence. Not reviewed: the site's label is OPEN (page last edited 01 April 2026; the tab, with one comment on the claim), the site's commentary does not mention the claim, and no outside review is known. Not refereed: the write-up has no journal or arXiv version. The theorem statements above are those of the write-up and the companion note; their proofs have not been reviewed.