Price: Perturbed Lobatto nodes

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theorem_1: Claimed in an unsigned write-up credited to GPT-5.5 Pro: for some epsilon > 0 and n_0, the least integral over [-1, 1] of the sum of the squared Lagrange cardinal polynomials of n nodes in [-1, 1] is at most 2 - (1 + epsilon)/n for every n >= n_0, witnessed by the roots of the integral of P_{n-1} + a P_{n-3} for a fixed small a > 0.

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Perturbed Lobatto nodes, preprint (2026), 11 pp., posted by Liam Price. The print has no author line, no date, no acknowledgment and no bibliography, and names no AI system; its information dictionary has no author entry and a creation timestamp of 14 October 2026, 22:19 UTC, consistent with the compile that produced the copy read here. The attribution is the erdosproblems.com thread's: Liam Price linked the project in the Problem 1131 discussion thread on 26 April 2026 with the words "GPT-5.5 Pro claims a disproof of the claimed asymptotic here", as the claim page records. No notice is printed in the eleven pages, which state no license; the hosting site's terms speak for the site, not the paper: Overleaf's terms page (https://www.overleaf.com/legal), as read on 2026-10-02 for the card of Price's counterexample to Problem 346, says "We don't claim any ownership of your stuff" and grants readers of a shared project no license; the term is unstated.

Theorem 1 (p. 1) concerns $M_n$, the least value over nodes $x_0,\dots,x_{n-1}\in[-1,1]$ of $\int_{-1}^1\sum_{i=0}^{n-1}l_i(x)^2\,dx$, where $l_i$ is the Lagrange cardinal polynomial of $x_i$: there are $\varepsilon>0$ and $n_0$ with $M_n\le2-(1+\varepsilon)/n$ for every $n\ge n_0$, so that $M_n\ne2-(1+o(1))/n$. Section 1 (pp. 1--2) puts $N=n-1$ and takes as nodes the $N+1$ roots of $Q_{N,a}(x)=\int_{-1}^x(P_N(t)+aP_{N-2}(t))\,dt$, with $P_m$ the Legendre polynomial normalized by $P_m(1)=1$ and $a$ a real parameter; for $N\ge3$ and $|a|\le1/100$ these roots are real, simple and include $\pm1$, and at $a=0$ they are the Legendre--Lobatto nodes. Section 2 (pp. 2--6) writes the integral $I_N(a)$ at these nodes as the trace of the inverse of the Legendre Christoffel kernel matrix, splits that matrix exactly into a positive diagonal part and a rank-three part, and reaches $I_N(a)=2-\operatorname{tr}(\mathcal A(a)^{-1}C(a)B(a))$ with explicit $3\times3$ matrices ((28), p. 6). Section 3 (pp. 6--8) proves $|I_N''(a)|\le C/N$ for $|a|\le a_0$, with $C$ and $a_0>0$ absolute ((29), (53)). Section 4 (pp. 8--10) computes $I_N(0)=2-2/(2N+1)$ ((55)) and $I_N'(0)=-4(N+1)/(3(N-1)(2N+1))$ ((64)). Section 5 (pp. 10--11) combines them into $n(2-I_N(a))=1+2a/3+O(a^2)+o(1)$ ((66)) and fixes a small $a>0$. The Theorem 1 page states the node family and the uniform expansion exactly.

At $a=0$ the nodes are the roots of $\int_{-1}^xP_{n-1}(t)\,dt$, the extremal nodes of Fejér's theorem on the least maximum of the sum of squares, and the value (55), $2-2/(2n-1)$ with $n=N+1$, is the one computed on that page and recorded on the problem page as the upper bound of Erdős, Szabados, Varma and Vértesi. The write-up lowers that bound within one explicit family of perturbed nodes; it proves no lower bound for $M_n$.

Source: https://www.overleaf.com/read/mhmzxwcsqjyd, the project Price's post links. The copy read is the PDF that Overleaf's viewer compiled from it, downloaded on 2026-10-14. A project can be edited and recompiled at that address, so the copy is of 2026-10-14; no copy of the document as it stood when it was posted is known to the corpus.

Read status: claims checked. Theorem 1 (p. 1), the node family of Section 1 (pp. 1--2) and the displays (28), (29), (48)--(55) and (64)--(66) (pp. 6--10) were read clause by clause on the page images; the proofs were read for structure only. Of the proof, only the closing arithmetic was redone by hand: (55) from the three weight sums before it, (64) from (61) and (63), and (66) from (65) with $n=N+1$. No other step was checked, the rank-three formula (17) and the uniform bound (53) included, and nothing here is independently reviewed. Result page: theorem_1.

Bears on.

• #1131: Theorem 1 (p. 1) gives $n(2-M_n)\ge1+\varepsilon$ for every $n\ge n_0$, so the displayed asymptotic $\min I=2-(1+o(1))/n$ fails and the problem's second question is answered no; it proves no lower bound for $\min I$ and leaves the first question open. Neither $\varepsilon$ nor $n_0$ is evaluated. The write-up is the document behind the pending claim of Price.

Results.

• Theorem 1 (p. 1), with the node family of Section 1 (pp. 1--2) and the uniform expansion (65)--(66) (p. 10): $M_n\le2-(1+\varepsilon)/n$ for every $n\ge n_0$, witnessed by the $n$ roots of $Q_{n-1,a}$ for a fixed small $a>0$.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.