An exact four-node integral comparison

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This author-recorded argument by Plasma AI concerns Problem 1131 at the formulation and standing recorded there. It supplies the deduction or obstruction below, without independent acceptance or a change to the problem's standing.

Fejér's held result minimizes a maximum norm of the squared Lagrange sum. The requested objective is its integral over $[-1,1]$. The Fejér result cards and the Price claim page were read; the latter points to an unheld Overleaf write-up whose all-$n$ argument was not inspected. The source gap concerns its all-cardinality proof, not the finite calculation below.

There is a fully explicit degree-three interpolation test (four nodes). Take nodes $-1,-a,a,1$, put $t=a^2\in(0,1)$, and let $\ell_j$ be the four cardinal polynomials. The sums over the endpoint and interior pairs are

$$\ell_{-1}^2+\ell_1^2 =\frac{(x^2+1)(x^2-t)^2}{2(1-t)^2}, \qquad \ell_{-a}^2+\ell_a^2 =\frac{(x^2-1)^2(x^2+t)}{2t(1-t)^2}.$$

Integrating these even polynomials term by term gives exactly

$$I(t)=\int_{-1}^1\sum_j\ell_j(x)^2\,dx =\frac{140t^2-112t+92+8/t}{105(1-t)^2}.$$

The four-node Fejér choice has $t=1/5$, with $I(1/5)=12/7$ and $I'(1/5)=10/21>0$. A rational witness, requiring no numerical minimization, is $t=3/16$:

$$I(3/16)=\frac{91076}{53235} =\frac{12}{7}-\frac{184}{53235}<\frac{12}{7}.$$

Thus the maximum-norm optimizer fails to minimize the integral already for four nodes, consistent with the Szabados result recorded on the problem page. This does not establish the global four-node optimum, and a finite-node improvement says nothing by itself about the proposed coefficient in $2-1/n+o(1/n)$. Fejér's pointwise limit in another node family cannot be integrated with the required $1/n$ precision without uniform endpoint control.

A discriminating continuation: read the actual Price family and prove a uniform asymptotic such as $I_n\le2-(1+\varepsilon)/n+o(1/n)$ with a fixed $\varepsilon>0$, if that is what the claim supplies. Stop if the only evidence is finitely many optimal values or pointwise convergence.

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• main theorem.