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Claim. Fix $n\ge2$ and consider the node systems $-1=t_0<t_1<\cdots<t_n=1$ that contain both endpoints, the paper's convention and the site's canonical systems. For such a system let $\lambda_i$ be the maximum of the Lebesgue function $\sum_k|l_k(x)|$ on the gap $[t_{i-1},t_i]$, and call the system equioscillating when $\lambda_1=\cdots=\lambda_n$. Theorem 1 of de Boor and Pinkus (p. 295) shows that the map sending a system to its vector of consecutive differences $\lambda_{i+1}-\lambda_i$ is a homeomorphism onto $\mathbb R^{n-1}$, so exactly one system equioscillates; with Kilgore's theorem [Ki77], that a system minimizing the Lebesgue constant must equioscillate, its Corollary (p. 295) gives that the equioscillating system has a strictly smaller Lebesgue constant than every other system in this convention. Theorem 2 (p. 298) adds that no two distinct systems satisfy $\lambda_i(s)\le\lambda_i(t)$ for every $i$, so every system has $\min_i\lambda_i\le\lambda^*\le\max_i\lambda_i$, where $\lambda^*$ is the minimal Lebesgue constant. Among systems containing both endpoints the minimizing choice of nodes is therefore the unique system whose Lebesgue function equioscillates, as Bernstein [Be31] conjectured and Erdős added ([Er47], and Problems and results on the theory of interpolation. I, Acta Math. Acad. Sci. Hungar. 9 (1958), 381–388, as de Boor and Pinkus cite them). For the free nodes of Problem 1129 the minimizers are its affine images whose Lebesgue function at $\pm1$ stays at most $\lambda^*$, as the Convention paragraph explains. Kilgore and Cheney [KiCh76] had shown that an equioscillating system exists, and Kilgore [Ki77] that a minimizer equioscillates; the paper supplies uniqueness and the strict comparison. The source card is de Boor and Pinkus 1978. The problem asks to describe the minimizing choice, a question with neither a proved nor a disproved shape, so the claim value is answered; the site's label PROVED is recorded on the problem page.

Convention. The site lets the $x_i$ range over all of $[-1,1]$ and writes the equioscillation condition over the $n+1$ pieces cut out by the nodes and the auxiliary points $x_0=-1$ and $x_{n+1}=1$; the paper's theorem concerns systems containing both endpoints, and the site states the uniqueness for those canonical systems. The minimal value over all systems is the canonical minimum $\lambda^*$: the fundamental polynomials are invariant under an affine change of variable, so the Lebesgue constant of a system is at least that of its affine rescaling onto $[-1,1]$. Which systems outside the canonical convention also attain $\lambda^*$ is not part of the paper's statement, and its uniqueness is a statement about canonical systems. Rack and Vajda print the transfer (Rack and Vajda 2015, Theorems 2.5 and 5.2 with their proofs): an affine shrink of the optimal canonical system whose Lebesgue function at $\pm1$ stays at most $\lambda^*$ keeps its interior maxima and attains $\lambda^*$, and every optimal system rescales to the optimal canonical one; so for $n\ge3$ the free-node minimizers are not unique, as Luttmann and Rivlin (IBM J. Res. Develop. 9 (1965), Theorem 2, as Rack and Vajda cite it) had shown. Two third-party Lean 4 developments, linked above at their pinned commits, prove these facts from the paper's theorems and name de Boor and Pinkus as their mathematical source. The file in the lean-proofs repository, with Codex and GPT-5.6 Sol as its formal authors, calls itself a correction to the unconstrained formulation and proves erdos_1129: for three free nodes the minimal Lebesgue constant is $5/4$, attained both by $(-1,0,1)$ and by $(-49/50,0,49/50)$, so free-node minimizers are not unique. Collin Yuanjie Ren's JSP-000936 development, which the community database lists as the problem's Lean formalization as of its last update on 2026-09-16 and describes as AI-assisted, builds on the canonical de Boor–Pinkus formalization of randyxian08 and proves minimizer_iff_equioscillating_and_controlled_tails: a family of at least two distinct nodes in $[-1,1]$ minimizes the Lebesgue constant among all families of its size exactly when all interior gap maxima are equal and the Lebesgue function at $-1$ and at $1$ does not exceed that common maximum; every singleton family is optimal, and no uniqueness of free-node minimizers is asserted. The optimal canonical system is known explicitly only for $n\le4$: the three-node minimum $5/4$ is in Bernstein [Be31, p. 1027], and the four-node system is Rack's (1984 and 2013), as Rack and Vajda [RaVa15, Section 3] recall it.

Acceptance. Refereed: Journal of Approximation Theory 24 (1978), no. 4, 289–303, received 1977-04-01, in the issue dated December 1978 in the publisher's record, which dates this page. Reviewed: the site's curator, Thomas F. Bloom, labels the problem proved and records in its commentary that de Boor and Pinkus proved the existence of a unique minimizing choice, after the results of Kilgore and Cheney and of Kilgore (erdosproblems.com/1129, last edited 2026-01-23, accessed 2026-09-04). The paper's note added in proof records that Kilgore also proved Bernstein's conjecture, by a different argument, in a paper published in the same issue; that independent proof has its own accepted page, Kilgore 1978. The theorem statements are taken from the paper itself, whose proofs are known here in outline only; the proofs are not compiled in this wiki. Formalization: the two Lean 4 developments linked above are neither built nor audited in this repository, so no formalized evidence is listed.

Depends on. Nothing in this wiki; the result rests on the refereed paper linked above.