Chebyshev Extremizers of Squared-Gap Entro

Abstract

This work studies a finite-dimensional extremal problem for the squared-gap logarithmic functional Cn(a)=i<j∑(ai−aj)2log(ai−aj)2 on the centered unit configuration space Mn={a∈Rn:i∑ai=0, i∑ai2=1}. The main result proves that, for every n≥2, the unique global minimizers up to permutation are the normalized zeros of the Chebyshev polynomial Tn: a∈MnargminCn(a)=Sn⋅aCh,akCh=n2cos2n(2k−1)π. Equivalently, the Chebyshev configuration uniquely maximizes the Shannon entropy of the normalized squared pair-gap distribution pij=n(ai−aj)2. A second equivalent interpretation is obtained from the ordered power energy Ep(a)=i=j∑∣ai−aj∣p. On Mn, E2≡2n is completely degenerate, while ∂pEp(a)∣p=2=Cn(a). Thus the Chebyshev configuration is the unique minimizing selector of the first-order correction to the degenerate p=2 energy as p=2+ε, ε↓0. The proof uses an equal-nodal-average characterization of constrained stationarity, logarithmic gap coordinates, a Hardy-type positivity identity, P-matrix theory, Gale–Nikaido global injectivity, and an exact Hessian–Jacobian conjugacy. These ingredients yield both global uniqueness and nondegeneracy for every finite n. As an application, the paper studies the entropy–log-gap family Vn,β(λ)=i∑λilogλi+βi<j∑(λi−λj)2log(λi−λj)2, proving a singular symmetry-breaking transition at β=0, the universal weak-coupling scale e−1/(4β), Chebyshev selection of the angular profile, and uniqueness of the minimizing Sn-orbit for sufficiently small positive coupling. For the certified case (n,β)=(3,1), a directed-rounding MPFR/Krawczyk computation proves that there are exactly 13 interior stationary points and exactly six global minimizers, forming one permutation orbit. The all-n Chebyshev shape theorem and weak-coupling results are analytic. The n=3,β=1 classification is computer-assisted and accompanied by reproducibility scripts and certification output. The manuscript also includes a discussion of its relation to classical Chebyshev approximation, Fekete/Vandermonde problems, logarithmic potential theory, discrete power energies, and An−1 root-system projections. No exhaustive bibliographic-priority claim is made.