P42: Euclidean persistence and sharp stability for anisotropic eigenvalue-torsion products

Henry Zweiman. September 15, 2026.

AI-assisted research preprint. Not peer reviewed or independently verified. The complete proof, primary-source comparison and internal audit are available for examination and correction.

The proposed theorem identifies the Euclidean norm as the unique optimizer of the first eigenvalue times a power of torsional rigidity on a fixed ball: a maximizer for sufficiently small positive product exponents, and a minimizer for sufficiently large ones. It covers all normalized symmetric seminorms, every dimension n>=2 and every energy exponent p>1. The product deficit is equivalent to an angular average and controls the uniform distance with the optimal power (n+1)/2.

This addresses the ball cases of the Euclidean-optimality questions in Buttazzo and Fernandes Horta, Section 6. Their quadratic-seminorm ball result and classical convex symmetrization are credited. General fixed domains, asymmetric anisotropies, optimal product-exponent thresholds and endpoint-uniform p estimates remain outside the theorem. Both regimes and the nonlinear extension form one paper.

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