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arXiv · 2609.32973

Discreteness, finite rank, and stability of p-frame energy minimizers

Josiah Park
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Abstract

We study probability measures minimizing the energy with kernel $|x\cdot y|^p$ on the real unit sphere. For every positive rational $p=a/b$, each minimizer admits a finitely supported minimizing replacement. If $a/b$ is reduced and $b\ge d$, every minimizer on $\mathbb S^{d-1}$ is finitely supported. The latter assertion follows from a finite-rank algebraic lift, Nash curve selection, and a Wronskian multiplicity bound. For irrational exponents, we prove that a positive semidefinite restriction of finite rank cannot have infinite compact support. We also establish local stability and a quantitative support bound under positive-definite Hessians at every contact of every limiting minimizer. This hypothesis is stronger than finite support and does not prove unconditional openness of the discreteness regime. Finally, subtracting a single suitably scaled high even power produces a uniformly nearby kernel whose every minimizer has finite support. We identify the atomic-purity selection of these approximants and prove a target-centered recovery theorem, separating finite approximation from discreteness of the limiting problem.

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Josiah Park. 2026-09-26. Discreteness, finite rank, and stability of p-frame energy minimizers. https://arxiv.org/abs/2609.32973

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