Equidistant sets for additive kernels with finitely many negative squares

Abstract

We isolate a linear-algebraic principle underlying several recent bounds for equilateral sets and equidistant codes. Let κ be a symmetric kernel on a set S that can be written as a squared Hilbert distance minus a squared distance into a space of dimension at most q (equivalently, an anchored Schoenberg kernel with at most q negative squares). For the coordinatewise sum of κ on Sn we prove that any m points whose pairwise kernel values all equal d > 0 satisfy R = (d/2)Cm + X with X positive semidefinite of rank at most qn, and consequently m ≤ 1 + Σt rt, where rt is the affine rank of the positive features of the alphabet actually used in coordinate t. A stable version replaces exactness by an operator-norm error and a spectral-tail budget. Three applications. (1) For codes in Zqn that are equidistant with respect to the squared Lee weight, |C| ≤ (q/2)n + 1 for even q and |C| ≤ 2⌈(q-1)/4⌉n + 1 for odd q, with no linearity assumption; for q = 5 the bound |C| ≤ 2n+1 is attained by an explicit evaluation code for every n = (5k-1)/2. (2) For p ∈ (4k, 4k+2) we recover the centred-Gram identity used by Ge–Xu–Zhou and tighten the finite negative rank budget from 4kn to 2kn; this rank improvement alone does not imply a linear cardinality bound. (3) The sharp first-order asymptotics of the Schütte problem on 2 ≤ p ≤ 4: limN N(Rp(N) - 1) = 2/p. Contents. Manuscript (PDF and LaTeX source) plus the two verification scripts covering every finite numerical claim, with a SHA-256 manifest. Both re-ran green on 2026-09-03. The proofs are analytic; the scripts check the finite claims, the closed forms and the negative controls. Status. Author preprint, not peer reviewed. A priority audit found no prior statement converting a negative-square count into an equilateral-set cardinality bound; scope was arXiv plus the reference lists of the cited works, with MathSciNet and zbMATH not searched. A finite search is not a proof of priority. Corrections welcome. AI disclosure. Anthropic Claude and OpenAI Codex were used as research tools for strategy exploration, derivation suggestions, symbolic and numerical checks, adversarial review, code drafting and manuscript drafting; the scripts were re-run in a separate session from the one that produced them. The author takes full responsibility for the accuracy and originality of the manuscript.