Sharp ratios of negative-exponent Riesz capacities: the two-point and simplex conjectures, the exact threshold at exponent two, and sharp stability

Unrefereed research candidate. In one sentence: the regular simplex maximises the ratio of two negative-exponent Riesz capacities exactly when the smaller exponent is at least two, while two points always maximise the reverse ratio. For a compact metric space K and r > 0, Mr(K) is the largest possible mean of d(X,Y)r over independent random points X, Y of K with a common distribution, and Cr = Mr1/r; for Euclidean sets Cr is the Riesz capacity of exponent −r in the convention of Clark and Laugesen, who conjectured which shapes maximise the ratio of two such capacities. With 0 < s < t and γ = 1/s − 1/t, the paper proves Ct/Cs ≤ 2γ on every compact metric space (two points are extremal; the same short argument was already posted publicly in a GitHub issue on 27 October 2026) and Cs/Ct ≤ (n/(n+1))γ for compact sets in Rn when 2 ≤ s < t (the regular simplex is extremal; much of this range follows from Davies, Lim and McCann 2022, and the paper makes the overlap exact and identifies a wedge of exponents where only this argument applies). It then shows that exponent two is an exact threshold: for every n ≥ 2 and s < 2, splitting one vertex of a regular simplex beats the simplex. For 0 < s < 1 it gives the sharp lower bound for Ms/Ds in terms of the Hausdorff distance to a diametral pair, attained by three-point spaces, and stability results with optimal deficit exponents 1/s, 1/4 and 1/2. At s = 2 the simplex bound is a finite-exponent form of Jung's theorem. The archive contains the manuscript (PDF, Markdown source and generated LaTeX), the parameter-region figure, the bibliography, three verification scripts with their deterministic receipts (exact rational checks, high-precision closed-form checks and seeded randomised spot checks), the dated literature audit, a rating-free internal review summary, the superseded first version, CI and SHA-256 sums. Assurance: written analytic proofs; the computations are finite sanity checks, not proofs; internal AI-assisted review only (not peer review, independent reproduction or formal verification); Hille (1965) and Alexander–Stolarsky (1974) were not read; stability constants are explicit but not optimised. Creator: Anonymous. Text and figures CC0-1.0; code MIT.