Corner-vector designs on 𝕊³: the maximum degree is exactly 13

Abstract

For a>0 and s ∈ {0,…,n-1} let the generalized corner vector v_(a,s) ∈ 𝕊ⁿ⁻¹ be the unit vector along (a,1,…,1,0,…,0) with s ones. Its hyperoctahedral orbit v_(a,s)^(Bₙ) is called proper when a ≠ 1; a weighted spherical t-design which is a union of such orbits, with one positive weight per orbit, is a design of type [eq:type]. Tanino, Tamaru, Hirao and Sawa proved that t ≤ 15 for every n ≥ 4, exhibited 11- and 13-designs on 𝕊³ numerically, and asked in their Problem 6.1 whether a 15-design on 𝕊³ of this type with more than five proper orbits exists. We answer this negatively and in a stronger form: on 𝕊³ there is no 15-design of type [eq:type] at all, with any number of orbits, proper or not. Such designs are centrally symmetric, so a 14-design is a 15-design; the uniform bound therefore sharpens at n=4 from t ≤ 15 to t ≤ 13. The value 13 is attained: we give a 13-design with nine proper orbits and 656 points, all of whose parameters and weights are rational and exact, where the examples in the literature have four proper orbits and six printed digits. Hence the maximum degree of a design of type [eq:type] on 𝕊³ is exactly 13. The proof is linear-programming duality rather than elimination: in the squared coordinates a design is a point of the convex hull of four explicit moment curves, and the obstruction is a single integer functional on the eleven moment conditions which is strictly positive at every point of every curve. Its positivity amounts to three degree-7 polynomial inequalities, which we prove from exact factorizations; no statement here rests on an exhaustive search. Every theorem, proposition and lemma below other than the reduction to the moment system and two one-line corollaries has been formally verified in Lean 4 against Mathlib; the remarks, which report how the certificate was found, what was tried and abandoned, and what the published configurations do when recomputed, are labelled as lying outside that development.

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