An antipodal rational 5-design with 37 points, a 3-adic obstruction for the degree-five Hilbert–Kamke equations, and the nonexistence of two-orbit weighted 9-designs on 𝕊³

Abstract

A Chebyshev-type quadrature of degree m, or an m-design, for a probability measure w(t) dt on an interval I is a set of distinct points of I over which the unweighted average of every polynomial of degree at most m equals its integral. For the Chebyshev measure (1-t²)^(-1/2)dt/π on (-1,1), Mishima, Lu, Sawa and Uchida determined the spectrum of antipodal rational 5-designs with an even number of points completely, proved that none with an odd number 2N+1 ≤ 33 of points exists, exhibited one with 35 points, and asked for one with 37 points as the smallest open case. We answer that question: the 37 rational numbers {0} ∪ {± m/144}, where m runs over {3,11,18,25,39,81,91,99,111,117,122,123,125,127,133,137,141,143}, form an antipodal 5-design. We also record a congruence obstruction on the denominators of such designs which the purely 2-adic machinery of the source does not see, and which together with that machinery cut the search space by a factor of about 3 · 10⁴: writing the degree-five system in the homogeneous form Σ mᵢ²=c₂h², Σ mᵢ⁴=c₄h⁴, the identity x⁴=x² on ℤ/3ℤ forces 3 | h whenever 3 ∤ c₄-c₂, and then a mod 9 count forces the number of mᵢ prime to 3 to be divisible by 9. For the 37-point system this says that every common denominator of the points is divisible by 12, and that over the least one, 9 or 18 of the 18 positive numerators are prime to 3; the new configuration satisfies both, at the smaller of the two allowed counts. A second, independent part of the paper settles a question about weighted spherical designs raised by Tanino, Tamaru, Hirao and Sawa. They construct designs on 𝕊ⁿ⁻¹ as unions of hyperoctahedral orbits of generalized corner vectors v_(a,s), and report for n=4 that they could find no 11-design, and not even a 9-design, with two proper orbits. We show that none exists: for every pair of orbit types s₁,s₂ ∈ {0,1,2,3}, all a₁,a₂>0 and all positive weights, the four B₄-invariant harmonic conditions of degree at most 9 are inconsistent. Their own Theorem 5.7 already excludes three of the five cases of their classification, for every n ≥ 3; the content that is new here is the two remaining cases at n=4. The degree is sharp — the corresponding two-orbit 7-design system is solvable — and Schur's four-orbit 11-design satisfies the very same four equations, so "two orbits" cannot be raised to "four". Two errata in the printed formulas of that source are recorded along the way. Every theorem and proposition below is formally verified in Lean 4, and we state precisely what lies outside that guarantee.

Open review

This founding-collection manuscript received AI review before publication. Independent human review is open. Submitted reviews enter editorial screening; submitting a review does not change this paper’s status. Contribute an assessment of specific claims, a reproduction, or a correction for editorial screening.

Archived files

1. Version 2 · current (opens in a new tab)

   Source snapshot 2026-08-30 15:34 UTC

   File fingerprint

   61e16b1c9c48a7a9b93b13010cc7efa1047d22b4a605cf05774f581e8ccc5efc