Finite-Defect Rigidity and the Minimum Spherical 4-Design on the Two-Sphere

Abstract

We prove that every equal-weight spherical $4$-design on 𝕊² has at least twelve points. Since the regular icosahedron is a spherical $5$-design, this determines the exact minimumN₄(𝕊²)=12;equivalently, no such design has $9$, $10$ or $11$ points.The proof is part of a finite-defect theory. If a spherical 2m-design has corank c=N−dim P_m, its Naimark complement consists of unit vectors u_x∈𝕊^c−1 forming a spherical $2$-design and satisfying the exact couplingu_x· u_y=−(K_m^(d)(x· y))/c (x≠ y).This gives a pairwise kernel bound, an antipodal lower bound, Cayley-Bacharach information, and uniform lower bounds for multiplicative-relation spaces. In corank one the design splits into two equal spherical m-designs. We derive a residue formula for its signed Schoenberg coefficients; the coefficient of degree m+3 is negative exactly when 3≤ d≤ m+1, excluding corank one throughout that range. At strengths four and six the only examples in any dimension are the regular hexagon and octagon, respectively.In corank two the complement is a circle Gale frame. Multiplication by its phase forces at least d−1 linear-quadratic aliases and yields exact norm and socle identities in every dimension. For eleven nodes on 𝕊², two aliases produce a real harmonic cubic and a Hermitian quartic matrix. A matrix-valued Cayley-Bacharach argument eliminates the generic branch; the exceptional branch reduces to a Pauli normal form and contradicts the second moments. In dimensions d≥4 the corank-two problem remains open; we identify a forced quadratic socle as the obstruction to extending the present argument.

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BibTeX RIS

BibTeX

@misc{singh2026finitedefect,
  title = {{Finite-Defect Rigidity and the Minimum Spherical 4-Design on the Two-Sphere}},
  author = {Shalender Singh and Vishnupriya Singh},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31689},
  eprint = {2609.31689},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31689},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Finite-Defect Rigidity and the Minimum Spherical 4-Design on the Two-Sphere
AU  - Shalender Singh
AU  - Vishnupriya Singh
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.31689
UR  - https://arxiv.org/abs/2609.31689
AB  - We prove that every equal-weight spherical $4$-design on 𝕊^2 has at least twelve points. Since the regular icosahedron is a spherical $5$-design, this determines the exact minimumN_4(𝕊^2)=12;equivalently, no such design has $9$, $10$ or $11$ points.The proof is part of a finite-defect theory. If a spherical 2m-design has corank c=N−dim P_m, its Naimark complement consists of unit vectors u_x∈𝕊^c−1 forming a spherical $2$-design and satisfying the exact couplingu_x· u_y=−(K_m^(d)(x· y))/c (x≠ y).This gives a pairwise kernel bound, an antipodal lower bound, Cayley-Bacharach information, and uniform lower bounds for multiplicative-relation spaces. In corank one the design splits into two equal spherical m-designs. We derive a residue formula for its signed Schoenberg coefficients; the coefficient of degree m+3 is negative exactly when 3≤ d≤ m+1, excluding corank one throughout that range. At strengths four and six the only examples in any dimension are the regular hexagon and octagon, respectively.In corank two the complement is a circle Gale frame. Multiplication by its phase forces at least d−1 linear-quadratic aliases and yields exact norm and socle identities in every dimension. For eleven nodes on 𝕊^2, two aliases produce a real harmonic cubic and a Hermitian quartic matrix. A matrix-valued Cayley-Bacharach argument eliminates the generic branch; the exceptional branch reduces to a Pauli normal form and contradicts the second moments. In dimensions d≥4 the corank-two problem remains open; we identify a forced quadratic socle as the obstruction to extending the present argument.
ER  -

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