Radial Icosahedral Sampling and Its First Harmonic Obstruction: Exact degree-six activation and Euler–Maclaurin coefficients

Preprint — mathematical results and reproducibility package. Let X_f ⊂ S² be the complete frequency-f Class-I grid obtained by barycentrically subdividing the twenty planar faces of a regular icosahedron and radially projecting every node to the unit sphere. For the equal-node probability measure μ_f, this paper proves μ_f ⇀ μ_rad, where μ_rad is the pushforward of normalized planar face area and is not uniform spherical area. Nevertheless, every complete X_f is an equal-weight spherical five-design. The first A₅-invariant harmonic occurs in degree six. For an explicit normalization I₆, an exact depth-six Bernstein certificate over Q(√5) proves C₀ = ∫_{S²} I₆ dμ_rad > 0. Consequently, the first harmonic degree distinguishing μ_rad from uniform spherical probability σ is exactly six. An explicit simplicial Euler–Maclaurin calculation further gives μ_f(I₆) = C₀ + C₂ f⁻² + C₄ f⁻⁴ + O(f⁻⁶), with exact formulas for C₂ and C₄ and exact vanishing of the normalized f⁻¹, f⁻³, and f⁻⁵ coefficients. Closed-surface face–edge assembly accounts for the first cancellation, while barycentric S₃-symmetry accounts for the next two. A separate arithmetic section records results for K_Ω = Q(π, √5), including its quadratic presentation, intersections with prime cyclotomic fields, the generic Galois group of Xᵖ − Y, and an equivalent finite-Fourier rigidity statement. A minimal exact transport lemma shows how already-defined states and observable relations are preserved under bijective representation. The accompanying reproducibility materials use exact symbolic arithmetic, finite exact computation, and a rigorous interval certificate. Decimal calculations are used only for numerical confirmation and not as proof. No claim of novelty, priority, peer review, or journal acceptance is made.