Exact Positive Icosahedral Cubature on Fixed Radial Class-I Grids: Quadratic-Field Interoperability, Exact Replay, and Typed Compatibility with Papers I–V

Abstract

This preprint establishes exact positive cubature rules on fixed radial Class-I icosahedral grids X_f ⊂ S² for frequencies 1 ≤ f ≤ 9. Using full icosahedral reflection symmetry H₃, invariant-cubature reduction, and exact arithmetic in ℚ(√5), strictly positive orbit-constant probability rules are certified with exact algebraic degrees 5, 9, 11, 15, 17, 21, 23, 27, 29. For 1 ≤ f ≤ 8, the required invariant moment systems are square and yield unique rules. At f = 9, the exact 11 × 12 system has rank 11 and a one-dimensional affine solution family W(t). Its complete strict-positivity interval is determined exactly in ℚ(√5), together with a rational witness W₃₃₃ = 1/27 and exact solutions maximizing, separately, the minimum total orbit mass and minimum per-node weight. Exact nonzero residuals at the first omitted invariant degrees establish the stated degrees rather than merely lower bounds. The paper also gives an exact coefficient-field interoperability map between ℚ(√5) and the quadratic quotient algebra K_Ω = ℚ(T)[λ]/(λ² − Tλ − T²), using √5 = 2λ/T − 1. Thus every certified cubature coefficient a + b√5 admits an exact two-channel representation (a − b) + (2b/T)λ. This representation is explicitly separated from the geometric derivation of the cubature rules. Independently, conditional on the frozen strict enclosure of λ_Ω = φ/(φ + π), exact integer computation verifies ⌊λ_Ω d⌋ = ⌊339949771344778·d / 10¹⁵⌋ for every unsigned 15-bit state 0 ≤ d < 2¹⁵. A determinant-one Farey certificate proves that the first disagreement over all nonnegative integers occurs at d = 42,189,065. The cubature rules are integrated conservatively with the existing Papers I–V mathematical object: they use the same fixed Class-I node sets while defining distinct positive measures, admit exact OMEGA coefficient representations, and are preserved under lossless SRSD coordinate transport. These interfaces do not alter the mathematical origin or proof of the cubature results. The reproducibility payload includes exact verification scripts, machine-readable coefficients, certificates, a Julia replay kernel, frozen dependency records, and manuscript source. The deterministic checks reconstruct the moment systems and weights, verify positivity and first-failure residuals, certify the complete frequency-nine positive family and both max–min rules, verify the OMEGA coefficient translation and 15-bit floor contract, and check the global-object interfaces. The all-frequency positive fixed-Class-I cubature statement remains a conjecture for f ≥ 10. No claim of novelty, priority, peer review, or journal acceptance is made.