Exact Trigonometric Moments of the A2 Selberg Integral -

We prove that all trigonometric moments c_m(k) and cross-moments χj,l(k) of the A₂ Selberg integral with integer parameter k are rational functions of k. Two independent proofs are given: (1) an elementary Fourier argument valid for any root system, and (2) an integration-by-parts identity with the A₂ Dunkl operator, where the cot-singularity is regularized via sin²ᵏ·cot = sin²ᵏ⁻¹·cos. Seven explicit closed-form formulas are established (c₁, c₂, c₃, χ1,1, χ1,2, B₂, B₃), all with integer numerator coefficients and Pochhammer-type denominators, verified for k = 1, ..., 30. A complete table of nine exact rational values at k = 9 is provided, together with 30-digit high-precision numerical verification (10⁻⁷¹ relative accuracy via 100×100 Gauss-Legendre quadrature). The normalization integral is identified with the Dixon identity I₀ = (3k)!/(k!)³ (Dixon 1903), giving I₀ = 27!/(9!)³ = 227,873,431,500 at k = 9. Applications to the Hosotani integral on T⁵ with SU(3) gauge structure (γ = k = 9), the AGT correspondence at β = 9, and connections to Heckman-Opdam theory are discussed. Companion paper: "Spectral Geometry on T⁵ with SU(3)" (v58, DOI: 10.5281/zenodo.19291911).