Exact Trigonometric Moments of the A2 Selberg Integral -

We prove that all trigonometric moments cₘ (k) and cross-moments χ₉, ₋ (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 sin^2k·cot = sin^2k-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).