Depth-Three Sato-Tate Harmonics for Three Non-CM Elliptic Curves: Analytic Closure of the Final Triple-Cubic Mode

This paper studies the final triple-cubic mode in the finite depth-three Sato–Tate harmonic packet attached to three non-CM elliptic curves over Q. Let πᵢ denote the unitary cuspidal automorphic representation of GL₂(AQ) associated with Eᵢ. The target is the degree-$64$ Euler product L₃₃₃(u)=L\!(u,Sym³π₁³π₂³π₃). Under the standing pairwise independence hypothesis (H1), we prove that L₃₃₃(u) admits meromorphic continuation to C and is holomorphic and nonzero on Re(u)≥ 1. The analytic argument develops a restricted rank-$(4,4,4)$ form of the fiber-bundle Poisson strategy of Getz–Gu–Hsu–Leslie, combined with the Fourier–Poisson theory for affine Ψ-bundles of Getz–Gu–Hsu. On an explicit global restricted test class, we establish the source-to-dual three-stage transform, the required Fourier covariance, absolute projective-height reassembly, coefficient-one good-unramified local data, prescribed-point nonvanishing at exceptional places, and the Archimedean asymptotic-Schwartz and two-sided rapidity conditions required for Mellin continuation. These ingredients yield analytic continuation and edge pole-freeness for L₃₃₃ without assuming the ambient GGHL Poisson conjecture. Edge nonvanishing is obtained independently from an exact nonnegative character polynomial. Its character expansion has c₃₃₃=1,₀₀₀=3949/4000,1-c₀₀₀=51/4000>0, while every other nonconstant term is supported on at most two elliptic factors and is controlled by established symmetric-power and Rankin–Selberg theory. Positivity of logarithmic-derivative coefficients excludes a zero at $u=1$, and a de la Vallée Poussin amplifier, with 4-3c₀₀₀=4153/4000>0, gives zero-freeness on the full edge Re(u)=1. Assuming in addition the prior edge hypothesis (H2) for the other $62$ nontrivial depth-three modes, the result completes the finite $64$-member depth-three harmonic packet. The result is a finite-packet analytic theorem. It does **not** assert unrestricted Sato–Tate equidistribution, the ambient GGHL Poisson conjecture, or an automorphic GL₆₄ triple-product transfer.