Edge Nonvanishing for Triple Products of Higher Odd Symmetric Powers via Restricted Poisson Summation

Let $E_1,E_2,E_3/\mathbf{Q}$ be non-CM elliptic curves, and let $\pi_i$ be the associated unitary cuspidal automorphic representations of $\mathrm{GL}_2(\mathbf{A}_{\mathbf Q})$. For fixed odd integers $n_1,n_2,n_3\ge 3$, assume that the three curves are pairwise inequivalent under quadratic twist. We prove that $$L\!\left(u,\operatorname{Sym}^{n_1}\pi_1\otimes\operatorname{Sym}^{n_2}\pi_2\otimes\operatorname{Sym}^{n_3}\pi_3\right)$$ admits meromorphic continuation to $\mathbf C$ and is holomorphic and nonzero on the closed half-plane $$\operatorname{Re}(u)\ge 1.$$ The proof develops a fixed-rank restricted form of the fiber-bundle Poisson strategy of Getz–Gu–Hsu–Leslie. Rather than assuming the conjectural ambient total-space Poisson formula, we construct an explicit global test class for which the middle affine-$\Psi$ factors satisfy the Getz–Gu–Hsu asymptotic-Schwartz, cuspidal-image, and two-sided rapidity hypotheses. The argument establishes the mixed-rank covariance required on the adelic norm-one quotient, absolute projective-height reassembly, prescribed-point nonvanishing at exceptional places, and a good-unramified shell identity producing the target triple-product Euler factor with coefficient one. These inputs yield the required multivariable Mellin continuation and exclude poles on $\operatorname{Re}(u)=1$. Zero-freeness is obtained independently from a finite rational nonnegative character polynomial whose target coefficient is one and whose scalar coefficient is strictly smaller than one, together with symmetric-power rigidity, Newton–Thorne functoriality, and a de la Vallée Poussin argument. The result is fixed-rank-by-fixed-rank and makes no uniform-in-rank assertion. It does not prove the ambient Getz–Gu–Hsu–Leslie Poisson conjecture, construct an ambient total Fourier transform, or assert automorphy of the full tensor product on $$\mathrm{GL}_{(n_1+1)(n_2+1)(n_3+1)}.$$ **Related work** A rank-$(4,4,4)$ depth-three Sato–Tate application, including the triple-cubic mode and finite depth-three packet closure, is developed in the companion paper: *Depth-Three Sato–Tate Harmonics for Three Non-CM Elliptic Curves: Analytic Closure of the Final Triple-Cubic Mode*, Version v1.0, DOI: 10.5281/zenodo.22791457. **Version note** Version v1.0. Public preprint corresponding to the final submission-freeze manuscript.