Triple Symmetric-Power \(L\)-Functions in the Interior Rank Range: Meromorphic Continuation and Edge Nonvanishing

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 $\mathrm{GL}_2(\mathbf{A}_{\mathbf Q})$. Assume that the three curves are pairwise inequivalent under quadratic twist. For every fixed triple of integers $n_1, n_2, n_3 \ge 2$, this paper studies the triple symmetric-power $L$-function $$L\!\left(u, \operatorname{Sym}^{n_1}\pi_1 \otimes \operatorname{Sym}^{n_2}\pi_2 \otimes \operatorname{Sym}^{n_3}\pi_3 \right).$$ We extend the restricted fiber-bundle Poisson method previously developed for higher odd symmetric powers to the full interior range $n_i \ge 2$. The new local issues are the rank-three endpoint $\mathrm{GL}_3 \times \mathrm{GL}_1$ and the Archimedean parameters of even symmetric powers. We verify the rank-three toric inversion, the finite-sector Plancherel and Fourier interfaces, two-sided asymptotic-Schwartz estimates, global cuspidal-image regularization, mixed-rank covariance on the adelic norm-one quotient, absolute projective-height reassembly, prescribed-point exceptional local nonvanishing, the coefficient-one unramified factor, and the restricted fixed-$a$ Mellin continuation. The resulting restricted three-stage Poisson–Mellin argument gives meromorphic continuation of the target $L$-function and excludes poles on the line $\operatorname{Re} u = 1$, without assuming the ambient Getz–Gu–Hsu–Leslie total-space Poisson conjecture. A parity-free positive-character construction, with a strictly positive scalar gap already at the endpoint $d_i = n_i + 1 = 3$, then excludes zeros on the full edge. Consequently, for every fixed $n_1, n_2, n_3 \ge 2$, the target $L$-function is holomorphic and nonzero on the closed half-plane $$\operatorname{Re} u \ge 1.$$ The result is fixed-rank-by-fixed-rank; no uniformity in the symmetric-power ranks is asserted. The remaining geometric boundary corresponds to $n_i = 1$, equivalently rank $2$, which lies outside the affine-$\Psi$ family used here.