An Automorphic Scattering Realization of the Triple Symmetric-Cube L-Function

We construct a higher-rank automorphic scattering realization of the degree-64 triple symmetric-cube \(L\)-function attached to three pairwise geometrically non-isogenous non-CM elliptic curves over \(\mathbb{Q}\). Starting from two prior inputs—an intrinsic all-place completed \(L\)-function for the triple symmetric-cube tensor and a coherent weak automorphic realization on \(\mathrm{GL}_{64}(\mathbb{A}_{\mathbb{Q}})\)—we first prove a finite-Euler reflection-rigidity theorem. In the self-dual setting, equality of the Archimedean factor and all but finitely many finite Euler factors, together with a strict half-plane bound for the exceptional local divisors, forces equality of the completed \(L\)-functions. This removes the remaining finite-local scalar ambiguity and identifies the intrinsic completion with the completed standard \(L\)-function of the weak automorphic realization. For each cuspidal constituent, we then attach a maximal-parabolic Eisenstein family and use Shahidi normalization to isolate a canonical one-dimensional Whittaker quotient. On this quotient, the nontrivial zero divisor of the intrinsic degree-64 \(L\)-function is transported, with multiplicity, to a canonical Eisenstein scattering-pole divisor by \(\rho \mapsto \rho-1\). Each target pole produces a nonzero outgoing Eisenstein principal part whose negative Laurent coefficients span a finite-dimensional space of distributional generalized states. We further calibrate the Archimedean scattering parameter as a radial momentum coordinate and show that the generalized Riemann hypothesis for the intrinsic degree-64 target is equivalent to all target scattering poles lying at universal normalized momentum depth \(-1/2\). This is an exact spectral reformulation, not a proof of GRH. Finally, a maximal-parabolic Maaß–Selberg identity separates the truncated Eisenstein Gram operator into a truncation-length term, a scalar completed-\(L\) logarithmic-derivative term, and a residual operator-valued delay. The paper does not identify the full Eisenstein or resolvent resonance spectrum with the target divisor and does not assume cuspidality of the weak \(\mathrm{GL}_{64}\) realization.