An Intrinsic Completed Functional Equation for a Triple Symmetric-Cube Tensor

Let \(E_1,E_2,E_3/ Q\) be pairwise geometrically non-isogenous non-CM elliptic curves, with unitary-normalized cuspidal automorphic representations \(π_i\). This paper studies the degree-\(64\) tensor parameter\[R333,v=Sym^3φπ_1,v^3φπ_2,v^3φπ_3,v.\] We prove an intrinsic all-place analytic completion for the Euler product determined by the standard local factors \(L_v(u,R333,v)\). The completed function\[Λ₃₃₃(u)=∏_v L_v(u,R333,v)\]extends to an entire function of finite order and satisfies the standard contragredient functional equation\[Λ₃₃₃(u)=ε(u,R₃₃₃,ψ)\,Λ₃₃₃(1-u).\] The argument is a representation-scoped specialization of the higher-rank restricted triple-product and affine-\(Ψ\) framework to the symmetric-cube rank pattern \((4,4,4)\). The local normalization problem is reduced to lower-rank standard objects through normalized Levi contraction and the split Piatetski–Shapiro–Rallis core. Ramified finite places are treated by a combination of Levi reduction, octahedral automorphic-induction comparison, and explicit special-type comparison. The Archimedean normalization is recovered from the previously established fixed-rank calibration theorem. A key point is that the proof identifies the intrinsic local gamma factors at every place rather than only at almost all unramified places. Auxiliary nonvanishing yields entire continuation of the incomplete target Euler product, and the completed functional equation together with tempered local holomorphy removes the apparent poles introduced when the finitely many bad local factors and the Archimedean factor are restored. The result is strictly an analytic \(L\)-function theorem. No automorphic representation on \(GL₆₄\) is constructed or assumed, and no general \(GL_4^3\) triple-product functoriality theorem is claimed. Version v1.0.