Hybrid Restricted Triple-Product Functional Equations and Weak Tensor Transfers for Fixed Symmetric-Power Pairs

This preprint develops a representation-scoped hybrid form of the restricted triple-product method for arbitrary fixed ranks $r_1, r_2, r_3 \ge 2$, with at least one rank at least three. Higher-rank axes are treated by the affine-$\Psi$ transforms of Getz--Gu--Hsu, while each rank-two axis is replaced by the ordinary normalized $\mathrm{GL}_2$ Whittaker--Mellin functional equation. The construction retains the rank-independent outer oscillator and projective Lagrangian geometry of Getz--Gu--Hsu--Leslie, establishes an exact coefficient-one unramified formula, gives one- and two-boundary global reassembly, and contracts all hybrid patterns to a common split Piatetski--Shapiro--Rallis core. The analytic package includes auxiliary-only continuation clearing, canonical finite-place normalization, a Clebsch--Gordan real calibration, one fixed highly ramified character for the converse-theorem twist family, and constituentwise reassembly for the Booker--Krishnamurthy converse theorem. The argument does not invoke the conjectural total GGHL Poisson formula. As an application, assume weak cuspidal single-curve symmetric-power realizations $A / \mathrm{GL}_{a+1}(\mathbb A_{\mathbb Q})$ and $B / \mathrm{GL}_{b+1}(\mathbb A_{\mathbb Q})$. For $N=(a+1)(b+1)$, the paper constructs an isobaric automorphic representation $\Pi_{a,b} / \mathrm{GL}_N(\mathbb A_{\mathbb Q})$ whose local parameter equals $$\operatorname{Sym}^a\phi_{\pi_i,v} \otimes \operatorname{Sym}^b\phi_{\pi_j,v}$$ at every finite place outside a fixed finite set. Combined with the companion single-curve symmetric-power realizations, this yields weak tensor transfers for every fixed symmetric-power pair attached to two non-CM elliptic curves. The conclusion is deliberately weak: no compatibility at ramified finite places or at the real place is asserted for the final converse-theorem output.