Hybrid Restricted Triple Products, Archimedean Recovery, and Weak Tensor Transfers

Version 2.0 — major revision. We develop a representation-scoped hybrid restricted triple-product method for fixed ranks, combining the proved affine-\(\Psi\) Fourier–Poisson theory of Getz–Gu–Hsu on higher-rank axes with ordinary \(\mathrm{GL}_2\) Whittaker–Mellin theory on rank-two boundary axes. After normalized induction, all hybrid rank patterns contract to a common measure-normalized split Piatetski–Shapiro–Rallis core. The construction includes coefficient-one unramified normalization, auxiliary-only meromorphic continuation, an absolute split-core normalization, and a fixed-rank two-block scalar-rigidity mechanism. These ingredients yield unitary isobaric weak tensor transfers for every fixed pair \[ \operatorname{Sym}^a\pi_i\otimes \operatorname{Sym}^b\pi_j \] attached to non-CM elliptic curves over \(\mathbb Q\). A stable-twist argument, together with global functional equations and the Archimedean local converse theorem, recovers the expected real local parameter, so the resulting transfers are exact at the Archimedean place and weak only at a finite set of finite places. A principal new feature of Version 2.0 is a dimension-independent fixed-rank iteration. For a unitary isobaric representation \(\Pi\) on \(\mathrm{GL}_M\) and a unitary generic cuspidal representation \(B\) on \(\mathrm{GL}_q\), \(q\ge 3\), under temperedness at the real place and the existence of a common unramified tempered finite place, the method constructs a unitary isobaric weak tensor transfer on \(\mathrm{GL}_{Mq}\), exact at the real place. The proof combines rank-two calibration, a parameter-aware two-block contraction, Archimedean chart coverage, and a finite-family converse-theorem construction preserving the common tempered place. Applied to symmetric powers of non-CM elliptic curves, this gives every fixed symmetric-power triple except the all-boundary case \((1,1,1)\). In particular, the triple symmetric-cube transfer on \(\mathrm{GL}_{64}\) is recovered as a special case. Version 2.0 also substantially strengthens the proof interfaces for the restricted global functional equation, split-PSR normalization, cross-rank scalar propagation, reducibility-wall specialization, and exceptional finite-place normalization. The argument uses only the proved affine-\(\Psi\) Poisson theory; no conjectural ambient GGHL total Poisson formula is assumed. No compatibility is asserted at the excluded finite places, and no uniformity in \(M\) or \(q\) is claimed.