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

We develop a fixed-rank hybrid restricted triple-product method combining the affine-Ψ Fourier–Poisson theory of Getz–Gu–Hsu with ordinary GL(2) Whittaker–Mellin theory on rank-two boundary axes. A contraction to the split Piatetski–Shapiro–Rallis core gives canonical finite local factors and a fixed-rank residual-scalar rigidity statement. For symmetric-power lifts attached to non-CM elliptic curves, the method yields unitary isobaric weak tensor transfers for every fixed pair Sym^a π_i ⊗ Sym^b π_j, with the expected local parameter at all finite places outside a fixed exceptional set and exactly at the Archimedean place. A stable-twist argument recovers the real local parameter from highly ramified GL(1) twists, Deligne–Henniart stability, global functional equations, and an Archimedean local converse theorem. Iterating the pair construction and calibrating the remaining real factor at twist rank two then produces a unitary isobaric representation on GL(64) realizing the triple symmetric-cube tensor outside a finite set and exactly at infinity. The proof does not use the conjectural ambient triple-product Poisson formula or a general GL(16) × GL(4) tensor-product theorem. This v1.0 is the unified public preprint corresponding to the internal `v0.1r6 FinalEditorialFreeze`. It integrates and substantially strengthens two earlier public preprints by making the proof architecture self-contained with respect to the author's earlier application papers and by adding exact Archimedean recovery for both the fixed-pair transfers and the final GL(64) output.