Hybrid Restricted Triple Products and Coherent Weak Tensor Transfers: Archimedean Recovery, Boundary Completion, and Finite Tensor Closure

This reference paper develops a restricted converse-theorem route to **finite-exception weak tensor functoriality** for fixed automorphic inputs. The analytic construction is representation-scoped: auxiliary data are built only for the fixed representations and finite local sectors needed in the argument. The proof uses the proved affine-$\Psi$ Fourier–Poisson theory on genuine axes, normalized $\mathrm{GL}_2$ Whittaker–Mellin theory on rank-two boundary axes, and an internal fixed-radial continuation argument; it does **not** assume a conjectural ambient total Poisson formula. The analytic endpoint is formulated in the exact form needed for the Booker–Krishnamurthy converse theorem. For the tensor candidates constructed here and the required finite-unramified unitary cuspidal twists, the completed Rankin–Selberg $L$-functions are shown to be entire of finite order and to satisfy the standard dual functional equation. The proof includes a finite-sector Archimedean construction in the GGH Poisson domain, auxiliary-only continuation, split Piatetski–Shapiro–Rallis normalization, residual-scalar rigidity, stable-twist comparison, and Archimedean local recovery. The paper then closes the rank-two boundary. Under the stated temperedness and common-good-place hypotheses, this yields a boundary-free fixed-rank weak tensor operation for every positive rank, extends it to isobaric inputs, and proves independence of auxiliary choices, ordering, and parenthesization at the level of the resulting global isomorphism class. In particular, a fixed finite family $B_1,\ldots,B_k$ in the theorem's class has a unique unitary isobaric weak tensor realization of degree $$\prod_{i=1}^k \deg(B_i),$$ with the expected tensor parameter at the real place and outside a finite exceptional set. For symmetric powers attached to non-CM elliptic curves, the framework covers every fixed finite tuple in the stated range, including the all-boundary triple on $\mathrm{GL}_8$ and the triple symmetric-cube tensor on $\mathrm{GL}_{64}$. The paper deliberately does **not** claim arbitrary all-place ramified compatibility: normalization on the remaining exceptional finite supercuspidal sectors is isolated as a separate local problem. Version 3.0 is a major reference-master revision of the earlier Version 2.0 release. It integrates rank-two boundary completion and coherent finite tensor closure, expands the source-level verification of the admissible-data and local-normalization interfaces, adds end-to-end worked proof routes, and adopts a reader-oriented monograph format with explicit dependency maps and scope ledgers.