Weak Tensor Transfers, Finite Tensor Closure, and Coherence from Restricted Triple Products

We construct a finite-exception weak tensor operation for fixed automorphic representations of general linear groups by combining restricted triple-product harmonic analysis with a converse-theorem argument. The analytic construction uses the proved affine-$\Psi$ Fourier–Poisson theory of Getz–Gu–Hsu on genuine axes, ordinary normalized $\mathrm{GL}(2)$ Whittaker–Mellin theory on rank-two boundary axes, and an internal fixed-radial continuation argument. No conjectural ambient total Poisson formula for the broader triple-product fiber-bundle framework is assumed. After canonical finite-place normalization and Archimedean recovery, the completed twists required by the Booker–Krishnamurthy converse theorem are shown to be entire of finite order and to satisfy the standard dual functional equations for the finite-place-unramified twisting family needed by the converse theorem. The remaining rank-two boundary is closed by combining a reoriented hybrid construction with the classical $\mathrm{GL}(2) \times \mathrm{GL}(2)$ automorphic tensor product. The resulting weak tensor construction extends to unitary isobaric inputs and is coherent under permutations and parenthesizations. In particular, for any fixed finite family of unitary generic cuspidal automorphic representations that are tempered at infinity and simultaneously unramified tempered at one finite place, the construction gives a unique unitary isobaric weak tensor realization with the expected tensor parameter exactly at the Archimedean place and at every finite place outside a finite exceptional set. Applications include all fixed finite tensor products of symmetric powers attached to non-CM elliptic curves, including the all-boundary triple on $\mathrm{GL}(8)$ and the triple symmetric-cube tensor on $\mathrm{GL}(64)$. The theorem is deliberately finite-exception. No arbitrary ramified all-place local compatibility is asserted at the excluded finite places. This Version 1.0 is the journal-facing, theorem-critically audited edition of the work. A longer reference-master treatment containing expanded background, proof architecture, and verification routes is available separately as Zenodo DOI: 10.5281/zenodo.23160929.