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

Description We construct a finite-exception weak tensor operation for fixed automorphic representations of general linear groups by combining restricted triple-product harmonic analysis with converse-theorem methods. The analytic construction uses the proved affine-$\Psi$ Fourier–Poisson theory of Getz–Gu–Hsu on genuine higher-rank axes, ordinary normalized $\mathrm{GL}_2$ Whittaker–Mellin theory on rank-two boundary axes, and an internal fixed-radial continuation argument. No conjectural ambient triple-product Poisson summation formula 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. The remaining rank-two boundary is closed by combining a reoriented hybrid construction with the classical $\mathrm{GL}_2 \times \mathrm{GL}_2$ tensor product. The resulting weak tensor operation is available in every fixed positive rank, extends to unitary isobaric inputs, and is coherent under permutations and parenthesizations. More precisely, for any fixed finite family $$\pi_i \text{ on } \mathrm{GL}_{q_i}(\mathbb{A}_{\mathbb{Q}})$$ of unitary generic cuspidal automorphic representations whose Archimedean components are tempered and which are simultaneously unramified tempered at one finite prime, the construction produces a unique unitary isobaric automorphic representation on $$\mathrm{GL}_{\prod_i q_i}(\mathbb{A}_{\mathbb{Q}})$$ whose local Langlands parameter is the tensor product $$\bigotimes_i \phi_{\pi_i,v}$$ at the real place and at every finite place outside a finite exceptional set. The resulting global isomorphism class is independent of the order and parenthesization of the iterated weak tensor products. Applications include all fixed finite tensor products of symmetric powers associated with non-CM elliptic curves, the all-boundary triple on $\mathrm{GL}_8$, and the triple symmetric-cube tensor on $\mathrm{GL}_{64}$. The theorem is intentionally a finite-exception statement: no arbitrary ramified local compatibility is asserted at the excluded finite places. Version 2.0 Notes: This version substantially revises and strengthens the pole-exclusion layer of the proof. The earlier universal prescribed-point nonvanishing formulation is replaced by a separated-family argument applied only after deep local separation. The revised proof uses type-supported non-Archimedean data, fixed-sector Bernstein/Hecke regularization, and a new separated-family auxiliary nondegeneracy step before pointwise pole exclusion. The finite-exception weak tensor-transfer, finite tensor-closure, and coherence theorems are retained with this corrected and more robust proof architecture. Creator Lee Byoungwoo Independent Researcher, Daejeon, Republic of Korea Keywords automorphic representations; Langlands functoriality; weak tensor transfers; tensor product functoriality; converse theorems; Booker-Krishnamurthy converse theorem; triple product L-functions; restricted triple products; Poisson summation; affine Psi-bundles; symmetric powers; Archimedean local converse theorem; isobaric automorphic representations; finite tensor closure; coherent tensor products; non-CM elliptic curves; GL(64); Bernstein-Hecke regularization