A Conditional Weak $\mathrm{GL}(12)$ Transfer for the $(2,3)$ Symmetric-Power Tensor

We establish a conditional weak automorphic realization on $\\mathrm{GL}_{12}(\\mathbb A_{\\mathbb Q})$ of the tensor $$\\operatorname{Sym}^2\\pi_i \\boxtimes \\operatorname{Sym}^3\\pi_j,$$ under the packet-level and analytic hypotheses stated in the manuscript. The argument is organized around the Booker–Krishnamurthy converse theorem. Its required family of twists is supplied by combining classical Rankin–Selberg theory in rank one, the Kim–Shahidi $\\mathrm{GL}_2 \\times \\mathrm{GL}_3$ product in rank two, and a representation-scoped restricted Poisson construction for twist ranks $3 \\le d \\le 11$. The latter provides analytic continuation, finite-order growth, and the finite-place functional equations needed for the converse theorem. Two normalization problems are treated explicitly. At the bad finite places, highly ramified twisting, residual Laurent-unit rigidity, conductor alignment, and automorphic calibration identify the restricted terminal scalar with the standard local epsilon package. At the Archimedean place, a rank-three virtual automorphic calibration is combined with an exact $C=-\\tfrac{1}{2}$ cross-rank contraction to recover the standard gamma factor throughout the required twist range. The resulting automorphic representation agrees with the expected tensor parameter at all but finitely many finite places. Under an additional simple self Rankin–Selberg pole hypothesis, the automorphic output is cuspidal. The result is deliberately weak and conditional: it does not assert prescribed compatibility at the excluded ramified places, a general $\\mathrm{GL}_3 \\times \\mathrm{GL}_4$ tensor-product functoriality theorem, or the full ambient Poisson conjecture underlying the restricted construction.