A Weak GL(16) Transfer for the (3,3) Symmetric-Power Tensor

This preprint constructs a weak automorphic $\mathrm{GL}_{16}$ transfer associated with the tensor product of two symmetric-cube representations attached to non-CM elliptic curves over $\mathbf Q$. More precisely, writing $$\Sigma = \operatorname{Sym}^{3}\pi_i, \qquad \Theta = \operatorname{Sym}^{3}\pi_j,$$ it constructs an isobaric automorphic representation $\Pi_{33}$ of $\mathrm{GL}_{16}(\mathbf A_{\mathbf Q})$ such that, for every finite place $v$ outside a fixed finite set, $$\phi_{\Pi_{33},v} = \operatorname{Sym}^{3}\phi_{\pi_i,v} \otimes \operatorname{Sym}^{3}\phi_{\pi_j,v}.$$ The proof uses the Booker--Krishnamurthy converse theorem. For twists of ranks $3 \le d \le 15$, the required analytic properties are supplied by a representation-scoped restricted Poisson construction for mixed triples $(4,4,d)$. Canonical finite normalization is obtained from contraction to the split Piatetski--Shapiro--Rallis core, moving-frame covariance, and the two Tate roots in the GHL factorization. The real factor is calibrated in rank three using $$\operatorname{Sym}^{3}V \otimes \operatorname{Sym}^{3}V \simeq \operatorname{Sym}^{6}V \oplus \operatorname{Sym}^{4}V \oplus \operatorname{Sym}^{2}V \oplus \mathbf 1,$$ together with Newton--Thorne symmetric-power automorphy, and is propagated across the remaining ranks by exact cross-rank scalar rigidity. The new low-rank input is an unconditional boundary theorem for rank two. Since the literal GGH affine-$\Psi$ geometry does not exist at $r=2$, the construction retains the two genuine rank-four GGH factors and replaces the third factor by the ordinary $\mathrm{GL}_2$ Whittaker/intertwining functional equation on a boundary incidence space. A finite-stage cutoff-transfer construction realizes the coefficient-one degree-$32$ basic section, while full-stabilizer equivariance, projective reassembly, source-level Whittaker unfolding, stable local epsilon normalization, and auxiliary-only clearing close the global boundary functional equation. The rank-one twist is supplied by ordinary Rankin--Selberg theory. The final converse-theorem output is weak: no ramified finite-place or Archimedean compatibility is asserted.