Multidimensional Tensor Convolution and the Spectral Cutoff of Lattice Transitions

Ascends the single-frequency phase space of Parts XX-XXI to the product space prodq>3 T^1_q and studies the global twin-twin correlation R(j)=prod_q R_q(j). Main result: R(j) converges absolutely to a strictly positive constant. The mechanism is sharp. For a fixed lag j the constructive resonance states (State A, R_q=q/(q-2), needs q|j; State B, R_q=q(q-3)/(q-2)^2, needs q|3j+-1) deviate by O(1/q) but occur at only finitely many primes, all <= 3j+1. Every prime beyond is locked into State C, R_q=q(q-4)/(q-2)^2 = 1 - 4/(q-2)^2 = O(1/q^2), so the infinite tail is summable and the product converges. This legitimizes a spectral cutoff k*(epsilon): the full j=1 product is R(1)=S_quad/S_twin^2=0.3969, and 1% / 0.1% fidelity need only the primes up to 73 / 557. The resulting Spatial Transition Operator MA,B(j) = rho_B * prod_q Rq,A,B(j) is the Hardy-Littlewood joint singular series re-expressed as a conditional density: a quasi-periodic, small-prime-dominated, long-range correlation (a multi-frequency diffraction grating), exact at integer lags and equal there to the discrete sieve ratio. Closed-form throughout; no new prime data and no infinitude claim.