Minor-Arc Cancellation for the Dense Reciprocal Transform: A Complete Non-Local Theorem after Canonical Local-Conductor Extraction

This preprint proves a complete non-local theorem for the centered reciprocal defect arising from a dense two-dimensional lattice decomposition. The central structural ingredient is an exact GCD-contraction identity. If g=(h,r)g=(h,r), r=gqr=gq, and H=h/gH=h/g, then every gcd-resonant configuration contracts exactly to the same reciprocal defect at the reduced scale HH, together with an explicit elementary correction. This identifies qh(r)=r(h,r)q_h(r)=\frac{r}{(h,r)} as the canonical reciprocal conductor. After extracting the local range qh(r)≤(log⁡(3h))Cq_h(r)\le (\log(3h))^C, the genuinely non-local region is treated in two complementary regimes. Unbalanced dyadic boxes are controlled through multiplicative Fourier expansion, primitive-conductor decomposition of imprimitive characters, generalized Gauss sums, and the primitive multiplicative large sieve. The nearly balanced shell is handled using Wright's 2026 trilinear Kloosterman-fraction theorem, followed by Vaaler reconstruction. For every B>0B>0 and every fixed divisor-bounded coefficient class, the full centered defect equals its canonical local projection plus OB,κ ⁣(h(log⁡(3h))−B).O_{B,\kappa}\!\left(h(\log(3h))^{-B}\right). The theorem is deliberately restricted to the dense reciprocal lattice model. It does not claim to solve the universal classical minor-arc problem, does not evaluate the remaining local projection, and does not by itself imply binary Goldbach. The application-specific local analysis and the associated Face-Residue Cancellation problem are treated separately in a companion preprint.