Correlation-Generated Factorability in a Generic Sector of Folded Quadratic Kloosterman Sums

We isolate a factorability mechanism that appears after a TT^* expansion of a folded quadratic-root Kloosterman sum. Starting from a two-variable sum with modular inverse phase and a smooth radial phase, the off-diagonal correlation for two short variables r₁,r₂ has a common difference d=r₂-r₁. An exact inverse-difference identity and additive reciprocity transform the finite phase to an inverse trace function with modulus M=q r₁r₂, while the entire radial oscillation may be absorbed into the deformation factor of the arithmetic exponent-pair framework of Wu and Xi. Thus factorability is generated by correlation variables rather than assumed of the original sieve modulus q. In the squarefree pairwise-coprime generic sector, under the corresponding Wu--Xi admissibility hypotheses, the one-step arithmetic A-process yields a pairwise q-1/6 saving and, after TT^*, a normalized R-1/2+q-1/12 bound. Under the two-step admissibility hypotheses, the balanced endpoint gives a C-3/56+o(1) saving. We state the remaining non-generic gcd/squarefull bookkeeping separately. The result is a reduction theorem for the generic sector, not a complete Brun--Titchmarsh theorem.