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

Abstract

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 $r_1, r_2$ has a common difference $d = r_2 - r_1$. An exact inverse-difference identity and additive reciprocity transform the finite phase to an inverse trace function with modulus $M = q r_1 r_2$, 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.