Terminal High-Frequency Structure for Quadratic-Root Bilinear Sums: Reciprocal-Prime Decay, Residual Orthogonality, and Fixed-Core Twist Families

Abstract

We study the terminal high-frequency regime of a quadratic-root bilinear architecture in which the long prime variable has length $D = K^2$ and a reciprocal phase of size $K$ survives. This regime is deliberately separated from the low-frequency exact-Kummer theory treated previously by the author. Four structural results are established. First, by combining a Vaughan decomposition with an unrestricted bilinear monomial estimate of J. Wu, we prove the smoothed terminal reciprocal-prime bound $$\sum_{n} \Lambda(n) W\left(\frac{n}{y}\right) e\left(\frac{x}{n}\right) \ll y^{5/6+\varepsilon}$$ for $x \asymp y^{3/2}$, which becomes a normalized $K^{-1/3+\varepsilon}$ decay at $y = D = K^2$. Second, for the physical centered residual Fourier packet, we prove an exact two-scale energy kernel, a divisor expansion, and a reciprocal-chirp large sieve with squared-operator scale $K^{3/2+\varepsilon}$. Third, we prove a General Shift Orthogonality identity: every nonprincipal multiplicative shift has local autocorrelation $O(q^{-1/2})$, with exact annihilation for odd shifts; this tensorizes to squarefree moduli and admits a cross-modulus form. Fourth, for quadratic Kummer characters with a fixed signed rational norm core, the nonabelian component is constant up to finitely many classes: the family is a finite union of rational quadratic twists of fixed dihedral base forms, with relative rational conductor $\ll K$ in the short-input block. These results sharply reduce the terminal obstruction, but they do not prove a global sub-$3/4$ theorem: the full same-core reciprocal-prime coupling and the different-core noncollision sector remain open.