Exact Kummer Fiber Rigidity and Surface-Scale Energy for Quadratic-Root Bilinear Sums: A 1/2 Low-Frequency Threshold

Abstract

We study a quadratic-root bilinear architecture in which an early Cauchy--Schwarz step leads naturally to a $3/4$-type exponent scale. At the canonical relation $D=K^2$, we work in the low-frequency, short-Kummer-conductor range. The central arithmetic object is the weighted exact-character energy $$E_K=\sum_\xi W_\xi^2,$$ where $W_\xi$ is the total homogeneous-scaling weight of weighted-primitive slopes in one exact quadratic Kummer squareclass. The main structural input is exact fiber rigidity: a Kummer squareclass fixes the signed rational squarefree norm core $r$ and, up to finitely many bad-prime choices, the weighted primitive content $G$; moreover $\gcd(G,r)$ is supported on a fixed set of bad primes. After Simon's integral conic parametrization and exact gcd normalization, the positive-norm charts satisfy an exact quartic norm-circle identity. A harmonic normal form and primitive determinant repulsion then give a Harmonic Annular Spacing estimate. Localizing simultaneously in $G$, the primitive coordinate $\vert{}u_*\vert{}\asymp U$, and the norm core $\vert{}r\vert{}\asymp R$ yields the shell energy bound $$E(G;U,R) \ll_\varepsilon K^{1+\varepsilon} \left(\frac{R}{G}+\frac{\sqrt{RU}}{G^2}\right).$$ Summing the dyadic shells and treating the negative-norm sector separately gives the surface-scale estimate $$E_K\ll_\varepsilon K^{2+\varepsilon}.$$ Inserted into a factorized large-sieve splice, this proves $$\mathcal B^{\mathrm{low,short}}\ll_\varepsilon K^{3/2+\varepsilon},$$ corresponding to the endpoint $\beta=1/2$ in the normalization $D^{\beta+\varepsilon}K^{1/2+\varepsilon}$. The theorem is deliberately local in frequency and conductor; no full high-frequency or global WBRT statement is claimed.