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

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=K², we work in the low-frequency, short-Kummer-conductor range. The central arithmetic object is the weighted exact-character energy EK=∑_ξ W_ξ², where W_ξ 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 (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 u_* U, and the norm core r R yields the shell energy bound E(G;U,R) _ε K1+ε (R/G+√RUG²). Summing the dyadic shells and treating the negative-norm sector separately gives the surface-scale estimate EK_ε K2+ε. Inserted into a factorized large-sieve splice, this proves Blow,short_ε K3/2+ε, corresponding to the endpoint β=1/2 in the normalization Dβ+εK1/2+ε. The theorem is deliberately local in frequency and conductor; no full high-frequency or global WBRT statement is claimed.