Canonical bounded pair defects for roots of quadratic congruences

Abstract

Let f in Z[x] be an irreducible quadratic polynomial and let A_{q,f}(X) count solutions of f(n) = 0 mod q in 1 <= n <= X.For an unramified odd prime power q, the local root system contains either zero or two residues per period.We study the dyadic pair-count defect binom(A_{q,f}(2X),2) - 4 binom(A_{q,f}(X),2).Within a natural affine class of quotient-linear corrections, boundedness uniquely determines a canonical renormalization.The resulting local observable is universal and takes values only in {-3,-1,0,1,2,3}, uniformly over irreducible quadratics, unramified prime powers, and scales.The construction extends to every integer dilation, with explicit canonical coefficients.For centered quadratic families x^2-D, a degree-two identity on multiquadratic Frobenius splitting patterns yields a dimension-uniform bounded shell observable.We also give an exact logarithmic arithmetic-product identity for the single-polynomial defect.For negative D, the clean shell 4X/3 < p <= 2X has exact cancellation of the root-independent contribution; Homma's discrepancy theorem then yields logarithmic cancellation of size X/(log X)^eta for every eta < 4/9.The analytic estimate is an application of known prime-root discrepancy theory; the principal contribution is the canonical bounded pair-defect structure.