Power-Saving Center-Failure Counts for the Derivative Laguerre Inequalities of the Riemann Xi Kernel

Abstract

This preprint studies the asymptotic number of failures of the derivative Laguerre inequalities at the symmetry point of the standard Jacobi-theta kernel associated with the Riemann Xi function. Building on a previous transition analysis, it strengthens the known qualitative counting law by proving an explicit power-saving error term. Using the best available irrationality measure for the value of the Riemann zeta function at two, the paper obtains an unconditional exponent of 0.303746 plus an arbitrarily small positive loss. The proof exploits the exact quadratic arithmetic structure of the transition phase rather than treating its values as a generic sequence modulo one. Two extremal resonant indices determine a rational slope, after which an integer determinant confines all remaining resonances to a small family of rational lines. Each line is then governed by a factored quadratic congruence and an explicit Chinese-remainder structure. The paper also determines the exact local modulus controlling these congruence classes and identifies the arithmetic alignment responsible for the remaining loss in the error term. This provides a concrete route toward possible further improvements without requiring a stronger general irrationality measure.