Local Symmetry and Residual Decomposition of (s) Near Nontrivial Zeros

The Riemann Hypothesis (RH) conjectures that all nontrivial zeros of the Riemann zeta function ζ(s) lie on the critical line ℜ(s) = 1/2 . Traditional analyses rely on global analytic continuation and functional equations. We apply the Structural Residual Com- pensation (SRC) framework, modeling residuals via structured asymptotic expansions, to study local behavior near these zeros. Building on SRC’s success in global series acceleration (e.g., seventh-order for ζ(2)), we use Taylor expansions around simple zeros ρ=1/2 + itk, with perturbations s = ρ+ ε, yielding a compensator φρ(s) = ζ′(ρ)(s−ρ) and residual δρ(s) = O(ε2), even in ε. Numerical experiments confirm this second-order behavior, with δρ(s)/ε2 →ζ′′(ρ)/2. We conjecture universal regularity, reinforcing critical line symmetry consistent with the RH.