Parity-Reduced Transverse Wronskians, Higher Dipole Invariants, and Rigorous Continuum Certification for Riemann's Xi-Function

We prove that the transverse Wronskian W(d,t) = (1/2) ∂d|Ξ(t − id)|2 of Riemann's completed Ξ-function satisfies odd transverse parity W(−d,t) = −W(d,t) with identically vanishing even derivatives ∂d2kW(0,t) ≡ 0, so that the parity-reduced Wronskian W̃(d,t) := W(d,t)/d removes the d → 0+ boundary-layer degeneracy in the Speiser–Hinkkanen–Lagarias positivity criterion. Our proof combines a division-free Wigner–hyperbolic integral representation governed by the monotone kernel sinchyp(dy) = sinh(dy)/(dy) ≥ 1, closed-form Cauchy–Riemann identities for the higher transverse dipole invariants c2m−1(t) and non-singular partial derivatives ∇W̃, and dual directed-rounding interval evaluators based on Euler–Maclaurin summation and Jacobi-theta quadrature. Applying two-dimensional interval mean-value enclosures, the integer argument principle, and continuum Lipschitz bounds with IEEE-754 rounding control, we establish strict positivity of the leading dipole moments c1(γk), c3(γk), c5(γk) > 0 at the first ten non-trivial zeros and certify the absence of off-critical zeros across 4,987,101 continuum cells covering [0, 0.50] × [0, 55]. This repository contains the 15-page AMS-LaTeX manuscript, native pdfTeX PDF, 300-DPI vector figures, AI agent skill specification, and standalone directed-rounding interval and GPU verification suite (scripts/verify_paper2_parity_reduced_wronskian.py).