A Rigorous Finite-Dimensional Verifier for an Odd Pole-Free Riemann Positivity Criterion

We develop a rigorous numerical verification framework for a finite-dimensional positivity problem arising from an odd, pole-free formulation connected to the Riemann hypothesis. The computational objective is not to infer positivity from floating-point evidence, but to certify explicit matrix inequalities using interval arithmetic and independently checkable finite reductions. The central finite object is an endpoint matrix H_N(L) entering the exact odd pole-free Galerkin block through a diagonal congruence. Using rigorous ball arithmetic via Arb, a finite-interface bound and a high-block lower bound were independently certified. The interface bound satisfied U_final <= 1.01738391101161043, the pole-free high block was certified to satisfy DH,Hᵖᶠ >= 1.1785076903962208908... I, and the resulting Schur separation was approximately 0.1611237793846104656. These calculations establish the stated finite numerical inequalities. They do not constitute a proof of the Riemann hypothesis. The remaining mathematical burden lies in verifying the exact analytical bridge from the arithmetic source to the finite criterion and in proving that no omitted finite prime, pole, or archimedean contribution can invalidate the required positivity. Five outstanding analytical obligations are enumerated explicitly. The bundle includes all source code, certificate JSON files, and test suites required for independent reproduction of the certified bounds.