Prime Errors and the Quasi-Riemann Frontier A Certified Conditional Boundary and Its Parameter-Optimization Barrier

We refine the amplified reflected-numerator parameter comparison of Xu and Sang, conditional on thesame analytic inputs imported from OpenAI, to obtain the exact admissible half-planeRe s > 4374433248150000000000 = 0.87488664962.This improves the preceding certified parameter boundary 0.87488666 by 1.038 × 10−8, without claiming anew mean-value estimate. The continuous endpoint inequality is proved by an exact rational polynomialidentity and a negative-discriminant certificate, with all scalar admissibility conditions separately checked.Within the unchanged admissibility system and licensed moment range, we further prove the barrierBlim = 0.874886649616631419 . . .; the displayed boundary lies less than 3.37 × 10−12 above that barrier.These results sharpen the parameter geometry, not the underlying analytic estimates. The original294-page cumulative research manuscript is retained after this current-results introduction as a historicaltechnical record; its earlier targets and abstracts are superseded by the present scoped statement