Prime-Power Checkpoints for Suzuki's Riemann Zeta Screw Function: Rigorous Positivity Certificate through q = 10^10 and an Explicit Chebyshev-Memory Barrier

Abstract

The prime-power checkpoint theorem reduces the pointwise Riemann-hypothesis criterion associated with Suzuki's screw function g_zeta to complete interval minima of Psi = -g_zeta. We implement that reduction as a streaming directed-rounding certificate and close all 455,062,595 prime-power event intervals through q = 10^10, with no nonpositive lower enclosure. The smallest certified enclosure is 0.021498559834383781734... > 0 for the interval beginning at q = 34,186,367, so Psi(t) > 0 for 0 < t <= log(10^10). Exact event ordering, strict convex interval closure and directed MPFR arithmetic certify complete intervals rather than a sampling grid. The same function admits the exact normal form Psi(log x) = B(x) - I_R(x), separating an explicit archimedean barrier from signed Chebyshev-error memory. Because B(x) < 0 for x >= 2, RH forces I_R(x) < B(x) < 0: preserving sign history is therefore essential to the cancellation. A complementary zero-side bound explains how finite verified zero information can support a second finite positivity domain, but its reach depends on a separately certified positive margin of the retained zero sum and cannot remove the infinite-tail obstruction. The computation is a rigorous finite base theorem, not a proof of RH.