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

Suzuki's screw-function criterion reformulates the Riemann hypothesis as a pointwise positivity problem for a continuous real function. The prime-power checkpoint theorem reduces its arithmetic region to one constrained minimum on each interval between consecutive prime powers. Here we realize that reduction as a directed-rounding streaming certificate and rigorously certify all 455,062,595 prime-power event intervals through q = 10^10. No interval has a nonpositive certified lower bound. The smallest certified lower bound occurs for the interval beginning at q = 34,186,367 and equals 0.021498559834383781734344241384770946... > 0. Consequently, Ψ(t) > 0 for 0 < t ≤ log(10^10). The certificate uses a segmented prime stream, exact prime-power event ordering, convex interval closure, and directed MPFR arithmetic, so positivity is established over complete intervals rather than inferred from a numerical sampling grid. An exact Chebyshev reformulation further separates the unresolved tail into signed arithmetic memory and an explicit archimedean barrier with residual asymptotics B(x) = (1 − α) log x + (C − 4) + O(x^−5/2). Across the ten decimal scales of the certified domain, the drawdown-required fraction decreases while the smallest certified lower bounds remain nonmonotonic, exhibiting sparse critical-contact geometry without implying an asymptotic positive floor. This is a finite computer-assisted theorem, not a proof of RH; its analytic contribution is to recast the remaining infinite-tail question as an explicit signed-memory no-crossing problem.