Prime-power checkpoints for the Riemann zeta screw function: Plastic-Constant Convexity and a Restricted Legendre--Mangoldt Representation

Abstract

Building on Suzuki's pointwise criterion for the Riemann hypothesis, this paper reduces its continuous sign condition to one uniquely determined checkpoint on each interval between consecutive prime powers. The key geometric input is an exact factorization of the smooth archimedean reservoir curvature. Its sign is controlled by the cubic x^3 − x − 1, whose unique positive root is the plastic constant ρ < 2. The curvature transition at log ρ therefore precedes the first arithmetic event at log 2, placing every prime-power interval in a uniformly strictly convex regime. Each checkpoint margin admits a restricted Legendre–Mangoldt representation, an exact Bregman drawdown identity, explicit curvature bounds, and a two-state event recurrence. A streaming directed-rounding MPFR certificate, using exact prime-power event ordering and the same scalable proof kernel used for larger finite domains, rigorously certifies all 5,762,859 prime-power event intervals through q = 10^8. No interval has a nonpositive certified lower bound. The smallest certified lower bound is 0.0214985808236729410096... > 0 for the interval beginning at q = 34,186,367, and consequently Ψ(t) > 0 for 0 < t ≤ log(10^8). Suzuki's zero expansion supplies a complementary asymptotic constraint: under RH together with simplicity of the nontrivial zeros and rational linear independence of their positive ordinates, the checkpoint margins have infimum zero. Thus the finite theorem cannot be extrapolated to a uniform positive asymptotic floor. The remaining infinite checkpoint tail is the unresolved RH-equivalent obligation.