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

Abstract

Masatoshi Suzuki's Riemann-zeta screw function g_zeta has the associated real positivity function Psi = -g_zeta, and his pointwise criterion makes the Riemann hypothesis equivalent to Psi(t) >= 0 for every real t. We show that after the first arithmetic event its smooth reservoir is strictly convex because A''(log x) = (x^3 - x - 1) / (sqrt(x)(x^2 - 1)); the unique positive root rho of x^3 - x - 1 satisfies rho < 2. Hence every interval between consecutive prime powers has exactly one constrained minimizer, reducing the continuum criterion to one checkpoint inequality per interval. Each checkpoint admits restricted Legendre--Mangoldt and Bregman representations, an exact event recurrence, and explicit curvature bounds. Directed-rounding MPFR arithmetic rigorously certifies all 5,762,859 prime-power intervals through q = 10^8, with no nonpositive lower enclosure. The smallest certified lower enclosure is 0.0214985808236729410096... > 0 for the interval beginning at q = 34,186,367, so Psi(t) > 0 for 0 < t <= log(10^8). Under RH together with simple zeros and rational linear independence of their positive ordinates, Suzuki's zero expansion implies that checkpoint margins have infimum zero. The infinite checkpoint tail remains open.