Prime-power checkpoints for the Riemann zeta screw function: Plastic-Constant Convexity and a Restricted Legendre--Mangoldt Representation -- Part 1

Masatoshi Suzuki's Riemann-zeta screw function gζ has the associated real positivity function Ψ = −gζ, and his pointwise criterion makes the Riemann hypothesis equivalent to Ψ(t) ≥ 0 for every real t. We show that after the first arithmetic event its smooth reservoir is strictly convex because A″(log x) = (x³ − x − 1)/(√x(x² − 1)); the unique positive root ρ of x³ − x − 1 satisfies ρ < 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 = 10⁸, with no nonpositive lower enclosure. The smallest certified lower enclosure is 0.0214985808236729410096… > 0 for the interval beginning at q = 34,186,367, so Ψ(t) > 0 for 0 < t ≤ log(10⁸). Under RH alone, absolute uniform convergence of Suzuki's zero expansion makes Ψ uniformly almost periodic and forces checkpoint margins to have infimum zero; the series completion strengthens the methodological consequence to an unconditional no-positive-floor theorem at recovery witnesses. The infinite checkpoint tail remains open.