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

Building on Suzuki's pointwise criterion for the Riemann hypothesis, this paper reduces its continuous sign condition to one uniquely determined checkpoint on each prime-power segment. The key geometric input is an exact factorization of the smooth archimedean reservoir's curvature. Its sign is controlled by the cubic x³ − x − 1, whose unique positive root is the plastic constant ρ < 2. The curvature transition at log ρ therefore precedes the first prime-power event at log 2. Consequently, every arithmetic segment lies in the uniformly strictly convex regime. Each checkpoint margin admits a restricted Legendre representation, an exact Bregman drawdown identity, explicit curvature bounds, and a two-state event recurrence. A directed-rounding MPFR certificate proves positivity on all 5,762,859 event segments through 10⁸. The smallest certified lower bound is approximately 0.02149858082367. Suzuki's zero expansion provides a complementary asymptotic constraint. Under the additional assumptions of the Riemann hypothesis, simple nontrivial zeros, and rational linear independence of their positive ordinates, the checkpoint margins have infimum zero. Thus no proof can rely on a uniform positive asymptotic floor. The finite certificate closes the declared range only; the remaining infinite nonnegativity statement is equivalent to the Riemann hypothesis.