The Remaining Riemann-Hypothesis Tail: From Prime-Power Checkpoints to Smoothed von Mangoldt Bounds

Abstract

For Suzuki's Riemann-zeta screw function g_zeta, the associated real function Psi = -g_zeta satisfies RH if and only if Psi(t) >= 0 for every real t. A prime-power checkpoint theorem reduces the positive half-line to one constrained minimum between consecutive prime powers, and directed interval arithmetic verifies all such intervals through q = 10^10. This paper addresses the remaining tail. At every post-prime-power state capable of generating a new minimum, we prove the exact balance V_q = C_q - J_q, where C_q is a determined capacity and J_q is one nonnegative, logarithmically smoothed von Mangoldt sum. Thus the unresolved internal event timing enters through one scalar cost. We first bound that cost by the exact terminal load. In the finite-spectral regime, a weighted von Mangoldt prefix bound requiring only a finite verified range of zeta zeros is pinned to the exact reference state and clipped by the terminal bound; the shared archimedean normalization cancels exactly. The resulting adaptive upper bound is never weaker than either input and can be strictly sharper. A complete proof would require showing that this upper bound never exceeds the exact capacity at any future active event. That all-event inequality remains open. RH is not proved.