Understanding the Riemann Hypothesis: From the primes through Riemann's zeros and Suzuki's screw function to Mittermeier's prime-power checkpoints and the open infinite-tail problem

Abstract

Suzuki's screw-function criterion makes the Riemann hypothesis equivalent to the pointwise nonnegativity of the real function Ψ = −gζ. The prime-power checkpoint approach developed here starts from this equivalence and identifies an exact geometric structure in Ψ. The curvature of its smooth archimedean component admits a factorization whose sign is governed by the cubic x³ − x − 1. Its positive root is the plastic constant ρ, and the inequality ρ < 2 places every interval between consecutive prime powers entirely in the strictly convex regime. Consequently, each such interval has one uniquely determined minimum. Suzuki's continuum sign condition can therefore be expressed as an infinite sequence of exact, event-aligned checkpoint inequalities, one for each prime-power interval. The checkpoint margins admit a restricted Legendre–Mangoldt representation that compares a discrete arithmetic moment with a continuous archimedean cost. The same geometry yields an exact Bregman description of the loss incurred by an arithmetic event and an exact two-state event recurrence, so that the relevant interval minima are determined analytically rather than inferred from a sampling grid. Using directed MPFR interval arithmetic, this structure gives a rigorous finite certificate: Ψ(t) > 0 for every 0 < t ≤ log(10¹⁰), covering all 455,062,595 complete prime-power intervals in that range. For the remaining infinite tail, the event dynamics lead to an exact active-state reserve criterion and to a scalar balance in which the unresolved arithmetic contribution is a nonnegative, triangularly smoothed von Mangoldt loading. Endpoint pinning produces an exact archimedean cancellation, while the corresponding Chebyshev normal form exposes the signed arithmetic memory that is lost under sign-blind majorization. The remaining proof obligation is thereby isolated as a uniform, RH-independent one-sided bound for this smoothed loading at every future active state. That bound is not proved here, and the Riemann hypothesis therefore remains open. The handbook develops this chain self-containedly from the classical foundations of prime distribution and the zeta function to the checkpoint construction, the rigorous finite theorem, and the precise obstruction that remains on the infinite tail.