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

This handbook develops a first-principles path from the distribution of the primes and the zeros of the Riemann zeta function to Suzuki's screw-function criterion, which makes the Riemann hypothesis equivalent to the pointwise nonnegativity of the real function Ψ = −gζ. It then develops the prime-power checkpoint architecture that turns this continuum sign condition into an exact event problem. For the zeta function, the smooth archimedean curvature changes sign at the positive root ρ of x³ − x − 1, the plastic constant; because ρ < 2, every interval between consecutive prime powers lies entirely on the strictly convex branch and therefore contains one uniquely determined minimum. Restricted Legendre–Mangoldt duality, an exact Bregman event loss, and a two-state recurrence make these minima analytically accessible without a sampling grid. Directed MPFR interval arithmetic yields a rigorous finite theorem: Ψ(t) > 0 for every 0 < t ≤ log(10¹⁰), covering all 455,062,595 prime-power event intervals through q = 10¹⁰. Beyond this finite base, the workload Y = −Ψ′ and the active reserve reduce the tail to the exact balance Vq = Cq − Jq. Recovery geometry reduces each recovered busy period to its last active witness, while the service clock τ = A′(t) converts the between-event dynamics to the unit drift dY/dτ = −1. A Mellin-transform argument unconditionally excludes a terminal never-recovering episode; combined with Suzuki's unconditional sign changes of the weighted Chebyshev error, this yields infinitely many finite busy periods and recovery witnesses. The same potential has the exact classical representation Ψ(log X) = L − F(X), with L = π²/4 + 2G − 8, where F is an explicit weighted von Mangoldt expression. Under RH, the zero expansion gives the rigorous envelope 0 ≤ Ψ(t) ≤ 2μ, and uniform almost periodicity yields relatively dense return times at which Ψ is arbitrarily close to zero. Unconditionally, lim inf as t → ∞ of Ψ(t) is ≤ 0, so no completion strategy may rely on a fixed positive reserve floor. The remaining infinite obligation is therefore exactly localized, not solved: one must prove by RH-independent means that Jq ≤ Cq at every recovery witness beyond the certified base. This universal inequality remains open, and so does the Riemann hypothesis.