A Zero-Side Positivity Certificate for Suzuki's Screw Function through 4·10²⁰: Window Means and One-Sided Criteria — Companion to Part 5

Suzuki proved that the Riemann hypothesis (RH) is equivalent to the nonnegativity of Ψ = −g_ζ, the negative of the screw function of the Riemann zeta function. In classical terms, Ψ(log X) = C₀ − F(X), where F is a normalized first-order Riesz mean of Λ(n)/√n and C₀ = π²/4 + 2G − 8. We prove without assuming RH that Ψ(t) > 0 for 0 < t ≤ 47.48, that is, F(X) < C₀ for 1 < X ≤ 4·10²⁰. The proof combines Suzuki's unconditional zero expansion, the verification of RH up to height 3·10¹², an explicit bound for the zeros above that height, and a directed zero-side certificate built on 3000 certified zeros. Under RH, Ψ returns arbitrarily close to zero, so the pointwise inequality has no margin. Means of Ψ over windows of fixed logarithmic length h have one: under RH they lie in an explicit band [μ₋(h), μ₊(h)] with μ₋(h) > 0, and unconditionally RH holds if and only if they are bounded below. Averaging also damps the possible zeros off the line, and the same verification proves without RH that all means over unit windows are positive for 0 ≤ t ≤ 104.63. We also prove unconditionally that ∑_(n≤x) Λ(n)n^(−1/2) − 2√x + ζ′(1/2)/ζ(1/2) changes sign infinitely often, that RH holds if Ψ(t) ≥ −C_ε e^(εt) for every ε > 0, and that lim inf_(t→∞) Ψ(t) ≤ 0. Under RH alone, the mean rate of decrease of Ψ exists and is positive.