One-Sided Prime Criteria and Explicit Hermitian Approximations to the Zeta Resolvent

We prove that the Riemann hypothesis is equivalent to an eventual upper bound, or an eventual lower bound, forthe single real prime-defined function P (X) = 2−√X +∑n≤X Λ(n)(n−1/2 −X−1/2). Equivalently, P (X) ≤ 3/4 forevery X ≥ 2. Under zero-freeness of ζ on Re s > θ, 1/2 ≤ θ < 1, for the corrected prime approximation HX to H =−ζ′/ζ −1/(s−1), we establish |HX (θ)| < 2 and the explicit error bound |H(s)−HX (s)| ≤ (3|s|/θ)X−δ (1+|s−θ|/δ),where δ = Re s − θ > 0. This removes the unspecified prime-tail constant and yields a conditional two-nodepositivity certificate for every X ≥ 2272. An adaptive rational Pick construction produces positive finite Hermitianresolvent functions converging locally uniformly to the shifted completed zeta logarithmic derivative on the samesupported region. The one-sided equivalence is proved, but the prime inequality is not established independently;RH and extension to the critical line remain unresolved here.