Prime-Defined Approximations and Zero-Free Boundaries for the Completed Zeta Resolvent

We construct an entire function from finitely many von Mangoldtcoefficients whose locally uniform limit is the pole-regularized logarithmic derivativeof the Riemann zeta function on every fixed zero-free half-plane. The construction isexplicit, including its endpoint correction at the prime cutoff and its removable valueat s = 1. Assuming the recently announced zero-free result Re s > 7/8, it convergeson that entire half-plane, with a compact-set error determined by the classicalChebyshev remainder estimate. Under the completed-function coordinate changethis supplies a prime–gamma approximation to the shifted logarithmic derivativeappearing in a July 2026 Prime–Weil–Pick–Hermitian programme. We derive arational lower bound for a two-node target Pick matrix in the newly available partof the strip. Finally, we show that convergence of the same prime-defined sequencethroughout Re s > 1/2 is equivalent to the Riemann hypothesis, rather than anautomatic consequence of the seven-eighths premise. Finite arithmetic and numericalreplays distinguish exact identities, target positivity, computed approximations, andthe remaining global problem