One-Sided Prime Bounds, Signed Chebyshev Primitives, and Finite Pick Positivity Sharp RH-Equivalent Criteria, an Exact Prime Certificate, and the Global Spectral Obstruction

Let E(x) = ψ(x) − x and define the continuous prime-power functionP(X) = 2 − √X + ∑n≤XΛ(n)(n−1/2 − X−1/2) = 1 + 12∫ X1E(u)u−3/2 du.We give a unified account of the one-sided Mellin–Landau criterion RH if and only if P iseventually bounded above, together with two explicit fixed-threshold equivalents: P(X) ≤2 − √2 for every X ≥ 2 and P(X) ≤ P(4) for every X ≥ 4. The thresholds are exact endpointmaxima under RH, but neither is logically stronger than eventual boundedness. We establishthe latter equivalences using an integrated zeta explicit formula, an absolute zero-sum estimateunder RH, the classical first-zero height bound, and a one-sided Mellin–Landau converse.Unconditional global boundedness is not obtained.A separate computation certifies by outward-rounded integer arithmetic that P(X) < P(4)for every real 4 < X ≤ 2,000,001. The certificate counts 148,933 primes, 149,235 prime powersand 5,176 possible interior stationary intervals, with no unresolved interval comparisons; ahigh-precision independent sieve reproduces P(2,000,001) = −0.689616512417929746 . . .. Wegive the exact algorithmic enclosure and its limits. A signed-tail reformulation is identifiedexplicitly as another RH equivalence: absolute-value majorants cannot establish it, whilezero-phase cancellation in the integrated explicit formula becomes available after RH isassumed.Finally, an exact rational four-node Pick matrix is positive definite even though itsgenerating meromorphic function has an upper-half-plane pole. We prove the correspondingphenomenon for every prescribed finite node set, distinguishing interpolation existence fromglobal identification of the sampled function. The manuscript provides a single theoremsequence, portable verification programs and two-dimensional plots. It does not prove RH oran independent global spectral closure theorem