Deep Episodes in the Prime-Power Checkpoint Program: An Unconditional Terminal-Episode Theorem, an Exact Chebyshev Bridge, and Sharp Recovery Recurrence -- Part 5

Suzuki's pointwise criterion makes the Riemann hypothesis equivalent to the nonnegativity of the real function Ψ = −gζ. Prime-power checkpoint theory reduces this continuum condition to one constrained minimum per prime-power interval and rigorously certifies all intervals through q = 10¹⁰. The unresolved tail is then organized into active episodes, each represented by its last active event, or recovery witness, together with a possible terminal episode that never recovers. This paper proves unconditionally that the terminal alternative cannot occur. The checkpoint workload is exactly a shifted weighted Chebyshev error plus an explicit positive integrable remainder, and its Mellin transform has exact cancellations at s = 0 and s = 1/2. A Pringsheim–Landau argument combined with analytic continuation excludes an eventually active workload. Together with Suzuki's sign changes of the weighted Chebyshev error, this yields infinitely many finite recovered episodes and recovery witnesses. Hence, after the verified finite base, RH is equivalent to 𝒱_q ≥ 0, or equivalently 𝒥_q ≤ 𝒞_q, at every recovery witness. We also prove the exact classical bridge Ψ(log X) = (π²/4 + 2G − 8) − α log X − S(X) − ϱ∞(X), where α = ζ′(1/2)/ζ(1/2), S(X) = Σ_{n≤X} Λ(n)n^(−1/2) log(X/n) − 4√X, and ϱ∞(X) > 0 is explicit and O(X^(−5/2)). Thus checkpoint margins are exactly deficits from an explicit weighted-Chebyshev comparison level, so classical estimates transfer directly to the checkpoint problem. Under RH this level is approached relatively densely from below, whereas failure of RH forces arbitrarily late crossings above it. Finally, under RH Ψ is uniformly almost periodic with Bohr mean μ = −Ξ″(0)/Ξ(0) and lim inf Ψ(t) = 0 as t → ∞; unconditionally, lim inf Ψ(t) ≤ 0. Therefore no positive uniform reserve floor can close the tail. The remaining open problem is an RH-independent proof of 𝒥_q ≤ 𝒞_q at every recovery witness. The Riemann hypothesis remains open.