The only exit: the no-boundary state of a black-hole universe, II

Paper II of a two-part study extending the no-boundary programme of A smooth beginning for spacetime to the black-hole topology. Paper I of this pair, in its second version, established that the no-boundary saddles of the S1 x S2 universe are smooth caps of complex Schwarzschild-de Sitter, each closing on one of its two horizons, that the pair's weight is that horizon's boundary term (|Ψ|² = e^(S_tot_Nariai) at the Nariai slice), and that the coalescence exit from the fluctuation catastrophe is closed by an exact discriminant. This paper is about what remains. Its computed core is a bridge theorem: the mode weights of the circle sector, An = ∓n/2 on the two members of the conjugate pair, are exactly ∓(1/2)m1ω1—the Hartle-Hawking member prepares the exact instantaneous vacuum of every circle mode at the boundary, with no adiabatic corrections at any mode number and no dependence on the circle size (the complex lapse cancels identically), while its conjugate prepares the inverse-Gaussian anti-vacuum. The catastrophe is therefore not a property of the state but of the pair split: one member carries Bunch-Davies-type data, the other its conjugate, and the causal contour tracks the catastrophic one in exact agreement with the numerical flare Ghosh et al. found in the causal amplitude, and with the polarity of Feldbrugge-Lehners-Turok on S3. With coalescence closed and the bias member-resolved, the only road out is the horizon's dissipation, and the paper states the erasure programme in the series' transposed currency: the repository is the Nariai ledger, S_tot_Nariai = (2/3)S_dS—the black-hole universe's erasure capacity is poorer than de Sitter's by exactly one third; the state contains one collector—the horizon its cap closes on; the second lies behind the data surface, and Paper I's Birkhoff no-go says no datum brings it in; and the rate tower once only conjectured as the Nariai transpose of the smooth-beginning trilogy's gated Γ_L = (L+1)H—is now a derived law, gated by an independent Pöschl-Teller reduction and two numerical cross-checks: Γn = (n + 1/2)√(Λ) running in the overtone rather than the circle mode the original conjecture guessed. The classical Gaussian data fixed here is exactly what the one-loop determinant of Turiaci and Wu dresses; the join is stated as a prediction-shaped open problem. Corrections made en route—including one committed in this paper's own gate, and the one inherited from Paper I's version 2—are retained.