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 established that the no-boundary saddles of the S¹ × S² universe are smooth caps of complex Schwarzschild-de Sitter, each closing on one horizon, that the pair's weight is that horizon's boundary term, and that the coalescence exit from the fluctuation catastrophe is closed. This paper is about what remains. Its computed core, version 2's vacuum bridge—the Hartle-Hawking member prepares the exact instantaneous vacuum of every circle mode, its conjugate the exact anti-vacuum—is here shown to be a circle-sector theorem, and completed: with a certified mode instrument the sphere modes come out dressed, below the instantaneous vacuum by a law derived in closed form, A_{n,l} = -(1/2)√(n² + ul(l+1)) + (u+1)/(16√(ul(l+1))) + O(l⁻³) for every circle momentum n, whose constant term vanishes precisely on the plateau line that fixes the background weight; the l > 2 gravitons are two-dimensional fields on the cap of mass (l+2)(l-1)Λ (Cardoso-Lemos verified) whose master-variable weights are the same instrument at a shifted eigenvalue, and the rate tower Γ_n = (n + 1/2)√Λ is theirs too. The catastrophe remains a property of the pair split, now in every sector run: scalar, graviton, and below the rotational twist. The join with Turiaci and Wu, stated in version 2 as a prediction-shaped open problem, is closed at the classical level and located at one loop. Their exponent S_dS/3 - 8π²S_dS/27L² is the late-time limit of the exact closure action, with its finite-sphere correction (1 - r_dS²/b1² + ...) supplied and the coefficient exactly zero on the Nariai slice. The l=0 sector is Jackiw-Teitelboim gravity on the complex cap exactly: the closure action is linear in the dilaton datum, its coefficient is the cap's geodesic-curvature term cos θ_b = i√(u-1), and the Schwarzian classical exponent of Maldacena-Turiaci-Yang is that term's large-l truncation; the Schwarzian mode action itself is the isoperimetric deficit of the boundary curve on the complex sphere, ΔS̃ = (v-1)/(2 cos θ_b) Σ_{|n|≥2} (n²-1)|η_n|², with n=±1 the rigid motions. The l=1 sector is the Kaluza-Klein gauge field of the sphere—three copies of two-dimensional Maxwell with coupling b⁴—whose exact partition function on the cap is the SO(3) heat kernel at "time" 3Q/4, Q = N_E(b0⁻³ - b1⁻³)/(3Δb) in closed form; its late-time limit reproduces Turiaci-Wu's rotational coupling K = μ_N S_dS / 4π to seven digits, because the late-time saddle's tip sits at the Nariai radius. Corrections made en route are retained.