From Nariai Instability to Big Bang II: Exact Kantowski Sachs Birth Phase and Curvature-Locked Statistical Anisotropy

Paper II of the Nariai cosmogenesis program. Companion to From Nariai Instability to Big Bang: A Model for Universe Nucleation from Maximally Saturated Black Holes (Paper I; DOI: 10.5281/zenodo.20662636). In the Nariai cosmogenesis model, the daughter universe is born from the Bousso–Hawking anti-evaporation of a Nariai-limit black hole as an exact super-Nariai Schwarzschild–de Sitter geometry in its cosmological (Kantowski–Sachs) region: a transient anisotropic phase that isotropizes exponentially onto de Sitter space. This paper converts the correlated-signature program sketched in Paper I into a computed, falsifiable template chain parametrized by a single number, the e-fold budget ΔN between the birth throat and the horizon exit of the present Hubble scale. Main results: Exact background: the expansion-rate anisotropy of the birth phase is sourced by the anisotropic spatial curvature, decays as a−2, and is rigidly locked to the residual curvature parameter (ΔH = 3H Ωcurv); the Weyl (mass) hair decays as a−3. Modes and initial state: the mode equation reduces exactly to Bunch–Davies form on the late-time attractor; the Weyl hair contributes zero to the superhorizon power at first order; solving the initial-state problem at the birth throat with the exact geometry gives the physical response coefficient c = −1.00 ± 0.05, with state ringing far below cosmic variance. Quadrupolar template: g*(k) ≈ −3 |Ωk,0| (k0/k)2 — negative, k−2, and with no free amplitude. A future curvature detection fixes the required low-ℓ statistical anisotropy; any mismatch falsifies the channel. Honest negative result: the minimal Λ-vacuum gradient channel yields a dipolar modulation A ≲ 10−5, four orders of magnitude below the observed hemispherical asymmetry A ≈ 0.072 ± 0.022, killed independently by a non-perturbative amplitude requirement and by the Grishchuk–Zel'dovich quadrupole bound. Constrained spectator ceiling: a curvaton-like extension reaches at most Amax ≈ 0.030 (band 0.015–0.06), independent of Ωk,0, touching only the ~2σ low end of the observed value; saturation requires an occupation n1 ~ 103 in the n = 1 spectator harmonic. The instanton verdict (new in v1.0): the Hartle–Hawking occupation of the n = 1 spectator harmonic, computed on the Ginsparg–Perry S2×S2 throat instanton, is n1 ≈ 0.25 — a factor ~4×103 below the target, mass-independent for light fields, protected by the symmetry of the saddle and by the curvature gap of the n = 1 sector. The spectator channel is closed, and the model's dipolar modulation is capped at A ≲ 10−4 across all computed channels: a confirmed primordial hemispherical asymmetry at the observed level would be evidence against this framework. The falsifiable correlation set: Ωk < 0, a negative curvature-locked k−2 quadrupolar modulation, any primordial dipolar modulation sharing the same axis and k−2 shape with a k-independent amplitude ratio, and a common suppression scale — all functions of the single parameter ΔN. Methodological note. The paper follows the working discipline of Paper I: every claim is labeled as derived, borrowed, or heuristic; no free functions are introduced; and negative results are recorded as primary outputs. The source package contains the verification and numerical scripts (exact-background checks, mode-equation reductions, first-order kernels, throat-state evolution, dipole transfer, the constrained spectator optimum, and the throat-instanton saddle verification and occupation computation). Analytical derivations and numerical cross-checks were developed in an extended collaboration with Claude (Anthropic); all remaining errors are the author's.