The deficient cap: the no-boundary state of a black-hole universe, I

Paper I of a two-part study extending the no-boundary programme of A smooth beginning for spacetime to the black-hole topology. The preceding series argued that the no-boundary saddle of the S3 universe carries the complete thermodynamics of the de Sitter horizon in its own data, that the Feldbrugge-Lehners-Turok fluctuation catastrophe is the fingerprint of data placed on a branch point, and that the horizon's dissipation erases the resulting bias wherever its ledger can afford it. This paper begins the same programme on the black-hole topology: the no-boundary state of a universe with spatial sections S1 x S2. The anatomy inverts. Where the S3 saddle caps the geometry smoothly, the S1 x S2 saddles are refused a smooth cap: at the distinguished slice b = 1/√(Λ) their tips carry the exact complex conical deficit δ±/2π = 1 ± i√(u - 1), one full turn per member plus an imaginary part that is pure boost—the tip is a Lorentzian cone of surface gravity √(u - 1). The pair's total deficit is exactly 4π, independent of the boundary data, and 4π at the Nariai sphere is the two horizons' whole ledger: AN(δ+ + δ-)/8πG = S_tot_Nariai. Consistently, each member's action satisfies |Re IE| = S_tot_Nariai/2 exactly, so the pair carries |Ψ|² = e^(S_tot_Nariai): the black-hole universe squares to the Nariai ledger, not the de Sitter one, and its entropy lives in the deficit, not the cap. These statements are exact in a units dictionary derived and gated here (8πG=1 reduction = 8π x the standard Kantowski-Sachs Lagrangian, with H² ≡ Λ), and they reproduce, as their classical shadow, the large-circle statements of Turiaci and Wu: the wavefunction's boundary-size independence (exact on the slice, approached as 1/u off it) and the norm's S2 x S2 form. The rest of the anatomy is a mapped obstruction: the constant-lapse saddle's Misner-Sharp mass drifts (the saddle is isometric to no single Schwarzschild-de Sitter geometry), both horizon entropies ride that single complex mass—their gradients in the data are exactly parallel—so the two-horizon first law cannot even be posed on the closure family, and the honest differential identities are a half-share tangency at Nariai and an exact three-root period lemma β- + β+ = -β0. A short final section states what the deficient cap does to its perturbations—circle-mode indices exactly imaginary, mode weights An = ±n/2, member-antisymmetric and linear in n—and the coalescence-exit's exact closure, which Paper II will take up as the forced dissipative exit. Corrections made en route are retained in the record.