The smooth 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 transposes. The S1 x S2 closure saddles are smooth caps: each member of the conjugate pair is exactly a complex Schwarzschild-de Sitter geometry, of constant complex mass M(u,v) and complex Euclidean period, closing regularly on one of its two horizons—the closure datum p is the tip's opening angle, Θ/2π = -p identically, and the standard prescription p = -1 is the smooth cap. The cap's weight is the horizon boundary term: the member action is I = p S_tip - 16π²c1/s exactly, a horizon term plus a wall term, which at the Nariai slice reduces to I = -S_tip with S_tip = S_tot_Nariai/2, so the pair carries |Ψ|² = e^(S_tot_Nariai): the black-hole universe squares to the Nariai ledger, by the same smooth-horizon mechanism as the sphere squares to the de Sitter one. These statements are exact in a units dictionary derived and gated here, and they reproduce, as their classical shadow, the large-circle statements of Turiaci and Wu. The geometry contains one horizon; the second is virtual, behind the data surface, which plays the role of York's cavity wall. That is why the two-horizon first law cannot be posed on this or any related family: at fixed Λ both horizon areas are functions of the one mass—Birkhoff's theorem—and the three added-data candidates (the b=0 closure, the opening angle as data, horizon-anchored Dirichlet families) are each closed exactly: the empty static patch, the one-horizon Legendre pair (S_tip, Θ/2π) and the closed two-horizon geometry with I = -Σ(Θ_i/2π)S_i = P Vol and identically zero energy. A short final section states what the smooth cap does to its perturbations—circle-mode indices ±n/2, real and selective as on the sphere; mode weights An = ±n/2, member-antisymmetric and linear in n—and the coalescence exit's exact closure, which Paper II takes up. Version 1 of this paper reported a complex conical deficit at the tip, a drifting mass, and imaginary indices; all three were one sign error in the reconstruction of the saddle's geometry, invisible to every value-level gate. The error, its diagnosis and what it withdrew are retained in the record, with the nine earlier corrections.