The entropy census of no-boundary topologies: the weight, the horizon it does not need, and the survivors of the null state

The closing paper of the programme begun in A smooth beginning for spacetime and continued in The far side of the seam and the black-hole pair The deficient cap / The only exit. We assemble, gate, and in two respects correct a law relating the no-boundary weight of a universe to its spatial topology. The census: |Ψ[S³]|² ~ e^(24π²/Λ); |Ψ[S¹ x S²]|² ~ e^(16π²/Λ); |Ψ[L(p,1)]|² ~ e^(24π²/pΛ); a polynomial entropically naked norm for T³; and, for S¹ x Σ_g with Σ_g hyperbolic, a negative exponent -(2/3)(g-1)S_dS that deepens with the circle; every row has an owner in the literature or in the companion papers, and the lens ladder, reproduced here by two routes, is due to Fagundes and Vargas. Stated as a law: a real horizon-holding covering solution prices a positive extensive ledger; none prices a non-positive one. The paper's contributions begin where the table ends. First, the orbifold action behind the ladder is defined: the cap singularities of S⁴/Z_p are A_{p-1} points whose smoothing is topologically forced and whose forced (Ricci-flat) resolution contributes nothing—the ladder is Einstein-Hilbert-exact, while the resolved topology χ = 2p makes the deep ladder an amplifier of any Gauss-Bonnet coupling. Second, the law's naive reading is falsified: for p ≥ 3 the weight e^(S_dS/p) is the area of no causal horizon in the quotient—the image-observers' joint causal patch degenerates to a codimension-two set, the late-time event horizon is blind to the identification, and p = 2 is the last case where the causal and extensive notions coincide (a genuine RP² horizon). The corrected law is extensive: the weight is the covering geometry's horizon ledger per fundamental domain, and the quotients thereby adjudicate between the two readings of de Sitter entropy, siding with the Hilbert-space-dimension count against the causal-area count. Third, the census meets the null-state result of Cotler and Jensen: the sphere sector's zero is a division by the volume of the noncompact conformal group of the boundary sphere, and their own criterion—a compact residual group gives a finite norm—is made exhaustive by the Ferrand-Obata theorem: the round S³ is the unique closed carrier of the mechanism, gated group-theoretically here. Their closing picture makes the physical norm a sum over topologies; the survivor census computes that sum's leading terms, and the black-hole universe is its dominant one on the Euclidean contour. On the Lorentzian contour, as Godet has shown, the exponents invert and the torus wins by mapping-class democracy; the ranking is doubly contingent, on the inner product and on the contour class, and both scenarios are stated. Fourth, the Z_p-projected one-loop lens norms are computed through spin two: the Clifford-translation structure renders the projected spectra a finite character identity and the determinant shift ultraviolet-finite; at graviton level the ghost sector shrinks to a compact centralizer, the transverse-traceless spectra stay positive, and the Polchinski phase of the quotient partition functions is stable down the ladder; the lens norms do not vanish. A corollary fixes the arrow of time as a homology class (the mass-clock alternative is falsified and retained), and the naked torus is run to the end: it has the arrow's structure—a branch pair for the contour to choose—with no load, no law and no bank; its conjugate branch members carry identical perturbation weights, the vacuum's slope dressed by one boundary symplectic invariant, (1/2)Re[c1 P_c], the fibre's breathing at the boundary, derived and gated. Corrections made en route—thirteen, several of which became theorems, the last of which withdrew a claim—are part of the record.