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 full structure—a branch pair for the contour to choose, a member-odd perturbation split with no load, no law and no bank; its unfunded arrow watermarks every mode with one boundary symplectic invariant, -(1/2)Re[i c1 π_c], derived and gated. Corrections made en route—twelve, several of which became theorems—are part of the record.