Homothetic Valuation Spectrum of a Quasilocal B⁴ Filling and an Effective Residual Sector on Compact S³

Abstract

This work develops a geometric-variational framework for describing an effective residual sector in a compact cosmology while keeping the physical arena strictly four-dimensional: M = I × S³_A. The four-ball B⁴. A is introduced only as a quasilocal filling, geometric measure, and auxiliary valuation structure—not as an additional physical dimension or five-dimensional bulk. From the homothetic identity V₄/V₃ = A/4, the filling energy E₄(A, ξ) generates the universal reconstruction map E₄(A, ξ) → (ε_D, p_D, Q_D), with ε_D = E₄/V₃ = Aρ₄/4 and p_D = −dE₄/dV₃. The construction yields the background conservation identity, the inverse representation ρ₄(A) = 4ε_D(A)/A, and the local equation-of-state relation w_D(A) = −1 − (1/3)d ln ε_D/d ln A. It also generalizes to a d-dimensional boundary through w_d = (β − d − 1)/d for a filling density scaling as ρ_{d+1} ∝ A⁻ᵝ. The central result is the classification of the full homothetic valuation spectrum. Under Euclidean-invariant, continuous, and additive valuation assumptions for the auxiliary convex filling in R⁴, the admissible energy contains both the bulk intrinsic-volume contribution and the complete hierarchy of boundary curvature invariants. For the round homothetic representative, this produces ε_D(A) = γA/4 + α₀ + 3α₁/A + 3α₂/A² + α₃/A³, corresponding to the five equation-of-state branches w = −4/3, −1, −2/3, −1/3, 0. The boundary-only restriction γ = 0 removes the bulk phantom-like branch, while the additional endpoint reduction γ = α₁ = α₂ = 0 retains only the boundary-volume and Gauss-Kronecker terms. The arbitrary homogeneous background function is thereby reduced to the vacuum-dust form ε_D(a) = ε_q + ε_w a⁻³, with p_D = −ε_q. Importantly, these reductions are stated as explicit closure principles rather than consequences of topology alone. A minimal covariant four-dimensional completion realizes the selected sector as vacuum energy plus a conserved dust current, giving cₛ² = 0, π_μν = 0, and u^μ∇_μu^ν = 0. Thus, the endpoint-selected A⁻³ branch is perturbatively equivalent to cold pressureless matter at the effective level. The framework fixes the available scaling structure and the functional form of the reduced residual sector, but it does not determine the numerical amplitudes ε_q and ε_w, a unique microscopic origin, or a complete dynamics for A(τ). At the minimal observational level, the completion reproduces curved ΛCDM with an effective cold component; its potentially distinguishing content lies in the valuation classification, endpoint selection, compact S³ harmonic spectrum, non-minimal branches, and any future geometric principle capable of fixing the amplitudes.