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 classifying homogeneous residual sectors in a closed FLRW cosmology with spatial carrier Σ ≃ S³. The four-dimensional ball B⁴_A is used exclusively as an auxiliary quasilocal filling and valuation structure, not as a physical extra dimension or five-dimensional bulk. The homothetic identity V₄/V₃ = A/4 generates the reconstruction map E₄(A, ξ) → (ε_D, p_D, Q_D), where ε_D = E₄/V₃ = Aρ₄/4 and p_D = −dE₄/dV₃. This construction yields background conservation, the inverse representation ρ₄(A) = 4ε_D(A)/A, the local equation-of-state relation w_D(A) = −1 − (1/3)d ln ε_D/d ln A, and the general dimensional law w_d = (β − d − 1)/d. The universal map can represent every differentiable homogeneous density within the adopted class; its predictive content therefore begins only when an invariant restriction selects a finite family of admissible filling energies. The central result is the full homothetic valuation spectrum. Under continuous, additive, and Euclidean-invariant valuation assumptions for the auxiliary convex filling in R⁴, the round boundary density takes the finite form ε_D(A) = γA/4 + α₀ + 3α₁/A + 3α₂/A² + α₃/A³, producing the five equation-of-state branches w = −4/3, −1, −2/3, −1/3, 0. The work strengthens this classification by showing that these branches form a finite spectral subspace of the homothetic dilation operator D = A d/dA, with eigenvalues s = 1, 0, −1, −2, −3. Membership in the complete round valuation profile is characterized exactly by (D − 1)D(D + 1)(D + 2)(D + 3)ε_D = 0. Spectral projectors then extract five invariant branch amplitudes and provide sharp obstruction tests: the bulk branch is removed by I₁ = 0, while the minimal endpoint density is characterized by D(D + 3)ε_D = 0 ⇔ ε_D = ε_q + ε_w a⁻³. Thus the valuation spectrum becomes an operational differential classification rather than merely a list of fitted power laws. Inside a precisely declared minimal endpoint class—local and Diff(S³)-invariant, additive, free of local propagating endpoint degrees of freedom, curvature duplication, fixed external structures, and additional functional length scales—the endpoint energy is conditionally classified as E_end(A, Q) = σV₃(A) + μQ, where Q ∈ Z is an oriented topological charge. It induces ε_D(a) = ε_q + ε_w a⁻³ and p_D = −ε_q, while the dust scaling follows from the structural kernel p = 0 ⇔ E = const ⇒ ε ∝ a⁻³. A minimal covariant completion realizes this sector as vacuum energy plus a conserved pressureless current with cₛ² = 0, π_μν = 0, and u^μ∇_μu^ν = 0. Topology may quantize Q, but it does not determine the dimensional amplitudes σ or μ. At the homogeneous observational level the minimal completion is exactly degenerate with curved ΛCDM; genuine observational superiority would therefore require background integrability, independent compact-S³ spectral-geometric support, and robust penalized statistical improvement against the nearest competing cosmologies.