Finite Dilation Operator Algebra of Homothetic B⁴/S³ Valuations and Homogeneous Closure Tests for Valuation Dark Energy

Abstract

What happens when a geometric reconstruction is forced to become a finite, testable algebra rather than an arbitrary background fit? This work develops a finite dilation-operator framework for homothetic B⁴/S³ valuations in closed FLRW cosmology. The physical carrier remains the compact three-sphere S³, while B⁴ is used only as an auxiliary quasilocal filling and valuation structure, not as a physical fifth dimension. The universal map E₄(A) → (ε_D, p_D, Q_D), with ε_D = E₄/V₃ and p_D = −dE₄/dV₃, is exactly invertible at the homogeneous level and is therefore treated as a representation theorem rather than as predictive closure. Predictivity begins only after the Euclidean convex-valuation hypothesis restricts the admissible filling functionals. For the round homothetic B⁴/S³ geometry this restriction produces the finite dilation spectrum S_val = {1, 0, −1, −2, −3}, with 𝒟 = A d/dA and characteristic polynomial P_val(z) = (z−1)z(z+1)(z+2)(z+3). The resulting operator action factors through the finite semisimple algebra 𝒜_D ≃ ℝ[z]/P_val ≃ ℝ⁵, turning the residual background sector into a genuine finite spectral calculus with primitive projectors, minimal annihilators, invariant branch coordinates and exact membership tests. This algebraic structure yields more than a list of possible equations of state. For nonnegative branch energies, the logarithmic flow satisfies the exact convexity relation 𝒟² ln(ε_D/ε_*) = Var_π(s) ≥ 0, while the effective equation of state obeys 𝒟w_D = −Var_π(s)/3 ≤ 0; additional Bhatia–Davis, spectral-diameter and Hankel-rank bounds provide nonlinear consistency tests for positive mixtures. A filling-relative local curvature-measure boundary filter removes the bulk s = 1 branch and leaves S_∂ = {0, −1, −2, −3}. At the level of H²(a), the genuinely nonstandard quotient direction is then a⁻¹, corresponding to w = −2/3, while an a⁻² physical stress must be kept distinct from geometric curvature through Ω₂,val = C₋₂ − Ω_K^geom. The physically relevant boundary extension is therefore two-dimensional, (Ω_X, Ω₂,val), with the one-parameter valuation-dark-energy branch obtained as the nested subclass Ω₂,val = 0. The construction supplies exact dilation-closure null tests: an empirically reconstructed ε_D(A) either lies in the finite valuation image or produces nonzero obstruction amplitudes that falsify the assumed closure. The work also separates geometric classification from physical realization. Positive s = −1 and s = −2 branches admit independent canonical-scalar realizations, while s = −3 admits a conserved pressureless-current realization. For the nested canonical valuation-dark-energy branch, the background trajectory fixes the scalar potential, rest-frame sound speed, anisotropic stress and mass-flow invariant rather than leaving them as arbitrary phenomenological functions. A first Bianchi IX robustness result shows that weak homogeneous squashing of S³ preserves the dilation grading: [𝒟, Lᵢ] = 0 and [𝒟, A²Δ_IX] = 0, while anisotropy enters through compact harmonic mode mixing rather than through a new dilation spectrum. At the same time, several limits are proved sharply. Topology alone cannot determine nonzero dimensional amplitudes; a Chern–Simons or S⁴ gluing condition can constrain orientation or topological phase but cannot generate the vacuum amplitude without an independent scale. Likewise, global topological charge, local dust dynamics and dimensional coupling data are kept logically distinct. The result is therefore not a claim that topology by itself “produces dark energy,” nor that a successful homogeneous reconstruction already constitutes a complete cosmological model. Its central advance is the conversion of an initially arbitrary residual profile into a finite invariant operator problem with spectral projectors, positivity bounds, curvature-splitting diagnostics, source-existence realizations and falsifiable null tests. The hierarchy is kept explicit: reconstruction ≠ source ≠ amplitude ≠ transfer ≠ likelihood. A supplementary non-flat CLASS interface is supplied for the nested canonical VDE realization, but executed Boltzmann spectra and likelihood evidence are not claimed here. The work thus establishes a finite homogeneous invariant classification and a concrete route by which compact S³ geometry can be confronted with cosmological data rather than protected from them.