Finite Dilation Algebra of Round Euclidean B⁴/S³. Valuation Profiles: Exact Invariants, Homogeneous Gravity Closure, and Executed Null Tests

Abstract

What if the homogeneous dark-sector problem could be reduced from an arbitrary function of cosmic scale to a finite algebra of geometrically admissible profiles, with exact invariants that can be tested directly against numerical cosmology? This work develops such a construction for round Euclidean B⁴/S³ valuations in closed FLRW geometry. The physical spatial carrier remains S³, while B⁴ serves only as an auxiliary quasilocal filling and valuation structure, not as a physical extra dimension. The universal work map E₄(A) → (ε, p), with ε = E₄/V₃ and p = −dE₄/dV₃, is exactly invertible and therefore provides representation rather than prediction. Predictive restriction enters only after assuming continuous, rigid-motion invariant and additive Euclidean convex valuations on B⁴. On the round homothetic family this forces the five dilation weights S_val = {1, 0, −1, −2, −3}, so that every admissible homogeneous profile satisfies P_val(𝒟)ε = 0, where 𝒟 = A d/dA and P_val(z) = (z−1)z(z+1)(z+2)(z+3). The resulting polynomial action is the finite semisimple algebra 𝔄_D ≃ ℝ[z]/(P_val) ≃ ℝ⁵. Primitive projectors isolate each branch exactly, annihilator ideals identify active support, and derivative or sampled Hankel pencils reconstruct the finite spectral content. The finite spectrum supports a much richer invariant calculus than a conventional equation-of-state parametrization. For positive mixtures, the branch fractions form an exponential family with 𝒟² ln ε = Var_π(s) ≥ 0 and 𝒟w = −Var_π(s)/3 ≤ 0. Exact Bhatia–Davis remainders detect interior spectral support; normalized observable jets provide positivity certificates; Hankel inertia records the signs of branch amplitudes; and the Fisher–Rao orbit obeys the global rigidity bound L_F ≥ π, with equality precisely for two-branch support. Every positive two-branch orbit has a universal logistic/tanh normal form and constant Schwarzian {w, ln A} = −(s₊−s₋)²/2. Within the declared boundary-local curvature-measure class, the bulk s = 1 branch is removed, leaving {0, −1, −2, −3}. After quotienting the homogeneous expansion history by the standard vacuum, curvature-degenerate and dust powers, the unique nonstandard direction is a⁻¹, corresponding to w = −2/3. Crucially, an a⁻² physical stress is not identified with spatial curvature: exact coefficient projectors enforce the split Ω₂,val = C₋₂ − Ω_K^geom, making the distinction operational rather than semantic. With general relativity explicitly declared, the same machinery produces an Einstein–Dilation Spectral Closure of the homogeneous gravitational sector. The Friedmann constraint defines the physical density ledger E_F = F − Ω_K^geom a⁻², while the Bianchi identity gives P_F = −E_F − 𝒟E_F/3. Once H₀, Ω_K^geom, the spectral support and branch amplitudes are supplied, the finite projectors and curvature split leave no additional independent homogeneous GR datum. This is a closure of the homogeneous sector, not a microscopic derivation of matter or dark energy: reconstruction ≠ source action ≠ amplitude selection, and consistency testing ≠ independent spectral identification ≠ likelihood evidence. The framework also proves an important negative result: topology and valuation structure can determine the allowed functional spectrum, but cannot by themselves generate nonzero dimensional amplitudes. The amplitudes belong to the realized cosmological history and must be fixed by an additional physical principle or empirical calibration. This revision adds executed null tests to the theorem-level structure. A pinned closed-FLRW CLASS implementation reproduces the imposed ρ_X ∝ a⁻¹ law with maximum residual 1.88 × 10⁻⁹, while the supplied best-fit ledger returns C₋₁ = 0.174541, C₋₂ = −0.010000, and therefore Ω₂,val = 0 after removal of geometric curvature. The dark-sector Hankel pencil recovers the expected support {−1, 0} with positive inertia, while deliberately inserting the negative closed-FLRW curvature coefficient into the physical moment measure creates exactly one negative inertia direction—demonstrating numerically why curvature must be separated before physical positivity is assessed. The executed DESI DR2 BAO-only profile gives q_X = 1.364 and p_one-sided = 0.121, corresponding to only ≈1.17σ, so no detection is claimed. Because w_X = −2/3 is imposed in the executed source lift, recovery of s = −1 is correctly interpreted as a consistency test rather than independent discovery of the valuation root. The decisive next step is therefore a covariance-aware free-spectrum reconstruction in which the dilation exponents themselves are inferred from data before comparison with S_val. The result is a framework in which finite geometric classification → exact invariant closure → executed numerical null tests → empirical likelihood form a single auditable chain without collapsing distinct logical levels into one claim.