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

What if a geometric dark-sector reconstruction could be tested not only by fitting a background history, but by asking whether that history belongs to a finite invariant algebra fixed in advance by geometry? This work develops a finite dilation-operator classification for round Euclidean B⁴/S³ valuation profiles in closed FLRW cosmology. The physical spatial carrier remains S³, while B⁴ is used only as an auxiliary quasilocal filling and valuation structure, not as a physical extra dimension. The universal work map E₄(A) → (ε, p) is exactly invertible at the homogeneous level, so reconstruction alone is not predictive. Predictivity begins after imposing the Euclidean convex-valuation hypothesis. On the round homothetic family this reduces the arbitrary homogeneous profile to the five dilation weights S_val = {1, 0, −1, −2, −3}, annihilated by P_val(𝒟) = (𝒟−1)𝒟(𝒟+1)(𝒟+2)(𝒟+3) with 𝒟 = A d/dA. The dilation action therefore factors through the finite semisimple algebra 𝒜_D ≃ ℝ[z]/P_val ≃ ℝ⁵. Primitive projectors, minimal annihilators, normalized jets, Hankel pencils and positivity conditions turn the construction into an exact membership problem: a reconstructed residual history either belongs to the declared valuation image or produces measurable obstruction amplitudes. The finite spectrum also exposes the internal structure of the residual sector. For positive mixtures, 𝒟² ln(ε/ε) = Var_π(s) ≥ 0* and 𝒟w = −Var_π(s)/3 ≤ 0, giving nonlinear consistency conditions that are independent of a conventional parameter fit. A declared boundary-local filter removes the bulk s = 1 contribution, leaving the boundary spectrum {0, −1, −2, −3}. After separating the standard vacuum, curvature-degenerate and dust powers, the unique nonstandard homogeneous direction is a⁻¹, corresponding to w = −2/3. The analysis also shows why an a⁻² stress contribution cannot simply be identified with spatial curvature: the physical coefficient must be separated from the geometric term through Ω₂,val = C₋₂ − Ω_K^geom. Canonical source realizations exist for the s = −1 and s = −2 branches, while s = −3 admits a conserved pressureless-current realization. At the same time, the framework proves an important limit: topology and valuation structure determine the allowed functional spectrum, but not nonzero dimensional amplitudes. Reconstruction, source realization, amplitude selection and empirical identification therefore remain logically distinct levels. Version 2.0 adds an executed exact-invariant audit to this theorem layer. Using the supplied pinned CLASS closed-FLRW realization and DESI DR2 BAO-only artifacts, the numerical background reproduces ρ_X ∝ a⁻¹ with a maximum residual of 1.88 × 10⁻⁹, while the first closed S³ scalar mode agrees with the expected compact-mode value at a relative difference of 6.25 × 10⁻⁶. At the supplied best-fit arithmetic point, the exact projector returns Ω_X = 0.174541, the curvature split gives Ω₂,val = 0, and the Hankel-inertia diagnostic correctly identifies the negative closed-FLRW curvature coefficient as geometric rather than as a negative physical valuation branch. The executed BAO-only boundary likelihood test gives q_X = 1.364 and one-sided p = 0.121, so no detection is claimed. This distinction is essential: because w_X = −2/3 is imposed in the executed source lift, recovering the s = −1 scaling is consistency evidence, not independent spectral discovery. The next decisive step is therefore a covariance-aware free-spectrum reconstruction in which the dilation weights themselves are inferred from data rather than inserted beforehand. The result is a framework in which geometric classification → exact invariant tests → executed transfer consistency → likelihood evidence are connected without collapsing them into a single claim.