A Two-Sided Modular Origin for Dark Energy: Vacuum Filtering by Jones Conditional Expectation and the Geometric-Mean Closure Radius

We propose a two-step structural mechanism for cosmic acceleration in the Two-Sided Closure Theory (TSCT) framework. First, the physical vacuum is the image of the Jones conditional expectation E: A(W) → N_0 acting on the ambient QFT algebra, filtering out vacuum modes that do not commute with the closure-defect operator Δ̂. At Jones level k = 3, the commuting subalgebra is finite-dimensional with Jones entropy Γ_core ≈ 2.528 nats per closure cell. Second, the relevant closure scale is the geometric mean of the Hubble and Planck scales, R²_cl = (H_0 M_Pl)⁻¹, forced by consistency of modular flows on the two-sided closure pair. The surviving dark-energy density is ρ_Λ^TSCT = Γ_core H_0² M_Pl², reproducing the observed ∼10⁻¹²² M_Pl⁴ hierarchy up to the Jones entropy prefactor. The canonical equation-of-state line is w_0 = −160/169 ≈ −0.9467 and w_a = −[M:N]²(1+w_0) = −36/169 ≈ −0.213, the latter from the Jones-index rate theorem applied to the cosmological closure sector. The paper is self-contained; its predictions are assessable without reference to the broader TSCT programme.