Foundations of the Substrate Dark Sector: Derivation of F1 Cancellation, F2 Saturation, and the Cosmological IR Cutoff

The substrate framework's three big claims — the MOND scale a₀ = cH₀/(2π), the cosmological constant ρ_DE = (3/4)M_P²H², and the unification of dark matter, dark energy, and modified inertia within F2 — rest on three structural inputs whose status as derivations versus assumptions has been ambiguous. This paper derives each. First: F1 does not gravitate, not as a renormalization-condition input, but because time emerges from substrate perturbations (P1–P3), and the standard QFT vacuum-energy calculation requires time to define mode operators — F1 has no time, so QFT vacuum energy is not a physically defined quantity there. The dim-reg machinery, applied to the substrate field with IR cutoff k_min = H, makes this automatic: scaleless (F1) integrals vanish by analytic continuation while scale-ful (F2) integrals give finite results. Second: F2 saturates the Cohen-Kaplan-Nelson holographic bound because it is in thermal equilibrium with the cosmological horizon at the Gibbons-Hawking temperature T_GH = H/(2π); saturation is the equilibrium result, not an additional postulate. The horizon entropy S = A/(4ℓ_P²) = πM_P²/H² directly determines the accessible mode count. Third: the 2π in a₀ = cH₀/(2π) arises from the angular completeness of the ℓ = 1 great-circle mode on F2's bounding sphere — the same 2π that appears in dipole-radiation integration. The factor-of-2 round-trip kinematics is a separate physical condition, encoded in the squared-propagator denominator of the Schwinger-Keldysh kernel, that smooths the transition rather than setting the threshold. Together these three derivations move the substrate framework's foundations from "structural input" to "derived consequence of P1–P8 + standard QFT machinery," leaving the residual ρ_DE factor-of-3 coefficient as the principal remaining open item.