The Modal-Horizon Framework IV: Casimir-in-de Sitter, Closed-Form Spectral Weight, and the Geometric Origin of the Coupling Structure

This paper establishes the quantum field theoretic foundations of the Modal-Horizon Framework (MHF) coupling structure. All coupling parameters are derived from three geometric inputs with no free parameters. What the cosmic event horizon does to the vacuum. The cosmic event horizon acts as a causal filter on the Bunch-Davies quantum vacuum, suppressing modes above a characteristic wavelength. The coupling amplitude γc is the integrated spectral weight of this filter. The ℓ = 0 contribution has an exact closed form via the polylogarithm Li₄, verified to machine precision. The ℓ = 1 contribution (22% of the total) requires a new transcendental object — the Lerch-Si transcendent — not reducible to standard polylogarithms. Deriving γc = 0. 1801 without free parameters. Two independent arguments converge on the same value. First, the MHF action is linear in the background field, so the fluctuation field has no bare curvature coupling. The one-loop renormalisation group equation drives the coupling to the unique infrared fixed point ξ = 1/6 (conformal coupling), at which the filter gives γc = 0. 1801. Second, the same value follows from the SO (1, 4) group-theoretic boundary between principal and complementary series representations of the de Sitter symmetry group. Both routes are independent of any fitting to data. The geometric origin of Z = 1/2 and the bare coupling exponent. The bare coupling exponent nbare = 0. 843 was previously treated as an empirical input with no theoretical explanation. This paper shows it is a derived quantity. The observer sits at the centre of the cosmic event horizon sphere. The kinetic energy of the MHF scalar at the boundary is shared equally between the interior (the observer's accessible region) and the exterior (beyond the horizon, causally inaccessible). This interior/exterior kinetic sharing is the physical origin of the factor Z = 1/2 that appears throughout MHF. It gives nbare = n★ (1 + K/8πH) = 0. 8395, agreeing with the empirical value 2nₒbs = 0. 840 at 0. 06 per cent. The complete derivation chain. Together with Paper III, the full MHF coupling structure — n★, nbare, nA, nₒbs, γc, and nγc — is now derived from three geometric inputs: n★ = 3/4 from the heat kernel, K = 3H from the de Sitter extrinsic curvature, and Z = 1/2 from the closed-sphere kinetic sharing. No free parameters remain in the coupling structure. Scalar field mass and CMB-S4 prediction. The conformal fixed point fixes the MHF scalar field mass at mₑff c² = 1. 9 × 10⁻¹⁶ eV, with a Compton wavelength comparable to the Hubble radius. If the scalar thermalised above the QCD transition, this predicts ΔNₑff ≤ 0. 043 — a near-miss prediction for the CMB-S4 experiment. Open problems. Three open problems are stated explicitly: the second-order Hadamard calculation in Schwarzschild-de Sitter spacetime needed to rigorously derive the power-floor running coupling; the closed-form expression for the Lerch-Si transcendent; and the proof that the Dirichlet boundary condition at the cosmic event horizon is uniquely selected by the MHF action.