Quantum Horizon Gravity IV: Coupling Constants from the Quantum Vacuum at the Cosmic Horizon

The coupling constants of Quantum Horizon Gravity (QHG, formerly the Modal-Horizon Framework) — the theory in which the cosmic event horizon filters the Bunch–Davies vacuum and couples to matter — are derived from first principles with no free parameters, completing the QHG parameter structure. Three results establish this. First, the coupling amplitude γc has an exact closed form via the fourth polylogarithm: I_γ^(ℓ=0)(ν) = 6(2π)⁻⁴ Re[A Li₄(r₁)/r₁ + Ā Li₄(r̄₁)/r̄₁] + I_SB, verified to machine precision across ν ∈ [0.30, 0.85]. The ℓ=1 contribution (22 per cent of total) requires a new transcendental object, the Lerch-Si function S_Si(p) = p⁻¹ Σ pⁿ Si(2πn), not reducible to standard polylogarithms and constituting an open problem in analytic number theory. Second, γc = 0.1801 is derived without free parameters from two independent routes: the conformal coupling fixed point (ξ = 1/6, ν_IR = 1/2, the principal-series boundary of SO(1,4)) and the one-loop renormalisation group flow from ξ_bare = 0. Both routes agree because cos²(πν/2) = 1/2 at both the UV (ν = 3/2) and IR (ν = 1/2) fixed points — a non-trivial symmetry of the CEH filter. Third, the bare coupling n_bare = 0.8395 = 2n_obs is derived from the heat-kernel root n* = 3/4 with the Gibbons–Hawking–York extrinsic curvature correction. The physical origin of Z = 1/2 is that the kinetic energy of the QHG scalar at the closed CEH sphere is shared equally between the observer's accessible interior and the causally inaccessible exterior — an entanglement condition at a thermal causal boundary. Together, these results derive n*, n_bare, n_obs, γc, and nγc = 1/(4π) from three geometric inputs (n* = 3/4, K = 3H, Z = 1/2) with no free parameters. The coupling exponent n_obs = 0.420 and the power-floor exponent α = 3/8 are quantum numbers of the de Sitter vacuum, determined by the representation theory of SO(1,4) at the CEH boundary in precise structural analogy with atomic quantum numbers determined by SO(4) at the nucleus. The remaining open problem — deriving the power-floor functional form from the Nekrasov–Shatashvili partition function for the conformally coupled scalar in Schwarzschild–de Sitter spacetime — is stated as a precise conjecture for Paper V.