The Modal-Horizon Framework IV: Closing the MHF Parameter Structure — Exact Spectral Weight, Derived Coupling Amplitude, and the Origin of Z = 1/2

This paper completes the theoretical foundations of the Modal-Horizon Framework by deriving the last undetermined coupling quantities from first principles, leaving no free parameters in the MHF coupling structure. The coupling amplitude gamma_c has an exact closed form: the l=0 cosmic event horizon filter integral is expressible in terms of the fourth-order polylogarithm Li4 with complex argument, verified to machine precision across the parameter range. The l=1 contribution (22 per cent of the total) requires a new transcendental object, the Lerch-Si function S_Si(p) = (1/p) × sum over n≥1 of p^n × Si(2πn), not reducible to standard polylogarithms. The value gamma_c = 0.1801 is derived without free parameters from two independent routes: the conformal coupling fixed point (xi = 1/6, nu = 1/2) and the one-loop renormalisation group flow from xi_bare = 0 to xi_IR = 1/6. Both routes agree to 0.2 per cent because cos^2(π×nu/2) = 1/2 at both the UV (nu = 3/2) and IR (nu = 1/2) fixed points. The physical origin of the Z = 1/2 halving factor is established: the kinetic energy of the MHF scalar at the closed cosmic event horizon sphere is shared equally between the observer's accessible interior and the causally inaccessible exterior. This gives a unified derivation of n_obs = Z × n_bare = 0.4198 and gamma_c = Z × n_A = 3/16 from the same geometric fact. The conformal fixed point fixes the MHF scalar field mass at m_eff × c^2 = sqrt(2) × hbar × H0 = 1.9 × 10^−16 eV, predicting a contribution to the effective number of relativistic species of ΔN_eff ≤ 0.043 if the scalar thermalised above the QCD transition — a near-miss for CMB-S4. Three open problems are stated clearly: the rigorous derivation of the Dirichlet boundary condition at the cosmic event horizon from the MHF action; the closed-form expression for the Lerch-Si transcendent; and the second-order Hadamard calculation needed to derive the power-floor running coupling exponent from first principles. This deposit contains the analysis notebooks supporting the paper, including the exact spectral weight calculation (Theory NB48), the RG fixed point analysis (Theory NB57, NB60–61), the nbare derivation (Theory NB62), and the Tolman response coefficient calculation (Theory NB66–74).