Relational Information Dynamics: Deriving the Coupling Amplitude a_n from the Vacuum Variance of the Modular Hamiltonian

Paper 1 of the RID series derives g† = 2cH0,dS/π² as a conjectural effective relation and explicitly flags one heuristic step: the coupling amplitude a_n ∝ τ_n is asserted rather than derived. This paper derives a_n via two independent routes. Route 1 (primary): a_n ≡ √⟨0|x_n²|0⟩ = 1/(2nH) follows in two exact lines from the modular Hamiltonian K and canonical commutation relations. Route 2 (corroborating): the exact KMS two-point function G(τ) = −(1/2H) ln(1 − e−H|τ|) expands as Σ_n a_n e−nH|τ| with a_n = 1/(2nH), independently confirming the same coefficients from the Green's function alone. Three open items are stated explicitly: (i) the overhead functional form (a_n/τ_n)² is FDT-motivated but the γ = 1 normalisation remains open (Paper 5); (ii) τ_n = π/(2nH) is now derived as the first zero of the KMS two-point function G_n(s), with E = ω_n following from the correlator frequency rather than assumed; the residual open item is the identification of the first zero as the synchronisation timescale (Paper 5); (iii) the critical exponent ν = 1/2: its slope face is now derived and directly measured (deep-MOND log–log slope 0.54; deep- and transition-restricted fits 0.485/0.504; §8.7); a Bayesian Υ*-marginalisation (2026-07) gives ν = 0.452 [0.444, 0.458] at that stage; completing the hierarchy with per-galaxy distance and inclination priors (SPARC Table 1) gives ν = 0.503 [0.495, 0.509] (0.489–0.520 across prior widths): the exact-1/2 form is fully consistent, the Υ-only softness is a nuisance-omission artifact, and the residual caveat moves to the scale — the all-nuisance global g† is ridge/prior-dominated, so the joint fit constrains shape while the absolute-scale comparison is a nominal-calibration statement (§8.7). The paper closes the a_n heuristic and upgrades two of the three open items from assumed/conjectural to derived or physically grounded.