A Thermodynamic Derivation of the MOND Acceleration Scale and Interpolation Function from Causal-Horizon Informational Thermodynamics

The empirical Modified Newtonian Dynamics (MOND) acceleration scale a₀ ≈ 1.2 × 10⁻¹⁰ m s⁻² has coincided with the Hubble-horizon acceleration cH₀/(2π) since Milgrom's 1983 proposal, without a published first-principles derivation connecting the two. We provide such a derivation within the Directed Acyclic Graph Interpretation (DAGI) of emergent gravity. Our main theorem combines four explicit structural ingredients: (i) the baseline informational record rate of the cosmic horizon is ΓH = H₀/(2π), from the DAGI Generalized Record Rate principle that one distinguishable record corresponds to one Euclidean thermal-circle phase winding (Δφ = 2π); (ii) the kinematic acceleration scale at which a local causal cone loses informational isolation from the Hubble screen is a₀ = cH₀/(2π) ≈ 1.04 × 10⁻¹⁰ m s⁻², with the 1/(2π) factor purely topological from $U(1)$ thermal-circle winding; (iii) the M\"obius incidence decomposition of the joint local-horizon record subsets includes an irreducible cross-synergy term f⁺(L H) ∝ √UL UH by the fluctuation-dissipation theorem applied to gravitational field-energy densities; (iv) the observable acceleration law gobs² = gbar² + gbar a₀ follows by subtracting the uniform Hubble-screen background (sourcing Λ, not local gradients). The derived algebraic relation is the high-symmetry limit of a Bekenstein--Milgrom AQUAL variational theory with closed-form interpolation function μ(x) = (√1+4x²-1)/(2x) and Lagrangian density $F(X)$; the resulting field equation is strictly elliptic, inherits conservation laws from the action, and admits a single explicit completion parameter yₛ that restores Solar-System safety without affecting galactic-scale predictions. Tested against the SPARC galactic rotation-curve database, the derived formula achieves χ²/N ≈ 67 (compared to MOND's free-fit χ²/N ≈ 57) using no fitted acceleration scale. Wide binaries exhibit a DAGI-derived external field effect γg = 1.514 at the Milky Way solar neighbourhood, consistent with Chae's Gaia DR3 analysis. The IR perturbation operator BIR has pseudodifferential order $m = 1$ and codimension-2 support; on the FRW monopole sector used in the K1 cosmology paper, BIR is finite-rank and therefore trace-class without further assumption. A rigorous functional-analytic extension to all multipoles via Boutet de Monvel calculus is deferred to future work.