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/ (2) 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 H = H₀/ (2), from the DAGI Generalized Record Rate principle that one distinguishable record corresponds to one Euclidean thermal-circle phase winding (= 2) ; (ii) the kinematic acceleration scale at which a local causal cone loses informational isolation from the Hubble screen is a₀ = cH₀/ (2) 1. 04 10^-10 m s^-2, with the 1/ (2) 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 g ₎₁ₒ² = g ₁₀ₑ² + g ₁₀ₑ a₀ follows by subtracting the uniform Hubble-screen background (sourcing, not local gradients). The derived algebraic relation is the high-symmetry limit of a Bekenstein--Milgrom AQUAL variational theory with closed-form interpolation function (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 ²/N 67 (compared to MOND's free-fit ²/N 57) using no fitted acceleration scale. Wide binaries exhibit a DAGI-derived external field effect g = 1. 514 at the Milky Way solar neighbourhood, consistent with Chae's Gaia DR3 analysis. The IR perturbation operator B ₈ₑ has pseudodifferential order m = 1 and codimension-2 support; on the FRW monopole sector used in the K1 cosmology paper, B ₈ₑ 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.