For sixty years, Modified Newtonian Dynamics (MOND) has fit galaxy rotation curves with remarkable accuracy, yet its physical origin has remained unknown. The empirical Radial Acceleration Relation (RAR) of McGaugh et al. (2016) encodes this success in a single interpolation function μ(x) = 1 − exp(−√x), where x = g_bar/g* and g* = 1.20 × 10⁻¹⁰ m s⁻². No derivation of this specific functional form from first principles has previously been given. We observe that μ(x) is mathematically identical to the Poisson survival probability P(N ≥ 1) with mean occupation ⟨n⟩ = √x. We propose that this identity reflects a physical reality: baryonic matter excites quanta of the de Sitter Lag Field with mean occupation ⟨n⟩ = √(g_bar/g*), where g* is the Bunch-Davies thermal acceleration scale derived in the companion paper (Paper 2). This identification gives the first statistical physics interpretation of MOND — the observed transition from Newtonian to modified dynamics is the transition from quantum vacuum (few quanta excited) to classical field (many quanta excited). Combined with Paper 2's derivation of g* from the de Sitter temperature, this connects all three previously disconnected components of MOND — the acceleration scale, the transition shape, and the deep-field formula — to a single physical framework with zero free parameters. The External Field Effect, a challenge for linear field theories, is shown to emerge from the concavity of √x in the Poisson activation rate without any modification of the linear field equation. The derivation of ⟨n⟩ = √x from the 12DST action in de Sitter background is identified as the key open problem, equivalent to a complete first-principles derivation of MOND.