Distributed Field Processing: The Gravitational Constitutive Law

We derive the MOND interpolating function μ(x) = x/(1+x) from gravitational closure constraints. This is not a fit. It is a uniqueness theorem within the minimal closure class. The 40-year gap: Since 1983, modified gravity has organized galactic phenomenology around a single function that no one could explain. Dozens of μ(x) candidates have been proposed; all were empirical choices. The theoretical origin remained unknown. What we prove: When matter couples to the gravitational field, closure must be lossless (no wakes, no drag) and local (causality forbids global computation). Within the Minimal Single-Channel Saturating Closure (MSC) class, the simplest model consistent with these constraints, μ(x) = x/(1+x) is uniquely determined. No free parameters. No curve-fitting. What Milgrom discovered empirically in 1983 emerges as the unique solution to a constraint-satisfaction problem. The relativistic extension: We derive the 1PN metric from operational calibration constraints, obtaining Φ = Ψ (gravitational slip γ = 1). The optical-mechanical analog recovers full gravitational lensing from the scalar Lagrangian, establishing the metric constraints any covariant completion must satisfy. The reframing: "Dark matter" is phantom density: the field's nonlinear response to baryons, misread through Newtonian analysis. It is not a substance; it is an operator residual. Falsifiable predictions: Fast galactic bars (ℛ ≈ 1.0–1.4), external field effects in satellite galaxies, and baryon-only lensing predictions in clusters. Observation will adjudicate.