Distributed Field Processing: The Gravitational Constitutive Law (No Dark Matter Required)

We derive the MOND interpolating function μ(x) = x/(1+x) from first principles. This is not a fit — it is a theorem. 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 class, these two constraints uniquely determine μ(x) = x/(1+x). There are no free parameters. What Milgrom discovered empirically in 1983 now emerges as mathematical necessity. 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. Scope a theorem, not a fit Scalar Lagrangian: L(x) = ½x² − x + ln(1+x) — the variational foundation 1PN metric: Φ = Ψ, γ = 1 — derived from operational calibration (Postulate A4) Full gravitational lensing: θ = 4GM/bc² — via optical-mechanical analog Phantom density: "missing mass" as operator residual, not substance Falsifiable predictions: Fast galactic bars (ℛ ≈ 1.0–1.4) — no halo-wake friction External field effects in satellites — environmental dependence CDM lacks Baryon-only cluster lensing — Bullet Cluster forward-modeling pipeline Wide-binary anomalies at g < a₀ Relativistic status: This paper establishes covariant completion at 1PN (metric structure, lensing, γ = 1). What remains is the fully covariant action — the S = ∫ℒ√-g d⁴x that generates time-dependent dynamics (gravitational waves, propagation). The static/1PN framework here constrains any such action. Cosmology: CMB and structure formation require relativistic completion beyond this paper's scope. However, existing work (Skordis DFP constrains what the galactic limit of any such completion must look like.