A Local Medium Action for Density-Dependent Dust Gravity in Flat FLRW Cosmology

A local covariant parent action is constructed for dust coupled to an algebraic density field and an internal oscillator. In flat FLRW cosmology, it reproduces the complete nonlinear dust equations with density-dependent coupling κ(ρ) = κ₀ − λρ, for positive κ₀ and λ. With metric signature (−+++), this is κ(T) = κ₀ + λT. Algebraic elimination produces the required effective density. A chosen derivative-shaped oscillator coupling preserves a common zero-oscillator preparation, after which the parent equations determine the dust-clock normalization and recover both metric equations and matter exchange. Every smooth target history admits an explicit stationary lift. A general constitutive extension is given under a sufficient condition. The construction is presented as structural design followed by coefficient matching and direct variation. The paper identifies the choices of field content, couplings, initial preparation, response measure and limiting procedure. Within the selected algebraic class, the unknown coefficient functions are determined explicitly. A specified classical first-response construction relates the parent action to the metric-target pairing. The density response is integrated with its canonical Jacobian, and the surviving average-field weight is retained. Regulated continuous constraint limits and weighted Gaussian classical limits are evaluated in their declared stages. The record includes the English paper, LaTeX sources, Lean 4 proofs, source hashes and verification logs for 453 public theorems. The recovery theorem concerns smooth flat-FLRW histories with the stated preparation. The finite-width weighted quantum closed-time-path functional remains a further problem. AI assisted Lean verification and accelerated checks of the construction.