DAG — Gravity Sector: Linearised Perturbation Equations on a Friedmann–Robertson–Walker Background

We derive the linearised field equations for scalar and diagonal tracelessperturbations of a Friedmann–Robertson–Walker (FRW) background in DirectionalAnisotropy Gravity (DAG). Expanding the minimal four-parameter Lagrangian(anisotropy fields Ā_k, memory field M, induced metric gμν(Ā)) to firstorder around the homogeneous isotropic background, we obtain: (a) areduced-sector linearised system for the trace-mode perturbationΔ ≡ Σ_k δĀ_k, the memory perturbation δM, and matter (δ_m, θ_m), in whichthe metric back-reaction is shown to be a computable linear functional of Δ(guaranteeing algebraic closure) but whose explicit coefficients are not fullyevaluated; (b) an independent wave equation for the two diagonal traceless"breathing" modes h_k (Σ_k h_k = 0), which propagate at the metric null cone.Two structural features distinguish DAG from standard scalar-tensor perturbationtheory. First, the scalar metric perturbations Ψ and Φ are algebraically tiedthrough the induced metric, yielding a fixed gravitational-slip relationη_slip = 1 − ln A, which reduces to η_slip = 1 (general relativity withoutanisotropic stress) at A → 1. Second, in the sub-horizon regime the Poissonequation recovers the standard Newtonian form with G_eff = G, and themodified-gravity parameters become μ_DAG = 1/(1−ln A) and Σ_DAG = (2−ln A)/[2(1−ln A)], both reducing to unity at A → 1. These are conditionalconsequences of the reduced-sector derivation, testable in principle by currentand near-future surveys (Euclid, LSST, DESI); a definitive comparison with datarequires Boltzmann integration. The derivation is conditional on five statedapproximations (minimal coupling, diagonal ansatz, longitudinal gauge,linearisation, slow roll).