The actual material-gradient equation with its full drift error.

theorem AVenhance.Infra.Section4.iterate_classical_commuted_material_gradient_equation {b : ℝ → Homogenization.Vec 2 → Homogenization.Vec 2} {κ : ℝ} {F u : ℝ → Homogenization.Vec 2 → ℝ} {u₀ : Homogenization.Vec 2 → ℝ} (hsol : IsClassicalSol b κ F u₀ u) (hb : ContDiffOn ℝ (↑⊤) (fun (z : AmnrSpace) => b z.1 z.2) (Set.Ioi 0 ×ˢ Set.univ)) {t : ℝ} (ht : 0 < t) (w : List (Fin 2)) (x : Homogenization.Vec 2) (j : Fin 2) : spaceGrad (amnrMaterial b (fun (s : ℝ) => iterateSpatialWord w (u s)) t) x j = κ * spaceLap (fun (y : Homogenization.Vec 2) => spaceGrad (iterateSpatialWord w (u t)) y j) x + spaceGrad (iterateSpatialWord w (F t)) x j - spaceGrad (fun (y : Homogenization.Vec 2) => iterateWordMaterialError (fun (z : AmnrSpace) => b z.1 z.2) w (fun (z : AmnrSpace) => u z.1 z.2) (t, y)) x j

Differentiate the actual commuted equation. The drift error has a minus sign and is kept explicit for transfer to the preceding Hessian.