Transported profiles at auxiliary spatial orders, with material cutoff orders still checked against the literal source cutoff.

theorem AVenhance.Infra.Construction.material_aux_transportProfile_iteratedFDeriv_norm_le_of_A3 {β C : ℝ} (I : Ingredients β) {Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ} (hΦ : IsStreamSeq I Φ) (hreg : StreamRegularityBounds C I Φ) {m : ℕ} (hm : 1 ≤ m) (k l : ℤ) (t : ℝ) (x : Homogenization.Vec 2) (ht : |t - ↑l * tauPP β I.Λ m| ≤ tauPP β I.Λ m / 2 + tauP β I.Λ m) {N : ℕ} (n : ℕ) (hbudget : n ≤ N) : ‖iteratedFDeriv ℝ n (fun (y : Homogenization.Vec 2) => psi β I.Λ m k (I.xFlowInv hΦ m l t y)) x‖ ≤ Section4.amnrTransportSpatialConstant N * a β I.Λ m * epsilon β I.Λ m ^ 2 * (epsilon β I.Λ m)⁻¹ ^ n

Every actual transported profile derivative retains the fine spatial radius. Coarse flow derivatives are absorbed by the actual scale separation.