Analytic forcing bound for the actual differentiated flux.

theorem AVenhance.Infra.Section4.iterate_terminal_small_analytic_word_flux_bound {s : ℝ} (hs1 : s ≤ 1) {A : ℝ → Homogenization.Vec 2 → Matrix (Fin 2) (Fin 2) ℝ} {u v : ℝ → Homogenization.Vec 2 → ℝ} (hu : ContDiffOn ℝ (↑⊤) (fun (z : AmnrSpace) => u z.1 z.2) (Set.Ici 0 ×ˢ Set.univ)) (hv : ContDiffOn ℝ (↑⊤) (fun (z : AmnrSpace) => v z.1 z.2) (Set.Ici 0 ×ˢ Set.univ)) (w : List (Fin 2)) (hcoef : ∀ p ∈ iterateSpatialSplits w, ContinuousOn (fun (z : AmnrSpace) => iterateMatrixWord (A z.1) p.1 z.2) (Set.Ici 0 ×ˢ Set.univ)) {κ C B ρ b L : ℝ} (hκ : 0 < κ) (hL : 0 < L) (hg : 2 * (b / L) ^ 2 ≤ 1 / 2) (a : ℕ) (hD : ∀ (t : ℝ) (x : Homogenization.Vec 2), ∀ p ∈ iterateSpatialSplits w, ∀ (j k : Fin 2), |iterateMatrixWord (A t) p.1 x j k| ≤ κ * C * ρ * ↑p.1.length.factorial * b ^ p.1.length) (hE : ∀ p ∈ iterateSpatialSplits w, (spaceTimeGradNormSq fun (t : ℝ) => spaceGrad (iterateSpatialWord p.2 (v t))) ≤ B ^ 2 * (↑(p.2.length + a).factorial * L ^ p.2.length) ^ 2) : |∫ (z : AmnrSpace) in iterateTruncatedCell s, Homogenization.vecDot (spaceGrad (iterateSpatialWord w (u z.1)) z.2) (iterateWordFlux (A z.1) (v z.1) w z.2)| ≤ (κ / 8 * spaceTimeGradNormSq fun (t : ℝ) => spaceGrad (iterateSpatialWord w (u t))) + 32 * κ * C ^ 2 * ρ ^ 2 * B ^ 2 * (↑(w.length + a).factorial * L ^ w.length) ^ 2

Conditional coefficient jets and preceding increment energies give an actual flux estimate. No current energy or forcing estimate is assumed.