Uniform relative Hm bound on the first positive half-cell. The actual endpoint estimates use the one-letter mixed budget through terminal Jcut.

theorem AVenhance.Infra.Section5.RelativeError.relative_initial_Hm_uniform_of_gradient_profiles (β C₀ CA : ℝ) (hCA : 0 ≤ CA) : ∃ (CH : ℝ), 0 ≤ CH ∧ ∃ (Λ₀ : ℝ), ∀ (I : Ingredients β), I.Czeta ≤ C₀ → I.Cxi ≤ C₀ → I.Chat ≤ C₀ → Λ₀ ≤ ↑I.Λ → ∀ (Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (hΦ : IsStreamSeq I Φ), ∀ κ ∈ permissibleSet β I.Λ, ∀ (M : ℕ), 1 ≤ M → κ ∈ permittedInterval β I.Λ M → ∀ (m : ℕ), 2 ≤ m → m ≤ M → ∀ (g : Homogenization.Vec 2 → ℝ) (θprev : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ), IsClassicalSol (streamVel (Φ (m - 1))) (I.kappaSeq κ M (m - 1)) (fun (x : ℝ) (x_1 : Homogenization.Vec 2) => 0) g θprev → I.IsTIterates hΦ m (I.kappaSeq κ M m) (I.kappaSeq κ M (m - 1)) g θprev T → ∀ (S : ℝ), 0 ≤ S → (∀ r ≤ Jcut β, ∀ n ∈ Finset.range (Nstar β), MeasureTheory.eLpNorm (Section4.amnrSpatialGradientTensor I hΦ m (I.kappaSeq κ M m) (T (Nstar β)) n r) 2 (MeasureTheory.volume.restrict timeCube) ≤ ENNReal.ofReal (CA * S * (epsilon β I.Λ m ^ 2 / I.kappaSeq κ M m) / √(I.kappaSeq κ M (m - 1)) * epsilon β I.Λ (m - 1) ^ (-(1 + gamma β / 2)) * (tauP β I.Λ m)⁻¹ ^ r)) → ∀ t ∈ Set.Ioc 0 (tauPP β I.Λ m / 2), √(l2NormSq (I.Hm hΦ m (I.kappaSeq κ M m) (T (Nstar β)) t)) ≤ CH * epsilon β I.Λ (m - 1) ^ delta β * S

Actual normalized endpoint gradient profiles imply a uniform positive-time Hm estimate. Constants and the threshold precede the ingredient package.