Relative initial corrector at the epsilon-delta scale on the full Ingredients beta range, using the exact gradient-only initial trace.

theorem AVenhance.Infra.Section5.RelativeError.relative_initial_corrector_delta_uniform (β C₀ Ctrace : ℝ) : ∃ (Ccorr : ℝ), 0 ≤ Ccorr ∧ ∀ (I : Ingredients β), I.Czeta ≤ C₀ → I.Cxi ≤ C₀ → I.Chat ≤ C₀ → ∀ κ ∈ permissibleSet β I.Λ, ∀ (M : ℕ), 1 ≤ M → κ ∈ permittedInterval β I.Λ M → ∀ (m : ℕ), 2 ≤ m → m ≤ M → ∀ (Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ) (hΦ : IsStreamSeq I Φ) (g : Homogenization.Vec 2 → ℝ), ContDiff ℝ (↑⊤) g → ∀ (u : ℝ → Homogenization.Vec 2 → ℝ) (T : ℕ → ℝ → Homogenization.Vec 2 → ℝ), I.IsTIterates hΦ m (I.kappaSeq κ M m) (I.kappaSeq κ M (m - 1)) g u T → ∀ (S : ℝ), 0 ≤ S → (∀ (i : Fin 2), √(l2NormSq fun (x : Homogenization.Vec 2) => spaceGrad g x i) ≤ Ctrace * ↑(Nat.factorial 1) * (Ctrace / epsilon β I.Λ (m - 1) ^ (1 + gamma β / 2)) ^ 1 * S) → √(l2NormSq fun (x : Homogenization.Vec 2) => ∑' (k : ℤ), I.xiMK m k 0 * Homogenization.vecDot (I.chiTilde hΦ m (I.kappaSeq κ M m) k 0 x) (spaceGrad (fun (y : Homogenization.Vec 2) => T (Nstar β) 0 (I.xFlow hΦ m (lIdx β I.Λ m k) 0 y)) (I.xFlowInv hΦ m (lIdx β I.Λ m k) 0 x))) ≤ Ccorr * epsilon β I.Λ (m - 1) ^ delta β * S

The initial corrector has the required relative epsilon-delta scale on all physical Ingredients, without an extra lower restriction on beta.