Proof of the big-bound estimate (AVenhance.bigbound).

The statement below is identical to the one (copied from AVenhance/Section4/BigBound.lean); the statement file is not imported. The proof is Infra.Section5.Contracts.bigbound_closed: the big-bound estimate from the Section 5 residual identity (repaired decomposition, ten slot names) and every big-bound input produced at the datum amplitude, including Normie3CenteredSourceContract (normie3⁺, exact zero cell mean).

theorem AVenhance.Proofs.bigbound (β C₀ : ℝ) : ∃ (C : ℝ), ∀ (I : Ingredients β), I.Czeta ≤ C₀ → I.Cxi ≤ C₀ → I.Chat ≤ C₀ → C ≤ ↑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 → ∀ (R : ℝ), 0 < R → ∀ (θ₀ : Homogenization.Vec 2 → ℝ), ContDiff ℝ (↑⊤) θ₀ → IsZ2Periodic θ₀ → Homogenization.MeanZeroOn unitCube θ₀ → IsThetaAnalytic R θ₀ → ∀ (m : ℕ), mTheta0 β I.Λ R ≤ m → m ≤ M → ∀ (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) θ₀ θprev → I.IsTIterates hΦ m (I.kappaSeq κ M m) (I.kappaSeq κ M (m - 1)) θ₀ θprev T → ENNReal.ofReal (I.kappaSeq κ M m ^ (-(1 / 2))) * (∫⁻ (t : ℝ) in Set.Ioo 0 1, (hMinusOneNorm fun (x : Homogenization.Vec 2) => advDiffOp (streamVel (Φ m)) (I.kappaSeq κ M m) (I.ansatz hΦ m (I.kappaSeq κ M m) (T (Nstar β))) t x) ^ 2) ^ (1 / 2) ≤ ENNReal.ofReal (C * epsilon β I.Λ (m - 1) ^ delta β * √(l2NormSq θ₀))

Big-bound estimate proof. Its statement is identical to AVenhance.bigbound.