apply the classical trace bridge to the square cutoff

theorem AVenhance.Infra.Section5.RelativeError.classical_lowProjection_trace_bound {b : ℝ → Homogenization.Vec 2 → Homogenization.Vec 2} {κ B : ℝ} {g : Homogenization.Vec 2 → ℝ} {u : ℝ → Homogenization.Vec 2 → ℝ} {X : ℝ → Homogenization.Vec 2 → ℝ → Homogenization.Vec 2} (M : ℕ) (hb : Flow.SmoothPeriodicField b) (hdiv : ∀ (t : ℝ) (x : Homogenization.Vec 2), vecDiv (b t) x = 0) (hX : IsFlow b X) (hDb : ∀ (r : ℝ) (y : Homogenization.Vec 2), ‖Flow.jointSpatialFDeriv b r y‖ ≤ B) (hmax : max 1 B ≤ κ * (2 * Real.pi * √2 * ↑M) ^ 2) (hB : 0 ≤ B) (hκ : 0 < κ) (hM : 0 < M) (hsol : IsClassicalSol b κ (fun (x : ℝ) (x_1 : Homogenization.Vec 2) => 0) g u) (hg : ContDiff ℝ 3 g) (hgper : IsZ2Periodic g) : √(gradNormSq (spaceGrad (Ergodic.lowProjection g M))) ≤ 2 * Real.exp 1 * (2 * Real.pi * √2 * ↑M) * √κ * √(∫ (t : ℝ) in 0..1, classicalCellGradientEnergy u t)

The cutoff lowProjection g M is contained in the Euclidean frequency ball of radius 2π√2 M. The spectral identities therefore feed directly into the classical transport trace argument.