Positive-time spatial-word adapter for the actual endpoint gradient. The terminal correction needs one spatial letter, not a material trace.

theorem AVenhance.Infra.Section5.RelativeError.relative_initial_endpoint_gradient_of_words {β : ℝ} (I : Ingredients β) {Φ : ℕ → ℝ → Homogenization.Vec 2 → ℝ} (hΦ : IsStreamSeq I Φ) (m : ℕ) (κ : ℝ) (T : ℝ → Homogenization.Vec 2 → ℝ) (n r : ℕ) {μ : MeasureTheory.Measure Section4.AmnrSpace} (hμ : μ.AbsolutelyContinuous (MeasureTheory.volume.restrict (Set.Ioi 0 ×ˢ Set.univ))) (hreg : ∀ (i j k : Fin 2), ContDiffOn ℝ 1 (fun (z : Section4.AmnrSpace) => I.Amnr hΦ m κ n T r z.1 z.2 i j k) (Set.Ioi 0 ×ˢ Set.univ)) {B : ℝ} (hword : ∀ (i j k p : Fin 2), MeasureTheory.eLpNorm (Section4.amnrWord (fun (z : Section4.AmnrSpace) => streamVel (Φ (m - 1)) z.1 z.2) [some p] fun (z : Section4.AmnrSpace) => I.Amnr hΦ m κ n T r z.1 z.2 i j k) 2 μ ≤ ENNReal.ofReal B) : (∀ (i j k p : Fin 2), MeasureTheory.AEStronglyMeasurable (fun (z : ℝ × Homogenization.Vec 2) => Section4.amnrSpatialGradientTensor I hΦ m κ T n r z i j k p) μ) ∧ MeasureTheory.eLpNorm (Section4.amnrSpatialGradientTensor I hΦ m κ T n r) 2 μ ≤ ENNReal.ofReal (16 * B)

Scalar one-letter word bounds give the actual four-index spatial-gradient norm and its coordinate measurability on a positive-time source measure.