N9.3: Leray projection and pressure commutator control for near-diagonal terms

[Copilot]

Near-diagonal commutators involving the Leray projector and pressure were unaddressed in the envelope framework, leaving a gap in the absorption argument for Lemma 3.1 of NS D9.

Added: docs/analysis_ns/n9_v0.1_leray_pressure_commutator.tex

• §2 — Leray commutator decomposition: Splits [P, χ_j]u via Bony paraproduct into near-diagonal (|k−j| ≤ L) and off-diagonal tails. Near-diagonal bound (Lemma 1) follows directly from envelope slow-variation; off-diagonal tails (Lemma 2) decay as 2^{−δL} via the Coifman–Meyer theorem, suppressed by choosing L large.

• §3 — Pressure commutator: [∇p, χ_j] = (∇χ_j)p is a pointwise multiplication error. Lemma 3 bounds it at 2^{js}‖·‖_{L²} ≤ C·a_j using only Calderón-Zygmund estimates and the product estimate for s > n/2.

• §4 — Absorption into Lemma 3.1 (NS D9): Proposition 1 shows both commutator error terms in the localized energy identity are bounded by C·a_j², matching the order of the main nonlinear term. The envelope ODE closes with no nonlocal or non-summable remainder, and no extra hypothesis is needed beyond admissibility of the frequency envelope at s > n/2 + 1.

All three N9.3 checklist items are resolved; the note follows the n9_v0.1_*.tex naming convention.