N9.1: Explicit near-diagonal decomposition of Π_{j,x₀,k}−Π_j into commutator/paraproduct pieces

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Adds the N9.1 LaTeX note delivering an explicit finite decomposition of the near-diagonal operator difference δΠ_{j,x₀,k} := Π_{j,x₀,k} − Π_j for j∈[k, k+J], with tracked constants and D9 envelope insertion recipe.

New file

docs/analysis_ns/n9_1_v0.1_near_diagonal_decomp.tex

Decomposition

Via a first split on the localisation mismatch and Bony paraproduct expansion of (m−1)f (where m = χ_{x₀,k}), the difference reduces to four pieces:

δΠ_{j,x₀,k} f  =  T₁ + T₂ + T₃ + T₄

T₁  =  m · Πⱼ(𝖯_{(m−1)} f)       — low–high paraproduct
T₂  =  m · Πⱼ(𝖯_f (m−1))         — high–low paraproduct
T₃  =  m · Πⱼ(𝖱(m−1, f))         — resonant/diagonal
T₄  =  (m−1) · Πⱼ f               — direct localisation error

T₁–T₃ constitute the commutator C(m, Πⱼ)f; T₄ is the cutoff mismatch.

Estimates (L² bounds, uniform in x₀ and k)

Piece	Bound	J-dep
T₁	C(J+1)^{1/2} ‖f‖	O(√J)
T₂	C_J ‖f‖	poly
T₃	C_{χ,J} 2^{−kε} ‖f‖	poly
T₄	C ‖f‖	none

Master bound: ‖δΠ_{j,x₀,k} f‖_{L²} ≤ C(n,χ)(J+1)² ‖f‖_{L²}, uniform over all x₀∈ℝⁿ, k≥0.

D9 envelope insertion

Section 6 gives a concrete recipe: in any D9-envelope inequality ∂ₜ𝓔_k ≤ [NS terms] + R_k, replace R_k with the master bound above; the off-diagonal tails (j>k+J and j<k) are summable via Σ_{k≥0} 2^{−kε} < ∞.