N9.1 — Decompose Π_{j,x₀,k}-Π_j into explicit near‑diagonal commutator/paraproduct pieces for j∈[k,k+J]

[TOTOGT]

Goal
Write an explicit finite decomposition of the near‑diagonal difference into commutators and paraproducts with tracked constants, valid for j∈[k,k+J].

Deliverables

• A LaTeX note with the decomposition and estimates for each term.
• Clear statements of constants and uniformity in (x0,k).

Checklist

• Identify all near‑diagonal interaction types (low–high, high–low, resonant).
• Derive pointwise or L^2 bounds for each piece.
• Track constants and dependence on J.
• Link to NS D9 envelope framework for insertion.