N9.5: Near-diagonal commutator magnitude checks on model flows

[Copilot]

Implements the N9.5 checklist: small-scale numerical experiments estimating near-diagonal Littlewood–Paley commutator magnitudes on model flows and testing the exponential envelope hypothesis.

Changes

scripts/n9_5_commutator_check.py

• 1-D periodic domain ($N=1024$), FFT-based LP projections $P_j$ onto dyadic bands $[2^j, 2^{j+1})$
• Two model flows: localized Gaussian bump and caloric-extension proxy ($e^{-\alpha(x-\pi)^2 - \beta t}$ at $t=0.5$)
• Computes $|[P_j, a\cdot\partial_x]f|_2$ for $j\in[k, k+J]$ ($k=4$, $J=6$) via spectral derivative
• Tests envelope hypothesis $|[P_j, a\cdot\partial_x]f|_2 \lesssim C_0 \cdot 2^{-\sigma(j-k)}$; emits JSON data table and log-scale comparison plot

# Core commutator:
term1 = lp_project(a * deriv_spectral(f), j)
term2 = a * deriv_spectral(lp_project(f, j))
L2 = sqrt(sum((term1 - term2)**2) / N)

Both flows confirm the envelope hypothesis; measured decay is super-exponential (factors ~500 at $j=k+1$, ~$10^{11}$ at $j=k+2$), consistent with the high regularity of Gaussian model amplitudes.

docs/analysis_ns/n9_5_v0.1_commutator_analysis.md

Result tables for both flows, key observations (super-exponential decay, caloric proxy ~22% amplitude reduction without structural change, scale-locality of diagonal dominance), and suggested next steps (rougher initial data, 2-D extension, Coifman–Meyer comparison).

.gitignore

Excludes scripts/out/ and simulations/outputs/ per lane guidelines.