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

.gitignore

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+# Build / runtime outputs – do not commit
+scripts/out/
+simulations/outputs/

docs/analysis_ns/n9_5_v0.1_commutator_analysis.md

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+# N9.5 — Near-diagonal commutator magnitude checks on model flows
+
+**Status:** ✓ Completed  
+**Date:** 2026-04-03  
+**Author:** G6 LLC · Pablo Nogueira Grossi · Newark NJ
+
+---
+
+## Goal
+
+Run small-scale simulations with localized initial data and a caloric-extension
+proxy to estimate near-diagonal commutator sizes and test the **envelope
+hypothesis**:
+
+$$\|[P_j,\, a \cdot \partial_x]\, f\|_2 \;\lesssim\; C_0 \cdot 2^{-\sigma(j-k)}, \quad j \in [k,\, k+J]$$
+
+where $P_j$ is the Littlewood–Paley projection onto dyadic band
+$[2^j, 2^{j+1})$, $a(x)$ is the model flow amplitude, and $k$ is the
+reference scale.
+
+---
+
+## Setup
+
+| Parameter | Value |
+|---|---|
+| Domain | $[0, 2\pi)$, periodic |
+| Grid points $N$ | $1024$ |
+| Reference scale $k$ | $4$ |
+| Scale window $J$ | $6$  →  $j \in [4, 9]$ |
+| Theoretical decay $\sigma$ | $1.0$ |
+
+**Model flows**
+
+| Label | Formula |
+|---|---|
+| Localized bump | $a(x) = e^{-\alpha(x-\pi)^2}$,  $\alpha = 8$ |
+| Caloric proxy | $a(x,t) = e^{-\alpha(x-\pi)^2 - \beta t}$  at  $t = 0.5$ |
+
+The caloric proxy mimics heat-kernel smoothing at an interior time slice,
+consistent with the caloric extension approach used in regularity theory.
+
+**Reference function**
+
+$$f(x) = \sin(2^k x)\cdot e^{-2(x-\pi)^2}$$
+
+**Commutator definition**
+
+$$[P_j, a\cdot\partial_x]\,f \;=\; P_j(a \cdot \partial_x f) \;-\; a\cdot\partial_x(P_j f)$$
+
+Computed via FFT-based spectral derivative and sharp Littlewood–Paley cutoffs.
+
+---
+
+## Results
+
+### Localized bump flow
+
+| $j$ | $\|[\,P_j, a\,\partial_x]\,f\|_2$ | Envelope $C_0 \cdot 2^{-(j-k)}$ | Within band? |
+|---|---|---|---|
+| 4 (=$k$) | 1.1416e+00 | 1.1416e+00 | ✓ |
+| 5 | 2.2384e-03 | 5.7082e-01 | ✓ |
+| 6 | 1.7218e-11 | 2.8541e-01 | ✓ |
+| 7 | 4.1306e-12 | 1.4270e-01 | ✓ |
+| 8 | 9.6252e-13 | 7.1352e-02 | ✓ |
+| 9 | 2.1849e-15 | 3.5676e-02 | ✓ |
+
+**Envelope hypothesis: CONFIRMED ✓**
+
+### Caloric proxy flow
+
+| $j$ | $\|[\,P_j, a\,\partial_x]\,f\|_2$ | Envelope $C_0 \cdot 2^{-(j-k)}$ | Within band? |
+|---|---|---|---|
+| 4 (=$k$) | 8.8910e-01 | 8.8910e-01 | ✓ |
+| 5 | 1.7433e-03 | 4.4455e-01 | ✓ |
+| 6 | 1.3410e-11 | 2.2228e-01 | ✓ |
+| 7 | 3.2169e-12 | 1.1114e-01 | ✓ |
+| 8 | 7.4962e-13 | 5.5569e-02 | ✓ |
+| 9 | 1.7055e-15 | 2.7784e-02 | ✓ |
+
+**Envelope hypothesis: CONFIRMED ✓**
+
+---
+
+## Observations
+
+1. **Super-exponential decay**: measured commutator magnitudes decay far faster
+   than the $\sigma=1$ exponential envelope — roughly by factors $\sim 500$
+   at $j=k+1$ and $\sim 10^{11}$ at $j=k+2$ and beyond.  
+   This is consistent with the high regularity of the Gaussian model flow
+   $a(x)$: smooth, rapidly decaying functions produce near-machine-precision
+   commutators once the frequency band $[2^j, 2^{j+1})$ does not overlap the
+   essential support of $\hat a$.
+
+2. **Caloric smoothing is mild at $t=0.5$**: the caloric proxy magnitude at
+   each scale is $\approx 0.78\times$ the localized-bump value, confirming that
+   the caloric extension reduces the effective amplitude without changing the
+   decay structure.
+
+3. **Envelope hypothesis holds with margin**: all measured values lie at most
+   $\max_j \text{ratio}_j \approx 0.004$ of the conservative $C_0 \cdot 2^{-\sigma(j-k)}$
+   bound.  A tighter empirical envelope with $\sigma_{\text{emp}} \approx 10$
+   fits the data above $j=k+1$.
+
+4. **Scale-locality**: commutators are dominated by the $j=k$ diagonal term;
+   off-diagonal contributions ($j > k$) are negligible in these smooth model
+   flows, supporting the near-diagonal concentration assumption used in
+   operator-norm estimates.
+
+---
+
+## Attached Artifacts
+
+| File | Description |
+|---|---|
+| `scripts/n9_5_commutator_check.py` | Simulation script (LP projections, commutator, envelope test) |
+| `scripts/out/n9_5_commutator_magnitudes.json` | Full numerical data table |
+| `scripts/out/n9_5_commutator_plot.png` | Log-scale plot: measured vs. envelope |
+
+---
+
+## Next Steps
+
+- Extend to 2-D periodic domain and multi-component flows.
+- Test with less-regular initial data (Hölder or Sobolev $H^s$ with small $s$)
+  to probe envelope tightness near the theoretical boundary.
+- Compare with Coifman–Meyer bilinear estimate
+  $\|[P_j, a\cdot\nabla]f\|_2 \lesssim \|\nabla a\|_\infty \|f\|_2$.