# Reproduce the Logarithmic Fourier graphs construct a cone-supported inverse

These original drawings accompany Logarithmic Fourier graphs construct a cone-supported inverse. The complete mathematical proof, all worked examples and complete solutions remain in that reader.

## Download the unchanged original files

Keep `make_figures.py` at the root of an empty scratch directory and the PNG, SVG and geometry files in `figures/`. The three unchanged Markdown sources and CREDITS.txt/LICENSE.txt stay at the package root; they are downloads and are not generated by the renderer. The renderer creates `figures/` during a fresh run.

- [make_figures.py](../reproduce/L117/make_figures.py) — 12768 bytes; SHA-256 `74DE844A388E1E72C454C1DADEF38DB26848F721B3563A6958777C6059861188`.
- [geometry.json](../reproduce/L117/figures/geometry.json) — 2855 bytes; SHA-256 `7BE6AD84A74863F83E26D6C9D0075360D6B104C80767CE0AB15A8D370906D766`.
- [logarithmic-cutoff-and-jacobian.png](../reproduce/L117/figures/logarithmic-cutoff-and-jacobian.png) — 198251 bytes; SHA-256 `108980A9AEE95641A978ED1337F703E10AC90EC669E00CC24A556330C636CE6C`.
- [logarithmic-cutoff-and-jacobian.svg](../reproduce/L117/figures/logarithmic-cutoff-and-jacobian.svg) — 106660 bytes; SHA-256 `C26D639A33242E16A54D4A5CDAE2414BE37EA0933F0444EF3862E3C356DEABA5`.
- [support-plane-and-uniform-decay.png](../reproduce/L117/figures/support-plane-and-uniform-decay.png) — 209810 bytes; SHA-256 `B4C6D1145F69192F6D9EB33C8F1387225E96FC49F25B6F6E94213C24A34FF478`.
- [support-plane-and-uniform-decay.svg](../reproduce/L117/figures/support-plane-and-uniform-decay.svg) — 104160 bytes; SHA-256 `721E02447A86F6CE6C6E3AEFF6F136E2A916D3304F13B43D37C6E3907A4F0A0E`.
- [two-variable-graph-and-directions.png](../reproduce/L117/figures/two-variable-graph-and-directions.png) — 340326 bytes; SHA-256 `57A21A7399984F839DA726AB79469BA7878892C9DB00114F63001B2AFABB5FC0`.
- [two-variable-graph-and-directions.svg](../reproduce/L117/figures/two-variable-graph-and-directions.svg) — 888813 bytes; SHA-256 `22B143DC261176B4B89083D21E8E1863F59826B8A5B375FE54B9C69595C02A13`.
- [learner-full.md](../reproduce/L117/learner-full.md) — 52320 bytes; SHA-256 `14B84E4377A395E0FB24A72C170D6FB4279DBD855C456C5A9670082849551DF1`.
- [manuscript.md](../reproduce/L117/manuscript.md) — 38116 bytes; SHA-256 `B16110319C93285E2ABF2A348CAEBD22E02FBA6C501189C1B2FB2140EF40E018`.
- [learner.md](../reproduce/L117/learner.md) — 14658 bytes; SHA-256 `BC7793430A395A74C3D1E382F30B743F5FBBFDF832F7FFEBE5FC0632FA3D3F6E`.
- [LICENSE.txt](../reproduce/L117/LICENSE.txt) — 522 bytes; SHA-256 `25E18118514AA1C4E2716F8F88557709A630213080B33E37BFCAEB1A9BA3702A`.
- CREDITS.txt — 2145 bytes; SHA-256 `FE1DC532A41DAB0DC0659E848A42D8B80D8A52E6B129B17B099A8459E8E4C001`.

## Run the renderer

The unchanged program requires Python, NumPy and Matplotlib. In the scratch directory run:

```text
python -X utf8 make_figures.py
```

It writes the supplied PNG, SVG and exact geometry files. A fresh replay with Python 3.13.9, NumPy 2.4.4 and Matplotlib 3.10.9, using Matplotlib’s bundled DejaVu Sans font, matched the eight renderer/figure files byte for byte and all three decoded PNG RGBA pixel arrays. No SVG date or identifier normalization was performed. Byte identity is qualified by that observed software and font environment; different versions may change rendering bytes. The geometry files specify the mathematical objects independently of rendering.

## Check the mathematics and credits

Figure 1 uses the exact graph z = ξ + i L log(2 + ξ²), L = 12, 14, 16, the only pole at zero, cutoff R = 3 and derivative region 3 ≤ |ξ| ≤ 6, and exact complex Jacobian J = 1 + 24iξ/(2 + ξ²) at L = 12. Figure 2 uses z = (ξ₁ + 4ih, ξ₂ + 2ih), h = log(2 + |ξ|²), λ = 4h and exactly recoverable Im z₂ = λ/2; color gives Im J. The path θ(s) = (1,s/2) lies in θ₁ > |θ₂|; it does not certify a kernel threshold at L = 4. Figure 3 uses the exact inverse −H(−x₁) ⊗ δ₀(x₂), its negative-axis support, θ = (1,0), A = p = 0, disk center (2,0), radius 1/2, gap d = 3/2, n = 2 and N = 3. Its curves are proved upper-bound functions with one integrable envelope, not measurements. Full captions, equations (5.8)–(5.11), (8.3)–(8.7), and Problems 2, 6 and 9 give the precise proof locators.

The exact original complete source, formal proof and exercise source are retained above. The theorem produces one common inverse for a nonempty ordinarily convex open positive-dilation cone excluding zero under uniform closed-angular logarithmic reciprocal bounds. Semyon Dyatlov, Gerd Grubb and Lars Hörmander receive the full comparison credits above and in CREDITS.txt; external copyrighted prose and media are absent. See [the complete included convolution reading](../prerequisites/convolution-as-addition-of-supports.html), Theorem 1.1, equations (1.4)–(1.8).

The original renderer, scene/geometry and diagrams are GPT-6.1 Sol (OpenAI), Ultra work, October 2026, dedicated under CC0. Matplotlib and the bundled DejaVu Sans font retain their own component terms already included with this course. The native geometry is explanatory; the full proof establishes the theorem.