Reproduce the When zero smooth periods mean that a cycle bounds
These original drawings accompany When zero smooth periods mean that a cycle bounds. The complete mathematical proof, all worked examples and complete solutions remain in that reader.
Download the unchanged original files
Keep render_detection.py at the root of an empty scratch directory and the remaining files in figures/. The renderer creates figures/ during a fresh run.
- render_detection.py — 6418 bytes; SHA-256
78A341BC9C6E182678D003E00CFE9668B8029634248089CCF2B976DFB35376AE. - cycle-detection.geometry.json — 51787 bytes; SHA-256
503A6B20CD4C26882222EF18EFC8126996678D8E20A435ED9CBFEADE7D483798. - cycle-detection.png — 232421 bytes; SHA-256
13641EA222BC41BBE23FEFB8B53FA4A96E218302918EF9E0F41E33661A6F6984. - cycle-detection.svg — 116328 bytes; SHA-256
AA95739469C02F8C5B25659825C56D05E153D1D82EE9E601CB8DA5E794542F7D.
Run the renderer
The unchanged program requires Python, NumPy and Matplotlib. In the scratch directory run:
python -X utf8 render_detection.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 every supplied file byte for byte and all decoded PNG RGBA pixels. 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
The left panel has eight balls centered at exp(iπj/4), radius 3/5, and the positively oriented unit circle. The arc distance bound is 2 sin(π/16)<3/5. These balls cover the displayed cycle, not all of the punctured plane. The right panel records δ horizontally, (−1)ᵖdᵥ vertically, and finite elimination along (0,2)→(1,1)→(2,0). It is an algebraic grid, not a geometric circle deformation. See Figure L1, Worked example 1, CD2–CD5c and the full CD0–CD10 proof. Georges de Rham and Allen Hatcher receive exact historical and method credits in the reader.
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.