# Entropy at return times: reproducible figure

The diagram, data and rendering code are original course material, dedicated to the public domain under CC0-1.0. No source scan or source illustration is included.

The exact mathematics is proved in [the lesson](../../OA-FLOW-IE.html#IE-stopped): IE32, IE37 and IE42 give the word convention and information sum; IE56 gives the complete geometric law; IE3 and IE48 give the changes of clock. The tower is the separate unbounded-roof example in Diagnostic 9. The tail bar groups all n ≥ 7 and does not truncate the distribution.

Run `python -B render.py` with Python and Matplotlib. The renderer uses the already supplied `../typeiii-zero-decomposition/DejaVuSans.ttf` and `DejaVuSans-Bold.ttf`; their font license remains in that directory. The SVG embeds glyph outlines. To render from another working location, pass `--font-dir` pointing to those existing fonts. No duplicate font copies are required.

Historical attribution of Abramov's formula is recorded in reference 1 of [Tim Austin, Measure concentration and the weak Pinsker property](https://www.numdam.org/articles/10.1007/s10240-018-0098-3/). All displayed claims have complete proofs in the lesson.