# Figure components and reproduction

The original diagram, exact data, drawing code and newly authored explanatory text are dedicated to CC0-1.0 to the extent of any rights held. No source-book page or source illustration is reproduced.

Run python render.py in this directory with Matplotlib installed. SymPy, when present, verifies the exact noncommuting matrix diagnostic in addition to the required rational checks. The code uses DejaVu Sans and its display variant, with STIX mathematical glyphs, as distributed with Matplotlib. The accompanying FONT-LICENSE.txt contains both complete component licenses. The SVG embeds glyph outlines. The rendering uses Python 3.13.9 and Matplotlib 3.10.9; the embedded font components are DejaVuSans, DejaVuSans-Bold, DejaVuSans-Oblique, DejaVuSansDisplay and STIXGeneral-Italic.

All arithmetic shown is exact. The roof panel displays two local points on an aperiodic base orbit, not a finite dynamical system. The graph's horizontal arrows denote forming regular crossed products; every vertical arrow is a normal onto isomorphism with the normal inverse proved in the manuscript. The rational grid is a reproducibility diagnostic, not the proof of the all-parameter cocycle identity.

Proof locators are [From a central cocycle to a full return algebra](../../OA-FLOW-SR.html), Sections 2, 3 and 5–9, equations AC3, AC6–AC7, AC10, SR8–SR10, SR19–SR33 and SR39–SR41, and Diagnostics 2–4. Mathematical source: Takesaki, *Theory of Operator Algebras II*, XII.3.4–3.5, printed pp.389–392. The independently authored diagram incorporates the inverse-coordinate correction proved in the manuscript. Human-source copyrights and access terms remain unchanged.