# Haar averaging: original figure and reproduction terms

The original diagram `haar-integration.png` and `haar-integration.svg`, the renderer `render_haar_integration.py`, and the authored descriptions and arrangement in `haar-integration-data.json` are dedicated under **CC0 1.0 Universal**, to the extent of rights held. Project credit: **OA-FLOW course project contributors**, 6 October 2026. Numerical mathematical facts are not claimed as exclusive property. The [CC0 dedication and terms](https://creativecommons.org/publicdomain/zero/1.0/) apply to this original project expression.

The diagram was newly constructed from the four-block example proved in OA-FLOW-L31, Section 8, equations (L31.8.a)–(L31.8.f). It contains no source-book image or adapted third-party artwork. The general mathematical antecedent is M. Takesaki, *Theory of Operator Algebras II*, Proposition X.2.6, pp.263–265. The figure distinguishes the general matrix average from the scalar spectral formula and records the actual Haar normalization.

Reproduce the PNG, SVG and mathematical data with `python render_haar_integration.py --output OUTPUT_DIRECTORY`. The recorded bytes use Matplotlib 3.10.9 and NumPy 2.4.4. Fixed SVG IDs and omitted date metadata make the outputs reproducible. No TeX, network access, or external image is required. Exact integer and rational identities accompany the finite matrix diagnostics; the arbitrary-group and extended-positive results are proved in the lesson.

Font designs and outlines are excluded from the project CC0 dedication. Complete [DejaVu](LICENSE_DEJAVU.txt), [STIX](LICENSE_STIX.txt), and [Computer Modern/BaKoMa](BAKOMA_SECTION.txt) notices accompany the typography and fallbacks. Runtime libraries retain their own terms; their implementations are not bundled.