# Bounded perturbations and flow domains: figure terms

These three figures explain common domains for cocycle evolution, loss of a second derivative, and summability bounds for flow domains. The proof locators, labels, signs and plotted data accompany each figure.

The original figure expression in the following files, and the authored annotations and arrangement in `34-derivation-domain-data.json`, are dedicated under **CC0 1.0**:

- `34-cocycle-and-common-domain.png` and `.svg`;
- `34-lost-second-derivative.png` and `.svg`;
- `34-summability-and-domain-window.png` and `.svg`.

Figure licences, font components and exact file identities are listed in [the component notices](PRESERVED_COMPONENT_NOTICES.json). Numerical mathematical facts are not claimed as exclusive property.

The renderer `render_figures.py` is dedicated under **CC0 1.0**. Reproduce all six images and the coordinate diagnostics with `python render_figures.py --output OUTPUT_DIRECTORY`. It needs Python, Matplotlib and NumPy; it launches no TeX process. SVG IDs and date metadata are fixed. The two plots use separate vertical scales, logarithmic horizontal axes and floating-point samples; their exact infinite-series bounds are proved in Section 7.

Typography and runtime libraries retain their own terms. The rendered DejaVu, STIX and Computer Modern glyph notices are supplied in LICENSE_DEJAVU.txt, LICENSE_STIX.txt and BAKOMA_SECTION.txt. Font outlines are excluded from the project-expression grant. Library and font implementations are not bundled.

The mathematical antecedent identified in the diagrams is M. Takesaki, *Theory of Operator Algebras II*, Chapter XI, §3. The figures are original project artwork, not images taken from that source.

The editable SVGs and exact coordinate data accompany the rendered PNGs. The mathematical proofs and domain qualifications are given in the lesson.