Fractal Notebooks — Proof Knowledge Bundle¶

Appendix-level proofs for the quantitative results stated in the Fractal Notebooks documentation, primarily Chapter 1: History of Fractal Mathematics (docs/foundations/history.md). Each document is a type: Fractal Proof record that states the theorem, its assumptions, numbered proof steps, an explicit verification status distinguishing machine-checked steps from steps asserted on the authority of a named theorem, the historical source, and a cross-link to the matching concept in ../concepts/.

Verification conventions¶

Each proof step is tagged:

• [verified] — checked by symbolic or numerical computation (SymPy / NumPy) in the fractal-proof environment. The check is described in the step and reproducible.
• [asserted] — relied upon as the conclusion of a named, published theorem that is cited but not re-derived here (e.g. Moran's theorem, Hardy's theorem, the Fatou–Julia dichotomy). These are the load-bearing external results; the proof is only as sound as the citation.

A proof's overall verification_status frontmatter field is one of verified (all quantitative steps machine-checked; any [asserted] steps are standard textbook theorems), partially-verified (core computation checked, substantive structural claims asserted), or unverified (no step machine-checkable with available tools).

Proofs¶

• Cantor Set — Dimension, Measure, and Cardinality — \(D=\log 2/\log 3\), Lebesgue measure zero, uncountability.
• Koch Curve — Dimension, Perimeter, and Area — \(D=\log 4/\log 3\), infinite perimeter, snowflake area \(\tfrac{8}{5}A_0\).
• Sierpinski Triangle and Carpet — Dimensions — \(\log 3/\log 2\) and \(\log 8/\log 3\).
• Menger Sponge — Similarity Dimension — \(\log 20/\log 3 \approx 2.7268\).
• Barnsley Fern — Box-Counting Dimension — numerical \(\approx 1.83\); no closed form (OSC fails).
• Mandelbrot Set — Escape-Radius Criterion — \(|z|>2 \Rightarrow\) orbit escapes; radius-2 bailout.
• Mandelbrot Boundary — Hausdorff Dimension 2 — Shishikura's theorem (asserted, not machine-verifiable).
• Julia Sets — Fatou–Julia Connectedness Dichotomy — \(J_c\) connected \(\iff 0\in K_c\).
• Weierstrass Function — Nowhere Differentiability — continuity everywhere, differentiability nowhere.
• Self-Affinity — Spectral Exponent Relation — \(\beta = 2H+1\) for Gaussian self-affine processes.
• Metabolic Scaling — WBE Quarter-Power Law — \(B \propto M^{3/4}\) from area-preserving, space-filling networks.
• Complex Dimensions — Cantor String Poles — \(\omega_k = \tfrac{\log 2}{\log 3} + i\tfrac{2\pi k}{\log 3}\).

See log.md for change history.