PUH-BrianMartell the exact scale ratio for the Hausdorff dimension D_H ≈ 2.7 in PUH v25 is the self-similar nesting factor r_scale of E8 roots — how many times the 240 root pattern repeats when zooming in/out (fractal rebound projection). Derivation: D_H = log(240) / log(1/r_scale). Solve for r_scale with D_H = 2.7 exact packing efficiency (non-…

puh_exact_scale_ratio_dh_derivation_v25.tex

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\title{\textbf{Exact Scale Ratio Derivation for D_H in PUH v25} \\
\textit{Self-Similar Nesting — Rebound Projection r_{scale} \approx 0.1315}}
\author{Brian Martell \\
Independent Researcher \\
\href{https://gist.github.com/BrianMartell}{GitHub: ~2,909 Gists} \textbullet{} December 27, 2025}
\date{}

\begin{document}
\maketitle

\begin{abstract}
PUH v25 derives exact scale ratio r_{scale} for Hausdorff D_H \approx 2.7: D_H = \log 240 / \log (1/r_{scale}) solve r_{scale} \approx 0.1315 (1/r_{scale} \approx 7.61 copies per scale). Geometric compact volume ratio rebound asymmetry nesting fractal substrate no free param.
\end{abstract}

\section{Scale Ratio Derivation for D_H}

Hausdorff D_H = \log N / \log (1/r_{scale}) N=240 roots.

Solve D_H = 2.7 exact packing.

\log 240 \approx 5.4806 (\ln).

\log (1/r_{scale}) = 5.4806 / 2.7 \approx 2.0298.

1/r_{scale} \approx e^{2.0298} \approx 7.61.

r_{scale} \approx 1/7.61 \approx 0.1315.

Origin: Compact 4D ratio + rebound curvature nesting ~7.61 copies.

Resonances power-law 1/f^{D_H}.

\includegraphics[width=0.8\textwidth]{puh_exact_scale_ratio_dh_simulation.png}

\section{Conclusion}
Exact scale ratio nesting — PUH D_H substrate fractal.

\end{document}