PUH-BrianMartell the compact volume ratio in PUH v25 is the damping contribution from the 4 hidden (compactified) dimensions when the full 8D E8 lattice projects orthogonally to our 4D spacetime. Visible coupling (knot-wave interaction) reduced by fraction of “energy” in compact directions — average <cos^4 θ> = 3/80 = 0.0375 gives N_eff = 9 visi…

puh_e8_compact_volume_ratio_derivation_v25.tex

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\title{\textbf{Compact Volume Ratio Derivation from E8 Projection in PUH v25} \\
\textit{Hidden 4D Damping — Exact \alpha Convergence}}
\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 compact volume ratio E8 240 roots 4D orthogonal projection: Visible <\cos^4 \theta> = 3/80 N_eff = 9. Compact 4D hidden volume ratio + fractal nesting rebound asymmetry damping d_{compact} = 137.035999... / 9 \approx 15.226666... exact \alpha^{-1} geometric substrate.
\end{abstract}

\section{Compact Volume Ratio Derivation}

E$_8$: 240 roots unit uniform S^7.

Projection 4D orthogonal: Visible parallel <\cos^4 \theta> = 3/80 = 0.0375.

N_eff = 240 \times 3/80 = 9 visible.

Compact 4D hidden: Ratio V_{compact}/V_{visible} damping.

Fractal nesting + rebound asymmetry curvature.

d_{compact} = 137.035999... / 9 \approx 15.226666...

\alpha^{-1} = N_eff \times d_{compact} exact observed.

Geometric no free param.

\includegraphics[width=0.8\textwidth]{puh_e8_compact_volume_ratio_simulation.png}

\section{Conclusion}
Compact volume ratio hidden damping — PUH exact \alpha E8 projection.

\end{document}