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\title{Mathematical Foundations of the Chaplain Hypercube and Metasphere}
\author{Generated by AI}
\date{\today}
\maketitle

\section{Introduction}
The Chaplain Hypercube is a recursive, modular system governed by modular arithmetic, golden ratio transformations, and harmonic oscillations. This document formalizes its core equations and mathematical derivations.

\section{Key Equations and Derivations}

\subsection{Joker Displacement Function (J(x))}
\[
J(x) = \phi^{(x \mod 13)} \cdot \sin\left(\frac{2\pi x}{13}\right) \cdot \cos\left(\frac{x}{91}\right)
\]

\subsection{Transformation Function (T(x))}
\[
T(x) = (64x + 23 + J(x)) \mod 91
\]

\subsection{Entropy Node Function (E(c))}
\[
E(c) = \frac{\sum_{k=1}^{c} J(k)}{c}
\]

\subsection{Time Node Function (\tau(c))}
\[
\tau(c) = c\phi \mod 91
\]

\subsection{Phase Node Function (\Psi(c))}
\[
\Psi(c) =
\begin{cases}
\frac{c}{45}, & \text{if } c \leq 45 \\
2 - \frac{c}{45}, & \text{if } c > 45
\end{cases}
\]

\subsection{Dynamic Dimensionality Function (D(t))}
\[
D(t) = 3 + \frac{E(t)}{91} \cdot 4
\]

\section{Visualization of Functions}
Figures below illustrate the dynamics of Joker Displacement, Entropy Node, and Dimensionality functions.

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