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Chaplain_Hypercube_Documentation.tex

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+\documentclass{article}
+\usepackage{amsmath}
+\begin{document}
+
+\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.
+
+\end{document}