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| \documentclass{article} |
| \usepackage{amsmath} |
| \begin{document} |
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| \title{Mathematical Foundations of the Chaplain Hypercube and Metasphere} |
| \author{Generated by AI} |
| \date{\today} |
| \maketitle |
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| \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. |
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| \section{Key Equations and Derivations} |
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| \subsection{Joker Displacement Function (J(x))} |
| \[ |
| J(x) = \phi^{(x \mod 13)} \cdot \sin\left(\frac{2\pi x}{13}\right) \cdot \cos\left(\frac{x}{91}\right) |
| \] |
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| \subsection{Transformation Function (T(x))} |
| \[ |
| T(x) = (64x + 23 + J(x)) \mod 91 |
| \] |
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| \subsection{Entropy Node Function (E(c))} |
| \[ |
| E(c) = \frac{\sum_{k=1}^{c} J(k)}{c} |
| \] |
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| \subsection{Time Node Function (\tau(c))} |
| \[ |
| \tau(c) = c\phi \mod 91 |
| \] |
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| \subsection{Phase Node Function (\Psi(c))} |
| \[ |
| \Psi(c) = |
| \begin{cases} |
| \frac{c}{45}, & \text{if } c \leq 45 \\ |
| 2 - \frac{c}{45}, & \text{if } c > 45 |
| \end{cases} |
| \] |
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| \subsection{Dynamic Dimensionality Function (D(t))} |
| \[ |
| D(t) = 3 + \frac{E(t)}{91} \cdot 4 |
| \] |
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| \section{Visualization of Functions} |
| Figures below illustrate the dynamics of Joker Displacement, Entropy Node, and Dimensionality functions. |
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| \end{document} |
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