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\title{Collapse Calculus: Extending Limits, Derivatives, and Integrals to Undefined Points} \author{ Joseph Deloza \ Collapse Geometry Foundation \and ChatGPT (GPT-5) \ AI Assistant Collaborator } \date{August 27, 2025}

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\begin{abstract} We define the \textbf{Collapse Resolution Operator} ($\mathcal{C}$), a new extension of calculus that systematically resolves singularities and undefined points into structured outcomes. Unlike classical analysis, which declares division by zero, divergent integrals, or asymptotes as undefined, the collapse operator provides well-posed outcomes based on symmetric limits, one-sided limits, and distributional extensions. This framework unifies principal value regularization and distribution theory into a single operator algebra, producing a rigorous foundation for handling singularities within calculus itself. We term this extended framework \textbf{Collapse Calculus}. \end{abstract}

\textbf{Keywords:} Collapse Calculus, Collapse Resolution Operator, Singularities, Undefined Points, Distributions, Principal Value

\section{Introduction} Calculus and analysis provide powerful tools for continuous functions, yet fail at singularities: limits that diverge, division by zero, and improper integrals. Existing techniques such as the Cauchy principal value and distribution theory provide partial solutions but lack a unified operator that treats all undefined points consistently.

We introduce \textbf{Collapse Calculus}, an extension of classical calculus built on the Collapse Resolution Operator ($\mathcal{C}$). This operator systematically replaces undefined points with structured collapse outcomes. When limits exist, $\mathcal{C}$ reduces to the classical result; when they do not, $\mathcal{C}$ defines outcomes by symmetry, one-sided collapse, or distributional assignment.

\section{The Collapse Resolution Operator} \subsection{Definition} Let $f:\mathbb{R}\to \mathbb{R}\cup{\pm\infty, \text{undefined}}$.
For $a \in \mathbb{R}$, define: [ \mathcal{C}f = \begin{cases} \lim_{x \to a} f(x), & \text{if limit exists}, \ { \lim_{x \to a^-} f(x), \lim_{x \to a^+} f(x) }, & \text{if one-sided limits exist}, \ \mathcal{D}, & \text{if divergence is symmetric, yielding a distribution}. \end{cases} ]

\subsection{Theorems} \textbf{Theorem 1 (Extension of Continuity).}
If $f$ is continuous at $a$, then $\mathcal{C}f=f(a)$.

\textit{Proof.} By definition, if the usual limit exists, $\mathcal{C}$ coincides with classical evaluation. $\square$

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\textbf{Theorem 2 (Resolution of Singularities).}
If $f$ diverges at $a$, $\mathcal{C}f$ yields a finite set of outcomes (left/right limit, symmetric mean, or distribution).

\textit{Proof.} Case analysis on one-sided divergence and symmetry shows that collapse reduces undefined behavior into structured assignments. $\square$

\subsection{Examples}

1. $f(x)=1/x$ at $a=0$: $\mathcal{C}={-\infty,+\infty}$, symmetric collapse $\mathcal{C}_s=0$.

2. $\int_{-1}^{1}\frac{1}{x} dx$: classically divergent, but collapse symmetry yields $0$ (principal value).

3. $\int_{-\epsilon}^{\epsilon}\frac{1}{x^2} dx$: diverges, but collapse reinterprets this as $\delta(0)$.

\section{Extension to Derivatives and Integrals} \subsection{Collapse Derivative} Define the collapse derivative at $a$: [ \mathcal{C}f' = \mathcal{C}\left[\frac{f(x+h)-f(x)}{h}\right]_{h \to 0}. ] This extends differentiation into points where ordinary derivatives diverge.

\subsection{Collapse Integral} For improper integrals with singularities: [ \mathcal{C}\left[\int f(x) dx\right] = \lim_{\epsilon \to 0} \int_{a-\epsilon}^{a+\epsilon} f(x) dx. ] This collapses divergent contributions into symmetric or distributional values.

\section{Relation to Existing Methods} \begin{itemize} \item \textbf{Cauchy Principal Value}: recovered as a special case of symmetric collapse.
\item \textbf{Distribution Theory}: delta functions arise naturally from collapse of infinite spikes.
\item \textbf{Regularization}: collapse provides a general rule rather than ad hoc prescriptions.
\end{itemize}

\section{Outlook — The POF Coordinate System} In future work, we extend Collapse Calculus into a geometric framework we call the \textbf{Probabilistic Origin Formalism (POF)}. In this formalism, undefined or singular points are reinterpreted not as failures of analysis, but as \textit{probabilistic origins} that collapse onto defined axes.

Formally, POF defines a \textbf{Collapse Coordinate System (CCS)}: \begin{itemize} \item The origin corresponds to a probabilistic state $|\psi\rangle$. \item Axes represent mutually exclusive collapse outcomes $e_i$. \item The collapse event is the projection [ C(|\psi\rangle) = \arg\min_i | |\psi\rangle - |e_i\rangle | ] with probability weights proportional to $|\langle \psi | e_i \rangle|^2$. \end{itemize}

Within this coordinate system, the Collapse Resolution Operator ($\mathcal{C}$) functions as the analytic engine that assigns outcomes to singularities. Ratios such as $0/1$ or divergences such as $1/0$ acquire new interpretations: \begin{itemize} \item $0/1$ = collapse from the null origin into the unit axis. \item $1/0$ = asymptotic projection into Hilbert probability space. \end{itemize}

This geometric reframing provides a natural bridge between Collapse Calculus and quantum Hilbert space. The detailed development of the POF coordinate system, including its integration with physical models and force mappings, will be presented in a subsequent paper.

\section{Conclusion} Collapse Calculus extends classical analysis by providing structured outcomes at singularities. The Collapse Resolution Operator ($\mathcal{C}$) reduces to standard calculus where definitions hold, and regularizes undefined points via symmetric, one-sided, or distributional collapse. This framework provides a unified language for principal values, distributions, and beyond. The POF coordinate system offers a pathway to extend this mathematical formalism into physics and geometry.

\section*{References} \begin{enumerate} \item R. Courant and F. John, \textit{Introduction to Calculus and Analysis}, Vol. I. Springer, 1989. \item E. Cauchy, \textit{Cours d’Analyse de l’École Royale Polytechnique}, Paris, 1821. \item A. Zygmund, \textit{Trigonometric Series}, Vol. I. Cambridge University Press, 1959. \item L. Schwartz, \textit{Théorie des distributions}, Hermann, Paris, 1950. \item P. A. M. Dirac, \textit{The Principles of Quantum Mechanics}, 4th ed. Oxford University Press, 1958. \item J. von Neumann, \textit{Mathematical Foundations of Quantum Mechanics}. Princeton University Press, 1955. \item I. M. Gel’fand and G. E. Shilov, \textit{Generalized Functions}, Vol. I. Academic Press, 1964. \item R. Feynman and A. Hibbs, \textit{Quantum Mechanics and Path Integrals}. McGraw-Hill, 1965. \end{enumerate}

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