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\fancyhead[L]{\small\itshape Ne.org, Vol.~2, No.~1, March 2026}
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\title{%
  \vspace{-1.5em}%
  \large\textbf{All you need is Integrals for Debugging.}%
}

\author{%
  \textbf{Amlal El Mahrouss}$^{1}$\thanks{Corresponding author. Email: amlal@nekernel.org}
  \\amlal@nekernel.org, amlalelmahrouss@icloud.com\\[0.4em]
  \small $^{1}$Ne.org Journal
}

\date{%
  \small 3 March 2026
}

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\begin{document}


\twocolumn[{%
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  %% ---- Abstract -------------------------------------------
  \begin{center}
  \begin{minipage}{0.92\linewidth}
    \small
    \textbf{Abstract.}\enspace
    This development presents an approach of software debugging[1] in which we'll name 'Integrals of Debuggging' (ID).
    We propose a mathematical of which integrals are used to analyze, modify, and apply, software patches to programs.

    
    \medskip
    \textbf{Keywords:}\enspace
    Computer Science; Mathematics;
  \end{minipage}
  \end{center}
  \vspace{0.8em}
  \rule{\linewidth}{0.4pt}
  \vspace{1em}
}]

\section{Introduction}

Consider an open interval from $n$ to $p$ over a field $\mathbb{Code}$ which describes a program and its properties. What are the techniques in which we can analyze and audit its properties and behavior? We will try to impose a framework in this report.

\section{Definition of the LongReturn(x, t) function}
\label{sec:cl-example-integrals}

Let an integral of $\operatorname{LongReturn(x, t)}$[2] be defined where the value $n > 0$ and $p > 0, \quad n < p$:

\begin{equation}
    \operatorname{LongReturn(x, t)} \coloneqq \int_{n}^{p} \operatorname{FastReturn(x, t)} \cdot d \operatorname{t}
\end{equation}
We assume that $x$ and $t$ are always greater or equal than one, $\operatorname{LongReturn(x, t)}$ shall always be greater than $\operatorname{FastReturn(x, t)}$ for all such value in $\mathbb{Code}$.
This introduction of the LongReturn, will let us jump to our next section about its applications in Computer Analysis.

\section{Constraints}

However before starting, we shall provide the following conditions from [2]:
\begin{equation}
    \operatorname{LongReturn(x, t)} \geq \operatorname{FastReturn(x, t)}, \quad \operatorname{x} \geq 1
\end{equation}
The variable x shall also be defined in $\mathbb{Code}$ throughout the integral. We shall now proceed into the next section.

\section{Definition of the DerRet(x, t) function}

The following is the derivative of $\operatorname{LongReturn(x, t)}$, which equals to:
\begin{equation}
    \operatorname{DerRet(State, Time\_State)} \coloneqq \frac{\delta \operatorname{Travel(State)}}{\delta Time\_State} + Rem
\end{equation}
Such that the following is assumed:
\begin{equation}
    \quad State, \operatorname{Travel(State)} \in \mathbb{Code}
\end{equation}
Indeed, the function $\operatorname{Travel(State)}$ is equal to the program state at the variable $\operatorname{State}$ which represents the program's state.

\section{Applications}

One application of the $\operatorname{LongReturn(x, t)}$ is software debugging, more so, the break-point system of a debugger. The following section includes graphs to measure the efficiency of our approach in such process.

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