chore: update papers with better writing and formalism.

Signed-off-by: Amlal El Mahrouss <amlal@nekernel.org>

2 files changed, 30 insertions, 22 deletions

diff --git a/source/dn001.05/paper.tex b/source/dn001.05/paper.tex
index 176f22a..ab7ccc6 100644
--- a/source/dn001.05/paper.tex
+++ b/source/dn001.05/paper.tex
@@ -33,47 +33,47 @@ We will define their concepts alongside their properties. We assume the reader h
 	\rule[1cm]{17cm}{0.01cm}
 \end{center}
 
-\section{The Execution Product}
+\section{The Execution Product Formula}
 
-Let $\Gamma(x, y, z)$ be an execution product of variable arguments $x$, $y$, $z$ defined in compatible and composable execution context $E$: \\
+Let $\Gamma(x, y, z)$ be an execution product of variable arguments $x$, $y$, $z$ defined in compatible and composable execution context $\mathbb{E}$. \\
 Let the index $\alpha$ denote the current execution domain of an execution product.
-\\ Let the following formula:
+\\ \\ Consider the following formula:
 \begin{equation}
-\Gamma(x, y, z) = \prod_{\alpha=1}^{n}(x_{\alpha} \times y_{\alpha} \times z_{\alpha})+C    
+\Gamma(x, y, z) = \prod_{\alpha=1}^{n}(x_{\alpha} \times y_{\alpha} \times z_{\alpha})+\mathbb{U}  
 \end{equation}
- Be defined as the execution product. Where $C$ is the Unknown Execution of $\Gamma(x, y, z)$—now defined as $g(E)$.
+which we define as the execution product, where $\mathbb{U}$ is the Unknown Execution of $\Gamma(x, y, z)$, now defined as $g(\mathbb{E})$.
 
-\subsection{Properties}
+\subsection{Properties of $\Gamma$}
 
 \begin{enumerate}
 
-    \item $\Gamma$ as defined previously as the `execution product' shall always be valid within the execution context $E$.
-    \item The execution context $E$ shall not denote $\varnothing$.
+    \item $\Gamma$ as defined previously as the `execution product' shall always be valid within the execution context $\mathbb{E}$.
+    \item The execution context $\mathbb{E}$ shall not denote $\varnothing$.
 
 \end{enumerate}
 
-\section{The Unknown Execution}
+\section{The Unknown Execution Property $\mathbb{U}$}
 
-Let an `Unknown Execution' $U$ be defined regarding an execution product $\Gamma$. \\
-Let $N$ be defined as $\varnothing$.
+Let $\mathbb{U} \in \Gamma$ be the `Unknown Execution' as $\mathbb{U} = g(\mathbb{E})$.\\
+Let $\mathbb{V} = \varnothing$ denote the `Empty Execution', with $\mathbb{V} \notin \Gamma$.
 
-\subsection{Properties}
+\subsection{Properties of $\mathbb{U}$}
 
 \begin{enumerate}
 
-    \item $U$ shall not be equal to $N$.
-    \item The execution domain of $\Gamma$ shall not be $N$.
+    \item $\mathbb{U}$ shall not be equal to $\mathbb{V}$.
+    \item The Execution Domain of $\Gamma$ shall not be equal to $\mathbb{V}$.
 
 \end{enumerate}
 
 \section{Conclusion}
 
-Such properties are essential to define `Execution Theory''s as a mathematical construct as per defined in El Mahrouss, A.' paper. 
+Such properties are essential to define Execution Theory as a mathematical construct, as presented in El Mahrouss, A. (2026).
 
 \section{References}
 
 \begin{enumerate}
-    \item El Mahrouss, A. (2026). The Execution Semantics: On Axioms, Domains, and Authority. (v2.0.0). Zenodo. https://doi.org/10.5281/zenodo.18366611
+    \item El Mahrouss, A. (2026). The Execution Semantics: On Axioms, Domains, and Authority. Zenodo. https://doi.org/10.5281/zenodo.18470651
 \end{enumerate}
 
 \end{document}
diff --git a/source/wg03/paper.tex b/source/wg03/paper.tex
index 82f53fa..98dec56 100644
--- a/source/wg03/paper.tex
+++ b/source/wg03/paper.tex
@@ -4,6 +4,7 @@
 \documentclass[11pt, a4paper]{article}
 \usepackage{graphicx}
 \usepackage{listings}
+\usepackage{amsmath,amssymb,amsthm}
 \usepackage{xcolor}
 \usepackage{hyperref}
 \usepackage[margin=0.5in,top=1in,bottom=1in]{geometry}
@@ -58,9 +59,9 @@
 	\rule[1cm]{17cm}{0.01cm}
 \end{center}
 
-\section{Definition of a Program}
+\section{Definition of the $\lambda$-Execution}
 
-Let a program $P$ be:
+Let a program $\mathbb{P}$ be:
 
 \begin{lstlisting}
 extern printf;
@@ -70,21 +71,28 @@ const main()
     const written := printf("%s:13", "Hello, world!\n");
     return written;
 }
-\end{lstlisting} $P$ shall execute:
+\end{lstlisting} 
+$\mathbb{P}$ shall execute:
 
 \begin{lstlisting}
 $ Hello, world!
-\end{lstlisting} And return variant $written$, now defined as $W$, upon completion. Such program that we denote as $\Theta$ may be defined as:
+\end{lstlisting}
+and return variant $written$, now defined as $\mathbb{W}$, upon completion. Such program that we denote as $\Theta$ may be defined as:
 
 \begin{equation}
 	\Theta(x) = \lambda x.(P(x))
-\end{equation} Where $P(x)$ is $P$ with an argument of $x$. \\ Let $\Theta(x)$ be defined as the $\lambda$-Execution of a program $P$.
+\end{equation}
+where $P(x) = \mathbb{P}$ with an argument of $x$. \\ Let $\Theta(x)$ be defined as the $\lambda$-Execution of a program $\mathbb{P}$.
 
 \section{Definitions}
 
 	\item Nectar: A compiled systems programming language—currently studied in this paper.
 	\item $\lambda$-Execution: Formally defined as: $\Theta(x) = \lambda x.(P(x))$.
-	\item $W$: The return variant—an $\lambda$-Execution variable based on the $\lambda$-Execution result.
+	\item $\mathbb{W}$: The return variant—an $\lambda$-Execution variable based on the $\lambda$-Execution result.
+
+\section{Conclusion}
+
+
 
 \section{References}