feat: Add technical notes for WG05.

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

2 files changed, 68 insertions, 0 deletions

diff --git a/source/wg03/wg03.tex b/source/wg03/wg03.tex
index 73bcc99..830b20e 100644
--- a/source/wg03/wg03.tex
+++ b/source/wg03/wg03.tex
@@ -70,4 +70,9 @@ const main()
 }
 \end{lstlisting}
 
+\begin{enumerate}
+    \item NeKernel.org (2025), \href{https://nekernel.org/nekernel}{nekernel.org}
+    \item El Mahrouss, A. (2026). The Execution Semantics: On Axioms, Domains, and Authority. (v1.0.0). Zenodo. https://doi.org/10.5281/zenodo.18362375
+\end{enumerate}
+
 \end{document}
diff --git a/source/wg05/tn/tn001.05.tex b/source/wg05/tn/tn001.05.tex
new file mode 100644
index 0000000..19400cb
--- /dev/null
+++ b/source/wg05/tn/tn001.05.tex
@@ -0,0 +1,63 @@
+% AUTHOR: Amlal El Mahrouss
+% PURPOSE: WG05.TN001: The Mathematics of Execution.
+
+\documentclass[11pt,a4paper]{article}
+
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{amsmath,amssymb,amsthm}
+\usepackage{hyperref}
+\usepackage[margin=1in]{geometry}
+
+\title{The Mathematics of Execution.}
+\author{Amlal El Mahrouss\\
+\texttt{amlal@nekernel.org}}
+\date{January 2026}
+
+\begin{document}
+
+\bf \maketitle
+
+\begin{center}
+	\rule[1cm]{17cm}{0.01cm}
+\end{center}
+
+\begin{center}
+\begin{abstract}
+This paper covers the algebraic properties and concepts of `The Execution Semantics' paper by El Mahrouss, A (2026). It is intended as an additional resource over the aforementioned paper.
+\end{abstract}
+\end{center}
+
+\begin{center}
+	\rule[1cm]{17cm}{0.01cm}
+\end{center}
+
+\section{The Execution Product}
+
+
+
+Let $F(x, y, z)$ be an Execution Product of variable arguments $x$, $y$, $z$ defined in compatible and composable execution context $E$: \\
+Let the index $\alpha$ denote the current execution domain of an execution product.\\ 
+$F(x, y, z) = \prod_{\alpha}^{n}(x\times y \times z)+C$. Where $C$ is the Unknown Execution of $F(x, y, z)$—now defined as $g(E)$.
+
+\section{The Unknown Execution Property}
+
+Let an `Unknown Execution' $U$ be defined regarding an execution product $F$ \\
+Let `Null' be defined as $\varnothing$.
+
+\subsection{Properties}
+
+\begin{enumerate}
+
+    \item $U$ shall not be equal to Null.
+    \item The execution domain of $F$ shall not be Null.
+
+\end{enumerate}
+
+\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
+\end{enumerate}
+
+\end{document}
\ No newline at end of file