chore: some paper updates.

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

3 files changed, 16 insertions, 11 deletions

diff --git a/source/README.md b/source/README.md
index 8e81443..042ae8f 100644
--- a/source/README.md
+++ b/source/README.md
@@ -8,7 +8,7 @@
 
 - WG01: `PL/OS Group`.
 - WG02: `Systems Group`.
-- WG03: `Compiler Design Group`.
+- WG03: `Nectar Group`.
 - WG04-SPACE-RF: `Distributed Systems Group`.
-- WG05: `Execution Semantics Theory Group`.
+- WG05: `Execution Theory Group`.
 
diff --git a/source/wg03/wg03.tex b/source/wg03/wg03.tex
index 19accb6..34ac08c 100644
--- a/source/wg03/wg03.tex
+++ b/source/wg03/wg03.tex
@@ -74,20 +74,25 @@ const main()
 
 \begin{lstlisting}
 $ Hello, world!
-\end{lstlisting} And return $written$, now defined as $W$, upon completion. Such program that we denote as $\Theta$ may be noted as:
+\end{lstlisting} And return variant $written$, now defined as $W$, 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 $x$.
-
+\end{equation} Where $P(x)$ is $P$ with an argument of $x$. \\ Let $\Theta(x)$ be defined as the $\lambda$-Execution of a program $P$.
 \section{References}
 
 \begin{enumerate}
-  \item NeKernel.org (2026), \href{https://nekernel.org/nekernel}{https://nekernel.org}
-  \item NeKernel.org Primer (2026), \href{https://nekernel.org/nectar\_primer.html}{https://nekernel.org/nectar\_primer.html}
-  \item El Mahrouss, A. (2026). The Execution Semantics: On Axioms, Domains, and Authority. (v1.0.0). Zenodo. https://doi.org/10.5281/zenodo.18362375
+  	\item NeKernel.org (2026), \href{https://nekernel.org/nekernel}{https://nekernel.org}
+  	\item NeKernel.org Primer (2026), \href{https://nekernel.org/nectar\_primer.html}{https://nekernel.org/nectar\_primer.html}
+  	\item El Mahrouss, A. (2026). The Execution Semantics: On Axioms, Domains, and Authority. (v1.0.0). Zenodo. https://doi.org/10.5281/zenodo.18362375
 	\item El Mahrouss, A. (2026). The Mathematics of Execution. Zenodo. \\https://doi.org/10.5281/zenodo.18399140
 	\item Church, A. (1936). An Unsolvable Problem of Elementary Number Theory. American Journal of Mathematics, 58(2), 345–363. https://doi.org/10.2307/2371045
 \end{enumerate}
 
+\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.
+
 \end{document}
diff --git a/source/wg05/tn/tn001.05.tex b/source/wg05/tn/tn001.05.tex
index dba4083..176f22a 100644
--- a/source/wg05/tn/tn001.05.tex
+++ b/source/wg05/tn/tn001.05.tex
@@ -37,9 +37,9 @@ We will define their concepts alongside their properties. We assume the reader h
 
 Let $\Gamma(x, y, z)$ 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.
-Let the following formula:
+\\ Let 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})+C    
 \end{equation}
  Be defined as the execution product. Where $C$ is the Unknown Execution of $\Gamma(x, y, z)$—now defined as $g(E)$.
 
@@ -68,7 +68,7 @@ Let $N$ be defined as $\varnothing$.
 
 \section{Conclusion}
 
-Such concepts and properties are essential for `Execution Theory''s formality as per defined in El Mahrouss, A. (2026)' paper.
+Such properties are essential to define `Execution Theory''s as a mathematical construct as per defined in El Mahrouss, A.' paper. 
 
 \section{References}