chore: add definitions and better paper wording.

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

2 files changed, 20 insertions, 4 deletions

diff --git a/drafts/dn001.06/paper.tex b/drafts/dn001.06/paper.tex
index b1b6435..097033b 100644
--- a/drafts/dn001.06/paper.tex
+++ b/drafts/dn001.06/paper.tex
@@ -7,7 +7,7 @@
 \usepackage{amsfonts}
 \usepackage[margin=0.5in,top=1in,bottom=1in]{geometry}
 
-\title{$\mathbb{C}$ Index (DRAFT)}
+\title{The $\mathbb{C}$ Index Property}
 \author{Amlal El Mahrouss\\amlal@nekernel.org}
 \date{February 2026}
 
@@ -18,13 +18,13 @@
 	\rule[0.01cm]{17cm}{0.01cm}
 \end{center}
 \abstract{
-	This paper defines `objects' as indexed $n \in \mathbb{R}$ valued in $\mathbb{C}$. Definitions and properties are stated in this paper.
+	This paper defines the property of $\mathbb{C}$ Indices such that $n \in \mathbb{R}$ for all $\Omega_{n} \in \mathbb{C}$. Definitions and properties are stated in this paper.
 }
 \begin{center}
 	\rule[0.01cm]{17cm}{0.01cm}
 \end{center}
 
-\section{Definitions}
+\section{Definitions of $\mathbb{C}$ Indices}
 
 \subsection{Definition of $\Omega$}
 
@@ -41,4 +41,20 @@ Let $\alpha$ denote $\Omega_{n}$ with a remainder of $\mathbb{X}$ defined as the
 	\alpha = \Omega_{n} \cup \mathbb{X}
 \end{equation}
 
+\section{Properties of $\mathbb{C}$ Indices}
+
+\subsection{Properties of $\Omega$}
+
+\begin{enumerate}
+	\item $\Omega_{n} \cup \mathbb{X}$ such that $\mathbb{X} = \beta_{n}$.
+\end{enumerate}
+
+\subsection{Properties of $\alpha$}
+
+\begin{enumerate}
+
+	\item $\mathbb{X} = \beta_{n}$ such that $\beta_{n} \in \mathbb{C}$
+	
+\end{enumerate}
+
 \end{document}
diff --git a/source/wg05/paper.tex b/source/wg05/paper.tex
index da2fa29..1f49783 100644
--- a/source/wg05/paper.tex
+++ b/source/wg05/paper.tex
@@ -80,7 +80,7 @@ An execution domain is defined as previously stated in Section~2. An execution a
 
 A \emph{trait} is a set of formal rules defining the semantic concepts of an execution context.
 
-Let $A$ be an execution authority of type $T$, where $T$ is a trait of an execution context.
+Let $A$ be an execution authority of type $T$, such that $T$ is a trait of an execution context.
 
 \subsection{Properties}