%
\label{def:mode-a-catalog}
%   I.T05, I.T08
The following constructions in Books~I--II
are \textbf{Mode~A} (Same):

\medskip
\renewcommand{\arraystretch}{1.35}
\begin{center}
\begin{tabular}{@{}p{0.23\linewidth}p{0.23\linewidth}p{0.23\linewidth}p{0.23\linewidth}@{}}
\toprule
\textbf{Object} & \textbf{$\tau$ source} & \textbf{Orthodox}
  & \textbf{Identity witness} \\
\midrule
Natural numbers $\mathbb{N}$
  & I.D07 (orbit of $\alpha$)
  & PA
  & Canonical bijection
    $\rho^n(\alpha) \leftrightarrow n$ \\
Prime numbers
  & I.D19b
  & Fundamental theorem of arithmetic
  & Same set, same divisibility \\
$\pi$
  & Book~II Part~V (earned)
  & Classical transcendental
  & Same limit of solenoidal ratio \\
$e$
  & Book~II Part~V (earned)
  & Classical transcendental
  & Same limit $(1+1/n)^n$ \\
$\hat{\mathbb{Z}}$
  & I.D19 ($\hat{\mathbb{Z}}_\tau$)
  & Profinite completion of $\mathbb{Z}$
  & Same inverse limit \\
Boolean fragment
  & I.D21 (Truth4 restricted)
  & Classical logic $\{T, F\}$
  & $\{T, F\} \hookrightarrow \{T, F, B, N\}$ \\
Fundamental theorem
  & I.T19
  & FTA
  & Same unique factorization \\
\bottomrule
\end{tabular}
\end{center}

\medskip
\noindent
For each entry, the identity witness is a
canonical bijection or embedding
that preserves all structure
(order, operations, defining properties).