%
\label{def:mode-b-catalog}
%   II.T06, II.T07, II.D09, II.D12, II.D13, II.D14,
%   II.D22, II.D35, II.D36
The following constructions in Books~I--II
are \textbf{Mode~B} (Parallel):

\medskip
\renewcommand{\arraystretch}{1.35}
\begin{center}
\begin{tabular}{@{}p{0.2\linewidth}@{\;\;}p{0.2\linewidth}@{\;\;}p{0.2\linewidth}@{\;\;}p{0.2\linewidth}@{}}
\toprule
\textbf{Construction} & \textbf{$\tau$ version} & \textbf{Orthodox version}
  & \textbf{Key difference} \\
\midrule
Holomorphic functions
  & $\tau$-holomorphic on $H_\tau$, $\jj^2 = +1$
  & Classical holomorphic on $\mathbb{C}$, $i^2 = -1$
  & Sign: wave vs.\ Laplace \\
Constructive reals
  & $R_\tau$ (I.D84), ultrametric
  & $\mathbb{R}$ (Dedekind/Cauchy), Archimedean
  & Metric: $2^{-\delta}$ vs.\ $|x-y|$ \\
Set membership
  & Divisibility (I.D31)
  & $\in$ (ZFC)
  & $a \mid b$ vs.\ $a \in b$ \\
Topology
  & Stone space (II.D14), clopen
  & Manifold, connected, Hausdorff
  & Totally disconnected \\
BndLift
  & CRT propagation (II.D36)
  & Cauchy integral
  & Finite vs.\ contour \\
Laurent expansion
  & CRT coefficients (II.D35)
  & Cauchy coefficients
  & Constructive extraction \\
de Rham complex
  & $\tau$-de Rham (Book~II Part~X)
  & Classical de Rham
  & Discrete vs.\ smooth \\
Idempotent decomp.
  & $f = e_+ g + e_- h$ (holomorphic)
  & $f = u + iv$ (harmonic)
  & Both holomorphic vs.\ only harmonic \\
Metric
  & $d = 2^{-\delta}$ ultrametric (II.D12)
  & $|x - y|$ Archimedean
  & Agreement depth vs.\ magnitude \\
Code/Decode
  & Boundary coefficients (II.T35)
  & Laurent series
  & Without integration \\
\bottomrule
\end{tabular}
\end{center}

\medskip
\noindent
For each entry, the \emph{same axioms} column
is the set of defining properties that both versions satisfy.
The \emph{key difference} column identifies
the carrier-level divergence.