%
\label{def:mode-c-catalog}
%   I.T36, I.D76, I.D86, II.T40, II.T41, II.D13, II.D14, II.D68
The following constructions are \textbf{Mode~C} (Refused):
$\tau$'s axioms structurally block them.

\medskip
\renewcommand{\arraystretch}{1.35}
\begin{center}
\begin{tabular}{@{}r@{\;\;}p{3.2cm}@{\;\;}p{3.2cm}@{\;\;}p{3.2cm}@{}}
\toprule
\textbf{\#} & \textbf{Refused construction}
  & \textbf{Blocking axiom/theorem}
  & \textbf{What it protects} \\
\midrule
1 & Uncountable sets
  & K6 (Object Closure)
  & Categoricity (I.T08) \\
2 & $\varepsilon$-$\delta$ limits
  & Unique $\omega$ (K2)
  & Single infinity \\
3 & Cantor diagonal argument
  & K5 (Diagonal Discipline)
  & No cardinality hierarchy \\
4 & Unrestricted comprehension
  & K5, K6
  & Paradox avoidance \\
5 & Conformal maps
  & $\jj^2 = +1$ (I.T10)
  & Bipolar decomposition \\
6 & Non-constructive existence
  & Constructive witnesses
  & Decidability \\
7 & Connected open sets
  & Stone topology (II.D14)
  & Clopen partition \\
8 & Smooth structure
  & Ultrametric (II.D13)
  & Stage-finite geometry \\
9 & Arbitrary choice (AC)
  & Constructive witnesses
  & Effective procedures \\
\midrule
\multicolumn{4}{@{}l@{}}{\textit{Book~I specific:}} \\
10 & Non-Euclidean geometry
  & Parallel Postulate proved (II.T15)
  & Euclidean uniqueness \\
11 & Hyperoperators above tet.
  & K5 (saturation at level~3)
  & Diagonal discipline \\
12 & Star-autonomous duality
  & Thin category
  & Rigid morphism structure \\
13 & Arbitrary model theory
  & Categoricity (I.T08)
  & Unique model \\
\midrule
\multicolumn{4}{@{}l@{}}{\textit{Book~II specific:}} \\
14 & Cauchy integral formula
  & Wave PDE (II.D70)
  & BndLift replaces \\
15 & Riemann surfaces
  & $\jj^2 = +1$, no conformal
  & Wave holomorphy \\
16 & SCV dimensional ladder
  & No Archimedean engine
  & Simultaneous features \\
\bottomrule
\end{tabular}
\end{center}