%
\label{def:mode-d-catalog}
%   I.T11, I.D03, I.D18, I.D21, I.D34,
%   II.T06, II.T40, II.T41, II.T42, II.D68
The following constructions in Books~I--II
are \textbf{Mode~D} (Gained):

\medskip
\renewcommand{\arraystretch}{1.35}
\begin{center}
\begin{tabular}{@{}r@{\;\;}p{3.9cm}@{\;\;}p{2.2cm}@{\;\;}p{5.0cm}@{}}
\toprule
\textbf{\#} & \textbf{$\tau$-gain}
  & \textbf{Source}
  & \textbf{Why impossible in ZFC/PA} \\
\midrule
\multicolumn{4}{@{}l@{}}{\textit{Book~I gains:}} \\
1 & Categoricity
  & I.T08
  & L\"owenheim--Skolem prevents categorical first-order theories \\
2 & Rigidity $\Aut(\tau) = \{\id\}$
  & I.T07, I.T11
  & Most structures have non-trivial automorphisms \\
3 & Finite axiom set (no schemas)
  & K0--K6
  & PA requires infinite induction schema \\
4 & Structural decidability
  & I.P07+
  & G\"odel sentences are undecidable \\
5 & $\dim = 4$ forced
  & I.D03
  & No orthodox axiom forces dimension \\
6 & Prime Polarity
  & I.T05
  & Invisible in orthodox number theory \\
7 & Algebraic lemniscate
  & I.D18
  & Requires $\jj^2 = +1$ \\
8 & Truth4 (4-valued logic)
  & I.D21
  & Classical logic is $\{T,F\}$ \\
9 & Explosion barrier
  & I.T13
  & Classical logic admits explosion \\
10 & Master constant $\iota_\tau$
  & I.D34
  & No canonical spectral constant in ZFC \\
11 & Presheaf characterization
  & Book~I Part~XVI
  & Requires categorical holomorphy \\
12 & Diagonal $=$ linear logic
  & Book~I Part~XVII
  & Independent characterization \\
13 & Safe self-enrichment
  & Book~II Part~VIII
  & Russell's paradox blocks na\"ive self-description \\
\midrule
\multicolumn{4}{@{}l@{}}{\textit{Book~II gains:}} \\
14 & Central Theorem
  & II.T40
  & No orthodox proof of $\mathcal{O} \cong A_{\mathrm{spec}}$ \\
15 & Boundary-first paradigm
  & II.R03
  & Orthodox: interior $\to$ boundary \\
16 & Inverted dependency
  & II.T06
  & Orthodox: topology before holomorphy \\
17 & Parallel Postulate proved
  & II.T15--II.T18
  & 2000 years of independence \\
18 & Liouville dodge
  & II.T41
  & Liouville theorem blocks non-constant bounded entire functions \\
19 & Categoricity of $\tau^3$
  & II.T42
  & Moduli spaces typically have dimension $>0$ \\
20 & Positive regularity
  & II.D49
  & Orthodox regularity defined negatively \\
21 & Holomorphic decomposition
  & II.L07
  & Orthodox $f = u+iv$ gives harmonic, not holomorphic, components \\
22 & Master switch (12-level)
  & II.D68
  & No orthodox single-parameter classification \\
23 & Structural incompatibility
  & II.T43
  & Internal to $\tau$ \\
\bottomrule
\end{tabular}
\end{center}