%
\label{def:mode-e-catalog}
%   I.T31, I.D07, I.D34, I.D84,
%   II.T06, II.T07, II.T15, II.T22, II.T27, II.T40,
%   II.D14, II.D22, II.D64
The following constructions in Books~I--II
are \textbf{Mode~E} (Earned):

\medskip
\renewcommand{\arraystretch}{1.35}
\begin{center}
\begin{tabular}{@{}r@{\;\;}p{3.0cm}@{\;\;}p{3.0cm}@{\;\;}p{3.5cm}@{}}
\toprule
\textbf{\#} & \textbf{Earned result}
  & \textbf{Orthodox status}
  & \textbf{$\tau$-derivation} \\
\midrule
\multicolumn{4}{@{}l@{}}{\textit{Book~I:}} \\
1 & Natural numbers $\mathbb{N}$
  & PA axioms (including $\infty$ induction schema)
  & Orbit of $\alpha$ under $\rho$ (I.D07) \\
2 & Composition associativity
  & Category theory axiom
  & Normal-form confluence \\
3 & Number tower $\mathbb{N} \to \mathbb{Z} \to \mathbb{Q} \to R_\tau \to H_\tau$
  & ZFC constructions (Dedekind, Cauchy)
  & Universal constructions from $\rho$ \\
4 & ABCD coordinates
  & External coordinate choice
  & Forced by hyperfactorization (I.T04) \\
5 & Hyperfactorization
  & No counterpart
  & $X = (A \hat{\phantom{x}} C)^B \cdot D$ unique \\
6 & Topos structure
  & Lawvere--Tierney axioms
  & Earned from HolFun monoid \\
7 & Global Hartogs
  & Cauchy integral proof (analytic)
  & CRT proof, constructive (I.T31) \\
\midrule
\multicolumn{4}{@{}l@{}}{\textit{Book~II:}} \\
8 & $\tau^3$ fibration
  & External fibration choice
  & Forced by ABCD (II.T03) \\
9 & Mutual determination
  & Separate proofs
  & Single derivation chain (II.T27) \\
10 & Self-enrichment
  & Grothendieck universes (new axiom)
  & K5-protected derivation \\
11 & $\tau$-manifold structure
  & Smooth manifold axioms
  & Earned from fibered product (Part~X) \\
12 & Tarski geometry (all 10 axioms)
  & 10 axioms postulated (Tarski 1959)
  & Derived from ultrametric (II.T15--II.T18) \\
13 & Induction
  & PA axiom schema ($\infty$ instances)
  & Earned from iterator ladder \\
\bottomrule
\end{tabular}
\end{center}