%
\label{def:five-comparison-modes}
Let $\mathcal{C}_\tau$ denote a construction
(definition, theorem, or property)
within Category~$\tau$,
and let $\mathcal{C}_{\mathrm{orth}}$ denote
its closest orthodox counterpart (if one exists).
The \textbf{comparison mode} of $\mathcal{C}_\tau$
is one of:

\medskip
\renewcommand{\arraystretch}{1.35}
\begin{center}
\begin{tabular}{@{}c@{\;\;}l@{\;\;}p{7.5cm}@{}}
\toprule
\textbf{Mode} & \textbf{Name} & \textbf{Criterion} \\
\midrule
\textbf{A} & Same &
  $\mathcal{C}_\tau$ and $\mathcal{C}_{\mathrm{orth}}$
  are identical objects
  with a canonical bijection preserving
  all relevant structure. \\
\textbf{B} & Parallel &
  $\mathcal{C}_\tau$ and $\mathcal{C}_{\mathrm{orth}}$
  satisfy the same defining axioms
  but live on different carriers.
  Structural isomorphism holds at the property level
  but fails at the element level. \\
\textbf{C} & Refused &
  $\tau$'s axioms structurally block
  $\mathcal{C}_{\mathrm{orth}}$.
  The orthodox construction violates
  a $\tau$ axiom (K0--K6)
  or requires tools $\tau$ lacks. \\
\textbf{D} & Gained &
  $\mathcal{C}_\tau$ achieves a property
  that is provably impossible,
  independent, or false in ZFC/PA.
  No orthodox counterpart exists. \\
\textbf{E} & Earned &
  $\mathcal{C}_\tau$ derives a result
  that orthodox mathematics takes as axiom
  or obtains via a fundamentally different proof. \\
\bottomrule
\end{tabular}
\end{center}

\medskip
\noindent
The modes satisfy:
\begin{enumerate}
    \item[\textup{(i)}]
          \textbf{Exhaustiveness.}
          Every construction $\mathcal{C}_\tau$
          in Books~I--II
          has a unique primary mode.

    \item[\textup{(ii)}]
          \textbf{Mutual exclusivity.}
          A construction is Mode~A
          if and only if the objects are literally identical;
          Mode~B if the axioms match but the carriers differ;
          Mode~C if the orthodox construction is blocked;
          Mode~D if the $\tau$-construction
          has no orthodox analogue;
          Mode~E if the content matches
          but the proof status differs.

    \item[\textup{(iii)}]
          \textbf{Compositionality.}
          When a construction has aspects
          of multiple modes,
          the primary mode captures
          the most significant structural difference.
\end{enumerate}