\label{thm:enriched-bi-square-comparison}
The four bi-squares in the scaling chain---algebraic (I.T41), topological (II.T49), enriched (Definition~\ref{def:enriched-bi-square}), and computational (Definition~\ref{def:computational-bi-square}, Ch.~58)---are structurally identical in the following sense:
\begin{enumerate}
\item[\emph{(Shape.)}] Each is a $2 \times 3$ pasted diagram with the same combinatorial shape: left square = tower coherence, right square = spectral naturality, pasting = the defining constraint.
\item[\emph{(Maps.)}] The structural maps at each level are the images of the algebraic maps under the successive enrichment functors: $\operatorname{Enr}_{01}$ from $\Elayer{0}$ to $\Elayer{1}$, and $\operatorname{Enr}_{12}$ from $\Elayer{1}$ to $\Elayer{2}$.  No new structural maps are introduced; the existing maps are transported.
\item[\emph{(Pasting.)}] The pasting identity upgrades at each level while preserving its algebraic core:
\begin{center}
\begin{tabular}{llll}
\toprule
\textbf{Level} & \textbf{Left Square} & \textbf{Right Square} & \textbf{Pasting} \\
\midrule
$\Elayer{0}$ (I.T41) & Tower coherence & Spectral naturality $\chi_\pm$ & $\alpha_p \wedge \alpha_q = \alpha_{p \times q}$ \\
\addlinespace
$\Elayer{0} \to \Elayer{1}$ & Holomorphic & Boundary values & $\mathcal{O}(\tau^3)$ \\
(II.T49) & extension & on $\Lemniscate$ & $\cong A_{\mathrm{spec}}(\Lemniscate)$ \\
\addlinespace
$\Elayer{1}+$ & Sector tower & Langlands & Finite \\
(this chapter) & coherence & functoriality & factorization \\
\addlinespace
$\Elayer{2}$ (Ch.~58) & TTM execution & CRT witnesses & Product-Meet \\
 & & & Collapse \\
\bottomrule
\end{tabular}
\end{center}
\end{enumerate}