\label{def:complete-dependency-chain}
The \emph{complete dependency chain} of Category~$\T$:
\begin{enumerate}
\item \textbf{Five generators.}
      $\{\alpha, \pi, \gamma, \eta, \omega\}$
      (Book~I, Part~I); the minimal-alphabet theorem forbids fewer.

\item \textbf{Seven axioms.}
      $\KAxiom{0}$--$\KAxiom{6}$
      (Book~I, Parts~II--III); each is the unique constraint
      compatible with five generators.

\item \textbf{Four orbits.}
      Progression on the seed~$\rho$ produces exactly four orbits
      (Book~I, Part~V, ABCD Orbit Theorem).

\item \textbf{ABCD coordinates.}
      The peel map $\Phi(x) = (A, B, C, D)$ assigns every
      $\T$-object a quadruple (Book~I, Part~V).

\item \textbf{Boundary ring.}
      $H_\tau = \mathbb{Z}[\,\jj\,]/(\jj^2 - 1)$,
      forced by the fiber~$T^2$
      (Book~I, Part~VII; Book~II, Part~III).

\item \textbf{Central Theorem.}
      $\mathcal{O}(\tau^3) \cong A_{\mathrm{spec}}(\Lemniscate)$
      (Book~II, II.T40).

\item \textbf{Enrichment ladder.}
      $\mathcal{F}_E$
      (Theorem~\ref{def:enrichment-functor}, Ch~4;
       Definition~\ref{def:layer-template}, Ch~5)
      produces
      $\Elayer{0} \subsetneq \Elayer{1}
       \subsetneq \Elayer{2} \subsetneq \Elayer{3}$,
      saturating at $\Elayer{3}$
      (Theorem~\ref{thm:saturation-e3}, Ch~7).

\item \textbf{4+1 sector template.}
      Four ABCD orbits plus one mixed sector partition every
      $\Elayer{1}$~datum
      (Definition~\ref{def:four-plus-one-decomposition}, Ch~10).

\item \textbf{Spectral algebra.}
      $\mathfrak{A}_\tau$ collects the eight spectral forces
      (Definition~\ref{ch:spectral-algebra-complete}, Ch~20).

\item \textbf{Millennium clusters.}
      Spectral purity (Definition~\ref{def:ch25-spectral-purity}, Ch~25),
      BSD coherence (Theorem~\ref{thm:bsd-coherence-theorem}, Ch~47),
      and five further readings (Parts~IV--VI)
      follow from~$\mathfrak{A}_\tau$ applied sector by sector.

\item \textbf{Enriched bi-square.}
      Book~I's bi-square (I.T41) lifts to~$\Elayer{1}+$
      (Definition~\ref{def:enriched-bi-square}, Ch~50),
      unifying all readings in one $2 \times 3$ pasted diagram.

\item \textbf{Enrichment tower assembly.}
      $\Elayer{0} \to \Elayer{1} \to \Elayer{2} \to \Elayer{3}$
      assembles into a single commutative diagram
      (Theorem~\ref{thm:enrichment-tower-assembly}, Ch~51).

\item \textbf{Computational collapse.}
      Product-meet collapse at $\Elayer{2}$
      identifies witness-search with composite-construction
      (Theorem~\ref{thm:product-meet-collapse}, Ch~58).

\item \textbf{Hinge.}
      The assembled tower, enriched bi-square, and computational
      collapse combine: \emph{the eight spectral forces are jointly
      determined by the five generators and no free parameter.}
      This is the Hinge Theorem (Ch~61).
\end{enumerate}