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\label{prop:passage-to-book-ii}
The following structures, earned in Book~I,
suffice for the Central Theorem of Book~II:
\begin{enumerate}
    \item $\tau^3 = \tau^1 \times_f T^2$
          with $\dim_\tau = 4$
          (Chapter~\ref{ch:dimension-fibration}).
    \item The algebraic lemniscate~$\mathbb{L}$
          (Theorem~\ref{thm:algebraic-lemniscate}, I.D18).
    \item Split-complex scalars with $\jj^2 = +1$
          (Definition~\ref{def:split-complex}, I.D20).
    \item Master constant $\iota_\tau = 2/(\pi + e)$
          (Definition~\ref{def:iota-tau}, I.D34).
    \item $\mathrm{Hol}(\mathbb{L})$
          with monoid and ring structure
          (Definition~\ref{def:hol-L}, I.D49).
    \item Spectral coefficients and decomposition
          (Definition~\ref{def:spectral-coefficients},
          Chapter~\ref{ch:spectral-decomposition}).
    \item Tower coherence (I.D46)
          and the $\tau$-Identity Theorem (I.T21).
    \item Global Hartogs (I.T06)
          and interior determination (I.C02).
    \item The earned topos (I.X02).
\end{enumerate}
No additional axioms beyond those of Book~I
are required.

The Central Theorem asserts
$\mathcal{O}(\tau^3) \cong A_{\mathrm{spec}}(\mathbb{L})$.
The left-hand side requires items~1, 3, 5, 7, 8;
the right-hand side requires items~2, 4, 6.
The isomorphism will be constructed in Book~II
using the earned categorical apparatus (item~9).