%
\label{thm:j-replaces-i}
In the holomorphic analysis of~$\tau^3$:
\begin{enumerate}
    \item[\textup{(i)}]
          The fiber involution $\sigma : B \leftrightarrow C$
          is the \textbf{unique} non-trivial automorphism
          of the fiber~$T^2$
          that preserves the ABCD decomposition
          and fixes the base~$\tau^1$.
          It generates a $\mathbb{Z}/2$ group.

    \item[\textup{(ii)}]
          Every $\mathbb{Z}/2$-equivariant
          algebra structure on the fiber
          is isomorphic to the split-complex ring
          $\mathbb{R}[\jj]$,
          with $\jj$ acting as~$\sigma$.

    \item[\textup{(iii)}]
          There is no continuous homomorphism
          $\mathrm{SO}(2) \to \mathrm{Aut}(T^2)$
          compatible with the ABCD decomposition.
          In particular, the imaginary unit~$i$
          \textup{(}generator of $\mathrm{SO}(2)$
          acting on~$\mathbb{R}^2$\textup{)}
          has no $\tau$-internal avatar.

    \item[\textup{(iv)}]
          The bipolar idempotents $e_\pm = (1 \pm \jj)/2$
          are the canonical sector projections:
          \[
              e_+ \;:\; z \longmapsto z_+ e_+,
              \qquad
              e_- \;:\; z \longmapsto z_- e_-,
          \]
          recovering the spectral decomposition
          of the boundary ring \textup{(I.D19, Book~I)}.
\end{enumerate}

\textbf{(i)}
The ABCD decomposition is a CRT product
(I.D17, Book~I).
Any automorphism preserving the D and A factors
must permute B and~C.
The permutation group of two elements is $S_2 \cong \mathbb{Z}/2$:
the identity and the swap~$\sigma$.

\textbf{(ii)}
A $\mathbb{Z}/2$-equivariant algebra
on a two-dimensional real vector space
with involution~$\sigma$
decomposes into $\pm 1$ eigenspaces.
The multiplication table
is determined by $\sigma$'s action:
if $\sigma$ has eigenvalues $\pm 1$
with one-dimensional eigenspaces,
the resulting algebra is
$\mathbb{R} \times \mathbb{R}
\cong \mathbb{R}[\jj]/(\jj^2 - 1)$.
This is the split-complex ring.

\textbf{(iii)}
$\mathrm{SO}(2)$ is connected.
$\mathrm{Aut}(T^2)$ preserving the ABCD structure
is $\mathbb{Z}/2$, which is discrete.
A continuous homomorphism from a connected group
to a discrete group must be trivial.
Hence no non-trivial $\mathrm{SO}(2)$ action exists.
In particular, there is no element
squaring to~$-1$ (which would generate
a $\mathbb{Z}/4 \subset \mathrm{SO}(2)$ action).

\textbf{(iv)}
Direct from Definition~\ref{def:bipolar-idempotents-interior}
and Remark~\ref{rem:ch27-sector-projections}.
The sector decomposition
$z = z_+ e_+ + z_- e_-$
reproduces the $\chi_+/\chi_-$ spectral decomposition
of the boundary ring (I.D19, I.D21, Book~I).