Interior j-Unit

The split-complex unit j: the automorphism of the fiber T^2 exchanging B and C coordinates, with j^2 = +1. Unlike i (rotation), j is an involution (polarity/reflection).

Interior j-Unit

Summary

The split-complex unit j: the automorphism of the fiber T^2 exchanging B and C coordinates, with j^2 = +1. Unlike i (rotation), j is an involution (polarity/reflection).

Statement

%
\label{def:j-unit}
The \textbf{split-complex unit}~$\jj$ is the
automorphism of the fiber $T^2$
that exchanges the B and C coordinates:
\[
    \boxed{%
    \jj \;:\;
    (D, A, B, C)
    \;\longmapsto\;
    (D, A, C, B).}
\]
Equivalently, $\jj$ is the generator
of the $\mathbb{Z}/2$ involution
$\sigma : T^2 \to T^2$
that swaps the two fiber directions.
The defining properties are:
\begin{enumerate}
    \item[\textup{(i)}]
          $\jj^2 = 1$ (the identity),
    \item[\textup{(ii)}]
          $\jj \neq \pm 1$
          (distinct from the identity and negation),
    \item[\textup{(iii)}]
          $\jj$ preserves the D and A coordinates
          and exchanges B and~C.
\end{enumerate}

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-02.jsonl line 72
• Manuscript source: 2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part05/ch27-j-replaces-i.tex lines 138-166

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookII.Transcendentals.JReplacesI
• Name: j_unit

Dependencies

• Canonical: I.D20, I.T10, II.D14

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.