ω-Coupling Sector

The fifth generator ω does not produce an independent sector. Instead, it mediates coupling: characters with non-trivial projection in both B and C lobes simultaneously. The structural reason for 4+1, not 5.

ω-Coupling Sector

Summary

The fifth generator ω does not produce an independent sector. Instead, it mediates coupling: characters with non-trivial projection in both B and C lobes simultaneously. The structural reason for 4+1, not 5.

Statement

%
\label{def:omega-coupling-sector}
The \emph{$\omega$-coupling sector} is
\begin{equation}\label{eq:ch10-omega-sector}
    \Sector{\omega} \;=\;
    \bigl\{\,
        \chi \in \Char(\Lemniscate)
        \;\big|\;
        P_B(\chi) \neq 0
        \;\text{and}\;
        P_C(\chi) \neq 0
    \,\bigr\}.
\end{equation}
It consists of all characters
whose projections onto both
the $B$-lobe ($\gamma$-stratum) and
the $C$-lobe ($\eta$-stratum)
are simultaneously non-trivial.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-03.jsonl line 27
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part02/ch10-the-four-plus-one-decomposition.tex lines 363-382

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Sectors.Decomposition
• Name: omega_coupling_5

Dependencies

• Canonical: III.D11

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.