4+1 Sector Decomposition

Five generators yield four primitive sectors (α→D, π→A, γ→B, η→C) plus one coupling sector (ω). Char(L) = D-sector ⊔ A-sector ⊔ B-sector ⊔ C-sector ⊔ ω-sector. Induced by Langlands₀, not postulated.

4+1 Sector Decomposition

Summary

Five generators yield four primitive sectors (α→D, π→A, γ→B, η→C) plus one coupling sector (ω). Char(L) = D-sector ⊔ A-sector ⊔ B-sector ⊔ C-sector ⊔ ω-sector. Induced by Langlands₀, not postulated.

Statement

%
\label{def:four-plus-one-decomposition}
The \emph{$4{+}1$ sector decomposition}
of the boundary character space is the partition
\[
    \Char(\Lemniscate) \setminus \{0\}
    \;=\;
    \underbrace{
        \Sector{D}
        \;\sqcup\;
        \Sector{A}
        \;\sqcup\;
        \Sector{B}
        \;\sqcup\;
        \Sector{C}
    }_{\text{four primitive sectors}}
    \;\sqcup\;
    \underbrace{
        \Sector{\omega}
    }_{\text{one coupling sector}}\,.
\]
The four primitive sectors are generated
by the four primitive generators
$\alpha$, $\pi$, $\gamma$, $\eta$
via their dominant ABCD coordinates.
The coupling sector is generated
by the crossing generator~$\omega$
via its simultaneous $B$-$C$ mixing.
The count is $4 + 1$, not~$5$:
the $\omega$-sector is structurally distinct
from the primitive sectors.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-03.jsonl line 26
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part02/ch10-the-four-plus-one-decomposition.tex lines 475-507

Lean / Formalization Notes

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

Dependencies

• Canonical: III.T05, III.D11, III.D14

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.