σ-Fixed Character

A boundary character χ is σ-fixed if σ(χ) = χ under complex conjugation in the split-complex codomain. σ-fixed characters form a sublattice of ℤ². In spectral trichotomy, σ-fixed = balanced B/C content.

σ-Fixed Character

Summary

A boundary character χ is σ-fixed if σ(χ) = χ under complex conjugation in the split-complex codomain. σ-fixed characters form a sublattice of ℤ². In spectral trichotomy, σ-fixed = balanced B/C content.

Statement

\label{def:sigma-fixed-character}
A boundary character $\chi = (m, n) \in \operatorname{Char}(\mathbb{L}) \cong \mathbb{Z}^2$ is \emph{$\sigma$-fixed} if $\sigma(\chi) = \chi$, i.e., if $(n, m) = (m, n)$. Equivalently, $\chi$ is $\sigma$-fixed if and only if $m = n$. The $\sigma$-fixed characters form the \emph{diagonal sublattice}
\begin{equation}
\operatorname{Char}(\mathbb{L})^\sigma \;=\; \bigl\{\, (n, n) \;\bigm|\; n \in \mathbb{Z} \,\bigr\} \;\cong\; \mathbb{Z} \;\subset\; \mathbb{Z}^2.
\label{eq:ch41-diagonal-sublattice}
\end{equation}

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-03.jsonl line 110
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch41-sigma-fixed-characters-and-sector-addressability.tex lines 69-77

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Physics.Hodge
• Name: sigma_fixed_check

Dependencies

• Canonical: III.T14, III.D13, III.D23

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.