Lemniscate Characters

Characters chi: L -> Z_hat_tau[j] as ring homomorphisms from the bipolar spectral algebra to split-complex scalars. Fundamental characters chi_+ and chi_- project onto B-sector and C-sector.

Lemniscate Characters

Summary

Characters chi: L -> Z_hat_tau[j] as ring homomorphisms from the bipolar spectral algebra to split-complex scalars. Fundamental characters chi_+ and chi_- project onto B-sector and C-sector.

Statement

%
\label{def:lemniscate-characters}
A \textbf{lemniscate character} is a ring homomorphism
\[
    \boxed{%
    \chi : H_\tau \to \hat{\mathbb{Z}}_\tau[j]}
\]
that satisfies:
\begin{enumerate}
    \item \textbf{Additivity.}
          $\chi(x + y) = \chi(x) + \chi(y)$
          for all $x, y \in H_\tau$.
    \item \textbf{Multiplicativity.}
          $\chi(x \cdot y) = \chi(x) \cdot \chi(y)$
          for all $x, y \in H_\tau$.
    \item \textbf{Unitality.}
          $\chi(1) = 1$.
\end{enumerate}
The set of all lemniscate characters is denoted
$\mathrm{Char}(\mathbb{L})$.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-01.jsonl line 87
• Manuscript source: 2nd-edition/book-i-categorical-foundations/02_mainmatter/part11/ch43-lemniscate-characters.tex lines 98-119

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookI.Boundary.Characters
• Name: Tau.Boundary.chi_plus

Dependencies

• Canonical: I.D18, I.D27, I.D20

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.