Spectral Polarity

For each primitive sector S_g, the spectral polarity pol(S_g) = ‖P_{χ₊}(S_g)‖/‖P_{χ₋}(S_g)‖. D: χ₊-dominant, B: χ₊-dominant, C: χ₋-dominant, A: BALANCED (pol = 1). Only the A-sector has equal polarity.

Spectral Polarity

Summary

For each primitive sector S_g, the spectral polarity pol(S_g) = ‖P_{χ₊}(S_g)‖/‖P_{χ₋}(S_g)‖. D: χ₊-dominant, B: χ₊-dominant, C: χ₋-dominant, A: BALANCED (pol = 1). Only the A-sector has equal polarity.

Statement

%
\label{def:spectral-polarity}
The \textbf{spectral polarity} of a primitive sector~$\Sector{g}$
is the ratio
\begin{equation}\label{eq:ch12-polarity-def}
    \mathrm{pol}(\Sector{g})
    \;=\;
    \frac{\lVert P_{\chi_+}(\Sector{g}) \rVert}%
         {\lVert P_{\chi_-}(\Sector{g}) \rVert},
\end{equation}
where $\lVert \cdot \rVert$ denotes the rank
of the projected module at each finite depth~$d$.
A sector is called
\begin{itemize}
    \item \textbf{$\chi_+$-dominant} if $\mathrm{pol}(\Sector{g}) > 1$,
    \item \textbf{$\chi_-$-dominant} if $\mathrm{pol}(\Sector{g}) < 1$,
    \item \textbf{balanced} if $\mathrm{pol}(\Sector{g}) = 1$.
\end{itemize}
The polarity is well defined because
the spectral decomposition is canonical
(Book~I, I.T12)
and the sector projections are depth-independent
(Chapter~\ref{ch:four-plus-one-decomposition}).

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-03.jsonl line 32
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part02/ch12-the-parity-bridge-theorem.tex lines 86-110

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Sectors.LanglandsReflection
• Name: spectral_polarity

Dependencies

• Canonical: III.D13

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.