Bipolar Spectral Algebra

H_tau = A_tau^(B) x A_tau^(C) with bipolar idempotents e_pm = (1 pm j)/2. Canonical decomposition of every element into B-sector and C-sector. Zero divisors managed by diagonal-free discipline.

Bipolar Spectral Algebra

Summary

H_tau = A_tau^(B) x A_tau^(C) with bipolar idempotents e_pm = (1 pm j)/2. Canonical decomposition of every element into B-sector and C-sector. Zero divisors managed by diagonal-free discipline.

Statement

%
\label{def:bipolar-spectral-algebra}
The \textbf{bipolar spectral algebra} is the split-complex
extension of the boundary local ring:
\[
    H_\tau
    \;:=\;
    \hat{\mathbb{Z}}_\tau[j]
    \;=\;
    \hat{\mathbb{Z}}_\tau \oplus j \cdot \hat{\mathbb{Z}}_\tau,
    \qquad j^2 = +1.
\]
It carries the following canonical structure:
\begin{enumerate}
    \item \textbf{Bipolar idempotents:}
    $e_+ = \frac{1+j}{2}$, \quad
    $e_- = \frac{1-j}{2}$.
    \item \textbf{Idempotent properties:}
    $e_+^2 = e_+$, \;
    $e_-^2 = e_-$, \;
    $e_+ \cdot e_- = 0$, \;
    $e_+ + e_- = 1$.
    \item \textbf{Canonical decomposition:}
    every $z \in H_\tau$ decomposes uniquely as
    $z = e_+ \cdot z_+ + e_- \cdot z_-$
    with $z_\pm \in \hat{\mathbb{Z}}_\tau$.
    \item \textbf{Sector correspondence:}
    the $e_+$-sector corresponds to the B-channel primes;
    the $e_-$-sector corresponds to the C-channel primes.
\end{enumerate}

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-01.jsonl line 67
• Manuscript source: 2nd-edition/book-i-categorical-foundations/02_mainmatter/part07/ch30-bipolar-algebra.tex lines 386-417

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookI.Polarity.BipolarAlgebra
• Name: Tau.Polarity.SplitComplex

Dependencies

• Canonical: I.D25, I.T05, I.T10

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.