Label-Idempotent Compatibility

B-type primes correspond to e₊-dominant spectral coefficients, C-type to e₋-dominant, X-type balanced. Label_n is compatible with the split-complex idempotent decomposition of H_τ.

Label-Idempotent Compatibility

Summary

B-type primes correspond to e₊-dominant spectral coefficients, C-type to e₋-dominant, X-type balanced. Label_n is compatible with the split-complex idempotent decomposition of H_τ.

Statement

%
\label{prop:label-idem-compatibility}
For every odd prime~$p_i$
and every depth $n \geq i$:
\begin{enumerate}
    \item[\textup{(i)}]
          $\mathrm{Label}_n(p_i) = B$
          $\;\Longleftrightarrow\;$
          the spectral coefficient $c_+(e_i)$
          is a quadratic residue modulo~$p_i$
          and $c_-(e_i)$ is a quadratic non-residue.
    \item[\textup{(ii)}]
          $\mathrm{Label}_n(p_i) = C$
          $\;\Longleftrightarrow\;$
          $c_-(e_i)$ is a quadratic residue
          and $c_+(e_i)$ is a quadratic non-residue.
    \item[\textup{(iii)}]
          $\mathrm{Label}_n(p_i) = X$
          $\;\Longleftrightarrow\;$
          $c_+(e_i)$ and $c_-(e_i)$
          have the same quadratic character modulo~$p_i$.
\end{enumerate}
For $p_i = 2$,
the spectral coefficients coincide modulo~$2$,
and the label is~$X$.

Proof / Justification

No immediate manuscript proof block was extracted in this pilot run.

Source Context

• Registry source: book-03.jsonl line 52
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part03/ch18-the-internal-bipolar-classifier.tex lines 571-597

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Spectral.BipolarClassifier
• Name: label_idem_check

Dependencies

• Canonical: III.D23

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.