Internal Bipolar Classifier

Label_n: computable classifier mapping primes ≤ p_n to {B, C, X}. B-type = exponent/χ₊-dominant, C-type = tetration/χ₋-dominant, X-type = balanced. Replaces informal lobe language with computable predicates.

Internal Bipolar Classifier

Summary

Label_n: computable classifier mapping primes ≤ p_n to {B, C, X}. B-type = exponent/χ₊-dominant, C-type = tetration/χ₋-dominant, X-type = balanced. Replaces informal lobe language with computable predicates.

Statement

%
\label{def:internal-bipolar-classifier}
For each primorial depth~$n \geq 1$,
the \textbf{internal bipolar classifier} is the function
\begin{equation}\label{eq:ch18-label-def}
    \mathrm{Label}_n \;:\;
    \{p_1, \ldots, p_n\}
    \;\longrightarrow\;
    \{B, C, X\},
\end{equation}
defined by the Legendre symbol at the individual prime:
\begin{equation}\label{eq:ch18-label-cases}
    \mathrm{Label}_n(p_i) \;=\;
    \begin{cases}
        B & \text{if } p_i > 2 \text{ and }
            \bigl(\frac{2}{p_i}\bigr) = +1,
            \\[4pt]
        C & \text{if } p_i > 2 \text{ and }
            \bigl(\frac{2}{p_i}\bigr) = -1,
            \\[4pt]
        X & \text{if } p_i = 2.
    \end{cases}
\end{equation}
The label~$B$ is read as
``exponent-stratum dominant'' ($\chi_+$-dominant);
$C$ as ``tetration-stratum dominant'' ($\chi_-$-dominant);
and $X$ as ``balanced'' or ``mixed.''

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-03.jsonl line 50
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part03/ch18-the-internal-bipolar-classifier.tex lines 194-222

Lean / Formalization Notes

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

Dependencies

• Canonical: III.T10

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.