PolStream

PolStream(omega): infinite binary sequence recording channel dominance along the primorial spine. At each stage k, records whether prime p_k is B-dominant or C-dominant.

PolStream

Summary

PolStream(omega): infinite binary sequence recording channel dominance along the primorial spine. At each stage k, records whether prime p_k is B-dominant or C-dominant.

Statement

%
\label{def:polstream}
Let $p_1 < p_2 < p_3 < \cdots$ be the enumeration
of the internal primes $\mathbb{P}_\tau$.
The \textbf{polarized stream} is the sequence
\[
    \mathrm{PolStream}(\omega)
    \;:=\;
    \bigl(\mathrm{pol}(p_1),\;
    \mathrm{pol}(p_2),\;
    \mathrm{pol}(p_3),\;
    \ldots\bigr)
    \;\in\; \{+, -\}^{\mathbb{N}},
\]
where $\mathrm{pol}(p_k)$ is the polarity of $p_k$
(Definition~\ref{def:polarity-map}).
At each stage $k$, the PolStream records which channel
--- $B$ ($\gamma$, encoded as $+$)
or $C$ ($\eta$, encoded as $-$) ---
dominates the $k$-th prime along the primorial spine.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-01.jsonl line 246
• Manuscript source: 2nd-edition/book-i-categorical-foundations/02_mainmatter/part07/ch29-polarized-omega-germs.tex lines 295-315

Lean / Formalization Notes

• Formalization: planned
• Module: None
• Name: None

Dependencies

• Canonical: I.D26, I.T05

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.