Parity Bridge Theorem

The weak sector (A, π-generator) is the unique sector whose spectral polarity permits the E₁→E₂ transition — the computational bootstrap. Balanced polarity enables the code→execution→code cycle: χ₊ (code) and χ₋ (execution) in equal measure. Computation is native to E₂; the Parity Bridge enables it. Life is computation physically instantiated.

Parity Bridge Theorem

Summary

The weak sector (A, π-generator) is the unique sector whose spectral polarity permits the E₁→E₂ transition — the computational bootstrap. Balanced polarity enables the code→execution→code cycle: χ₊ (code) and χ₋ (execution) in equal measure. Computation is native to E₂; the Parity Bridge enables it. Life is computation physically instantiated.

Statement

%
\label{thm:parity-bridge}
Among the $4{+}1$ sectors of Category~$\tau$ at $\Elayer{1}$,
the A-sector~$\Sector{\pi}$ is the unique sector
whose spectral polarity permits the transition
from $\Elayer{1}$ (physics) to $\Elayer{2}$ (computation).
Specifically:
\begin{enumerate}
    \item[\textup{(i)}]
          The $\Elayer{1} \to \Elayer{2}$ transition
          requires a sector $\Sector{g}$ in which
          the operational closure cycle
          \[
              \text{code}
              \;\xrightarrow{\;\chi_+\to\chi_-\;}\;
              \text{product}
              \;\xrightarrow{\;\chi_-\to\chi_+\;}\;
              \text{code}
          \]
          closes without capacity loss.
    \item[\textup{(ii)}]
          Closure without capacity loss
          holds if and only if
          $\mathrm{pol}(\Sector{g}) = 1$.
    \item[\textup{(iii)}]
          $\mathrm{pol}(\Sector{g}) = 1$
          if and only if $g = \pi$
          (Proposition~\ref{prop:balanced-sector-uniqueness}).
\end{enumerate}
Therefore $\Sector{\pi}$ is the unique carrier
of the $\Elayer{1} \to \Elayer{2}$ transition.

Proof / Justification

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

Source Context

• Registry source: book-03.jsonl line 34
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part02/ch12-the-parity-bridge-theorem.tex lines 329-361

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Sectors.ParityBridge
• Name: parity_bridge_5_3

Dependencies

• Canonical: III.D17, III.P04, III.D08

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.