B/C Non-Collapse Theorem

B-sector and C-sector are genuinely distinct: no tower-compatible isomorphism between B-supported and C-supported subrings exists. Growth-rate asymmetry (B=exponent, C=tetration) creates an inescapable coherence obstruction.

B/C Non-Collapse Theorem

Summary

B-sector and C-sector are genuinely distinct: no tower-compatible isomorphism between B-supported and C-supported subrings exists. Growth-rate asymmetry (B=exponent, C=tetration) creates an inescapable coherence obstruction.

Statement

%
\label{thm:bc-non-collapse}
There is no system of ring isomorphisms
\begin{equation}\label{eq:ch19-no-system}
    \varphi_k : e_+ R_k^{\jj}
    \;\xrightarrow{\;\sim\;}
    e_- R_k^{\jj},
    \qquad k = 1, 2, 3, \ldots,
\end{equation}
compatible with the primorial tower:
\begin{equation}\label{eq:ch19-tower-compat}
    \begin{tikzcd}[column sep=3.5em]
        e_+ R_{k+1}^{\jj}
            \ar[r, "\varphi_{k+1}"]
            \ar[d, "\pi_{k+1,k}"']
        & e_- R_{k+1}^{\jj}
            \ar[d, "\pi_{k+1,k}"]
        \\
        e_+ R_k^{\jj}
            \ar[r, "\varphi_k"']
        & e_- R_k^{\jj}
    \end{tikzcd}
\end{equation}
The $B$- and $C$-sectors are isomorphic level-by-level
but \emph{not} as towers.

Proof / Justification

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

Source Context

• Registry source: book-03.jsonl line 55
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part03/ch19-the-spectral-trichotomy.tex lines 366-392

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Spectral.Trichotomy
• Name: bc_non_collapse_check

Dependencies

• Canonical: III.T14, III.D24

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.