Sector Independence

Every SectorFun is sector-independent by construction: B-output depends only on B-input, C-output depends only on C-input. The structural content of the split-CR equations.

Sector Independence

Summary

Every SectorFun is sector-independent by construction: B-output depends only on B-input, C-output depends only on C-input. The structural content of the split-CR equations.

Statement

%
\label{prop:sector-independence}
If $f : H_\tau \to H_\tau$ is D-holomorphic,
then in sector coordinates
$f$ decomposes as:
\[
    \boxed{%
    f(u, v) \;=\; \bigl(g(u),\; h(v)\bigr)}
\]
for functions
$g : \hat{\mathbb{Z}}_\tau \to \hat{\mathbb{Z}}_\tau$
and
$h : \hat{\mathbb{Z}}_\tau \to \hat{\mathbb{Z}}_\tau$,
each depending on a single sector variable.

Proof / Justification

By the sector-coordinate form of the split-CR equations
(Remark~\ref{rem:sector-form-cr}):
\begin{itemize}
    \item $\partial F_+ / \partial v = 0$
          implies $F_+$ is independent of $v$,
          so $F_+(u, v) = g(u)$
          for some function $g$ of $u$ alone.
    \item $\partial F_- / \partial u = 0$
          implies $F_-$ is independent of $u$,
          so $F_-(u, v) = h(v)$
          for some function $h$ of $v$ alone.
\end{itemize}
Therefore $f(u, v) = (g(u),\, h(v))$ in sector form.

Source Context

• Registry source: book-01.jsonl line 109
• Manuscript source: 2nd-edition/book-i-categorical-foundations/02_mainmatter/part13/ch49-d-holomorphy.tex lines 410-425

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookI.Holomorphy.DHolomorphic
• Name: Tau.Holomorphy.sector_fun_independent

Dependencies

• Canonical: I.D42

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.