Omega-Germ Transformer

An omega-germ transformer maps omega-tails to sector-pair values. Carries both a SectorFun (D-holomorphic structure) and a StageFun (stagewise evaluation for tower coherence).

Omega-Germ Transformer

Summary

An omega-germ transformer maps omega-tails to sector-pair values. Carries both a SectorFun (D-holomorphic structure) and a StageFun (stagewise evaluation for tower coherence).

Statement

%
\label{def:omega-germ-transformer}
An \textbf{$\omega$-germ transformer} is a function
\[
    \boxed{%
    T : \mathrm{OmegaTail} \to \hat{\mathbb{Z}}_\tau[j]}
\]
that takes a compatible tower
$(x_k)_{k \geq 1}$
(Definition~\ref{def:omega-tail}, I.D25)
--- where $x_k \in \mathbb{Z}/M_k\mathbb{Z}$
and $x_\ell \equiv x_k \pmod{M_k}$ for all $k \leq \ell$ ---
and produces a split-complex value
$T\bigl((x_k)\bigr) \in \hat{\mathbb{Z}}_\tau[j]$.

At each finite truncation depth $d$,
$T$ restricts to a function
\[
    T_d : \mathrm{OmegaTail}_d \to \mathbb{Z}/M_d\mathbb{Z}[j],
\]
where $\mathrm{OmegaTail}_d$ denotes the set of
depth-$d$ compatible towers
$(x_1, x_2, \ldots, x_d)$
and $\mathbb{Z}/M_d\mathbb{Z}[j]$ is the
split-complex extension of $\mathbb{Z}/M_d\mathbb{Z}$.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-01.jsonl line 110
• Manuscript source: 2nd-edition/book-i-categorical-foundations/02_mainmatter/part13/ch50-tau-holomorphic.tex lines 115-141

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookI.Holomorphy.TauHolomorphic
• Name: Tau.Holomorphy.GermTransformer

Dependencies

• Canonical: I.D25, I.D42

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.