Tower Coherence

Tower coherence: reduce(f(n, l), k) = f(n, k) for k <= l. A naturality condition on the primorial inverse system: reducing the output first, or reducing the input first and then applying, gives the same result.

Tower Coherence

Summary

Tower coherence: reduce(f(n, l), k) = f(n, k) for k <= l. A naturality condition on the primorial inverse system: reducing the output first, or reducing the input first and then applying, gives the same result.

Statement

%
\label{def:tower-coherence}
An $\omega$-germ transformer $T$ is \textbf{tower-coherent}
if for all $k \leq \ell$ and all
depth-$\ell$ omega-tails $t \in \mathrm{OmegaTail}_\ell$:
\[
    \boxed{%
    \pi_{\ell \to k}\bigl(T_\ell(t)\bigr)
    \;=\;
    T_k\bigl(\pi_{\ell \to k}(t)\bigr),}
\]
where $\pi_{\ell \to k}$ denotes the primorial reduction map
from stage $\ell$ to stage $k$
(reduction modulo $M_k$),
acting on both the source omega-tail
and the target split-complex value.

Equivalently: \emph{reducing the output first,
or reducing the input first and then applying $T$,
gives the same result.}

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-01.jsonl line 111
• Manuscript source: 2nd-edition/book-i-categorical-foundations/02_mainmatter/part13/ch50-tau-holomorphic.tex lines 191-212

Lean / Formalization Notes

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

Dependencies

• Canonical: I.D45, I.D29

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.