Stabilized ω-Germ

An ω-germ f is stabilized at primorial depth k₀ if Δ(f, k) = 0 for all k ≥ k₀. Equivalently, f is determined by its value at depth k₀ and extends uniquely to all deeper levels.

Stabilized ω-Germ

Summary

An ω-germ f is stabilized at primorial depth k₀ if Δ(f, k) = 0 for all k ≥ k₀. Equivalently, f is determined by its value at depth k₀ and extends uniquely to all deeper levels.

Statement

\label{def:stabilized-omega-germ}
Let $f$ be $\tau$-admissible fluid data on a clopen cylinder domain $U \subset \tau^3$ (Definition~\ref{def:tau-admissible-fluid-data}, Ch.~34). The $\omega$-germ of $H_{\mathrm{flow}}(f)$ at a point $x \in U$ is \emph{stabilized} if there exists a primorial depth $N_x \in \mathbb{N}$ such that for all $n \geq N_x$, the restriction
\begin{equation}
H_{\mathrm{flow}}(f)\big|_{\operatorname{Prim}(n)}(x) \;=\; H_{\mathrm{flow}}(f)\big|_{\operatorname{Prim}(N_x)}(x).
\label{eq:ch37-stabilization}
\end{equation}
We write $\omega\text{-}\mathrm{germ}(f, x)$ for the stabilized value, which is well-defined once the tower becomes eventually constant at~$x$.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-03.jsonl line 98
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch37-positive-regularity.tex lines 39-48

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Physics.PositiveRegularity
• Name: stabilized_germ_check

Dependencies

• Canonical: III.D39, III.D40

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.