\label{def:tau-admissible-gauge-data}
A \emph{$\tau$-admissible gauge datum} at primorial depth $k$ is a triple $\mathfrak{g}_k = (U_k, \, \phi_k^C, \, \mathfrak{c}_k)$ where:
\begin{enumerate}[(i)]
\item $U_k$ is a finite union of clopen cylinders in $\tau^3$ at level $\operatorname{Prim}(k)$;
\item $\phi_k^C \colon U_k \to \mathbb{Z}/\operatorname{Prim}(k)\mathbb{Z}$ is an $\omega$-germ assignment with spectral support in $\operatorname{Sector}(C)$: the ABCD extraction of $\phi_k^C$ has $A$-, $B$-, $D$-, and $\omega$-components identically zero;
\item $\mathfrak{c}_k$ is the C-sector extraction of $\phi_k^C$, decomposed into idempotent components
\begin{equation}
\phi_k^C \;=\; e_+ \cdot \phi_{k,+}^C \;+\; e_- \cdot \phi_{k,-}^C,
\label{eq:ch38-gauge-idempotent}
\end{equation}
with the $\chi_-$-dominant constraint: $\|\phi_{k,+}^C\|_{\operatorname{Prim}(k)} \le \|\phi_{k,-}^C\|_{\operatorname{Prim}(k)}$;
\item the C-coordinate extraction is \emph{bounded}: $\|\phi_{k,-}^C\|_{\operatorname{Prim}(k)} \le M \cdot \operatorname{Prim}(k)^{1/2}$ for a uniform constant $M$ independent of~$k$.
\end{enumerate}