\label{prop:earned-admissibility}
The following hold:
\begin{enumerate}
\item \textbf{Characters.}
The boundary characters $\chi_+ = \tfrac{1}{2}(1 + \jj)$ and
$\chi_- = \tfrac{1}{2}(1 - \jj)$ are $\tau$-admissible with
$W(\chi_\pm) = 1$.
\item \textbf{Identity.}
The identity function $\id$ is $\tau$-admissible with $W(\id) = 0$.
\item \textbf{Successor.}
The successor map $\rho(x) = x + 1$ is $\tau$-admissible with
$W(\rho) = 1$.
\item \textbf{Composition.}
If $f$ and $g$ are $\tau$-admissible, then $g \circ f$ is $\tau$-admissible with
\begin{equation}\label{eq:ch56-composition-width}
W(g \circ f) \;\leq\; W(f) + W(g).
\end{equation}
\item \textbf{CRT product.}
If $f_1, \ldots, f_k$ are $\tau$-admissible with widths $W(f_i) = w_i$,
then the CRT product $f_1 \times \cdots \times f_k$
(acting componentwise on the CRT decomposition) is $\tau$-admissible with
\begin{equation}\label{eq:ch56-crt-product-width}
W(f_1 \times \cdots \times f_k) \;=\; \max_{1 \leq i \leq k} w_i.
\end{equation}
\end{enumerate}
Consequently, the $\tau$-admissible functions form a compositionally closed class.