%
\label{thm:point-set-well-defined}
The set
\[
    \boxed{\tau^3
    \;:=\;
    \tau^3_{\mathrm{fin}}
    \;\cup\;
    \tau^3_{\mathrm{lim}}}
\]
is a well-defined set satisfying:
\begin{enumerate}
    \item[\textup{(1)}] \textbf{Finite part.}
          $\tau^3_{\mathrm{fin}}$ is in bijection
          with $\Obj(\tau)$ via the ABCD chart~$\Phi$
          (Hyperfactorization, I.T04).
    \item[\textup{(2)}] \textbf{Limit part.}
          $\tau^3_{\mathrm{lim}}$ is non-empty:
          the primorial tower produces
          at least one limit point
          (Proposition~\ref{prop:ch04-limit-nonempty}).
    \item[\textup{(3)}] \textbf{Closure under reduction.}
          $\tau^3$ is closed under the CRT reduction maps:
          for every point $x \in \tau^3$ and every stage~$k$,
          the stage-$k$ projection
          $\pi_k(x)$
          is a well-defined $\tau$-admissible point.
\end{enumerate}

\emph{Clause~(1).}
By Proposition~\ref{prop:ch04-bijection},
every $n \in \tau\text{-Idx}_{\geq 2}$
maps to a unique $\tau$-admissible quadruple
$\Phi(n) \in \tau^3_{\mathrm{fin}}$,
and conversely,
every element of $\tau^3_{\mathrm{fin}}$
with $A \geq 2$ determines a unique index.
The two degenerate points are handled
by the convention of
Definition~\ref{def:tau-admissible-point}.
Hence $\tau^3_{\mathrm{fin}} \cong \Obj(\tau)$.

\emph{Clause~(2).}
Proposition~\ref{prop:ch04-limit-nonempty}
exhibited the primorial tower
$\{\Phi(P_k)\}_{k \geq 1}$
as a non-eventually-constant coherent family.

\emph{Clause~(3).}
For a finite point $\Phi(n) \in \tau^3_{\mathrm{fin}}$,
the stage-$k$ projection is
$\Phi(n \bmod P_k)$,
which is a well-defined finite $\tau$-admissible point
(the normal-form decomposition of $n \bmod P_k$
satisfies constraints (C1)--(C5)
since $n \bmod P_k \geq 0$ is a valid index).
For a limit point
$\{(A_k, B_k, C_k, D_k)\}_{k \geq 1}
\in \tau^3_{\mathrm{lim}}$,
the stage-$k$ projection is
$(A_k, B_k, C_k, D_k)$ by definition,
which is $\tau$-admissible by the coherence condition.
The tower compatibility
$\pi_{k,l} \circ \pi_{l,m} = \pi_{k,m}$
(CRT coherence, I.T18)
ensures that the projections are self-consistent.