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\label{prop:ultrametric-replaces-card}
The Cauchy completeness of $\mathbb{R}_\tau$
and the profinite completeness of $\hat{\mathbb{Z}}_\tau$
both derive from the same source:
the \textbf{ultrametric structure on omega-germs}.
Two completions ---
one Archimedean ($\mathbb{R}_\tau$),
one non-Archimedean ($\hat{\mathbb{Z}}_\tau$) ---
arise from the single convergence mechanism
of the primorial ladder.
Cardinality is unnecessary; convergence suffices.

The primorial ladder
$M_1 \mid M_2 \mid M_3 \mid \cdots$
defines a filtration on the integers:
\[
    \mathbb{Z} \twoheadrightarrow \cdots
    \twoheadrightarrow \mathbb{Z}/M_3
    \twoheadrightarrow \mathbb{Z}/M_2
    \twoheadrightarrow \mathbb{Z}/M_1.
\]
The inverse limit of this system
is $\hat{\mathbb{Z}}_\tau$, which is profinitely complete
by construction.
The CRT decomposition
$\hat{\mathbb{Z}}_\tau \cong \prod_p \mathbb{Z}_p$
(Chapter~\ref{ch:profinite-boundary-ring})
endows each factor $\mathbb{Z}_p$ with the $p$-adic ultrametric:
\[
    d_p(x, y) = p^{-v_p(x - y)},
\]
where $v_p$ is the $p$-adic valuation.
This is a non-Archimedean metric.

On the Archimedean side,
the constructive reals $\mathbb{R}_\tau$
are the Cauchy completion of
$\mathbb{Q}_\tau = \mathrm{Frac}(\tau\text{-Idx})$
under the standard absolute value.
The completeness of $\mathbb{R}_\tau$
follows from the explicit-modulus requirement:
every Cauchy sequence with computable modulus
determines a unique equivalence class.

Both completions arise from the same data ---
the arithmetic of $\tau$-Idx and its primorial filtration.
The Archimedean completion uses the ordinary metric;
the non-Archimedean completion uses the profinite metric.
No cardinality argument is invoked in either case.