The Jordan Chamber Lift: Higher-Dimensional Denominator Shells via the GCD Collapse

The one-dimensional divisor-filtered chamber
\[
        \left\{\frac{k}{n}:1\leq k\leq n\right\}
\]
has reduced denominator
\[
        q=\frac{n}{\gcd(n,k)},
\]
and the number of points with reduced denominator \(q\mid n\) is Euler's totient \(\varphi(q)\). This note records the natural \(d\)-dimensional Cartesian lift. In the chamber
\[
        C_d(n)=
        \left\{
        \left(\frac{k_1}{n},\dots,\frac{k_d}{n}\right):
        1\leq k_r\leq n
        \right\},
\]
define the reduced common denominator by
\[
        q_n(k_1,\dots,k_d)
        =
        \frac{n}{\gcd(n,k_1,\dots,k_d)}.
\]
Then the number of chamber points with reduced common denominator \(q\mid n\) is the Jordan totient \(J_d(q)\), and the shell conservation law is
\[
        \sum_{q\mid n}J_d(q)=n^d.
\]
Thus the original one-dimensional chamber is the case \(J_1=\varphi\). The note also separates four structures that should not be conflated: Cartesian cube shells, simplex or tetrahedral cuts, product or multiplication-grid geometry, and visible-lattice density. In particular,
\[
        \frac{J_d(q)}{q^d}
        =
        \prod_{p\mid q}\left(1-p^{-d}\right)
\]
is a finite coprime-to-\(q\) shell density, whereas
\[
        \frac{1}{\zeta(d)}
        =
        \prod_p\left(1-p^{-d}\right)
\]
is the infinite-prime visible-lattice density. These two quantities are related, but they are not the same object.