The Zeta-Exponent Chamber Lift: Dimensional Visibility, Jordan Shadows, and Valuation Mirrors

This note extends the primitive-ray chamber from the plane to arbitrary dimension.  A nonzero vector
$\mathbf x=(x_1,\ldots,x_d)∈ Z^d$ has a unique radial decomposition
$\mathbf x=g\mathbf q$, where $g=\gcd(|x_1|,\ldots,|x_d|)$ and $\mathbf q$ is primitive.  For $d\ge2$, the natural density of primitive vectors is
\[
  \prod_p(1-p^{-d})=\frac1{\zeta(d)},
\]
and the exact gcd layers obey the probability law
\[
  P(\gcd(x_1,\ldots,x_d)=G)=\frac{1}{\zeta(d)G^d}.
\]
We compare this infinite visibility density with its finite-modulus analogue, the Jordan totient
\[
  J_d(n)=n^d\prod_{p\mid n}(1-p^{-d}),
\]
whose normalized form is the finite divisor sum
\[
  \frac{J_d(n)}{n^d}=\sum_{r\mid n}\frac{\mu(r)}{r^d}.
\]
This single identity separates three ledgers: finite divisor support, infinite all-divisor support, and averaged divisor-frequency support.  Along primorial moduli the finite product descends to $1/\zeta(d)$, while ordinary averaging over all moduli introduces an additional divisor-frequency factor $1/r$ and gives
\[
  \lim_{x\to\infty}\frac1x\sum_{n\le x}\frac{J_d(n)}{n^d}=\frac1{\zeta(d+1)}.
\]
Finally, $k$-free integers provide a valuation mirror: the local obstruction $p^k\mid n$ has density $p^{-k}$, so the $k$-free density is $1/\zeta(k)$.  The paper is an expository atlas of classical density facts.  No new theorem about prime distribution is claimed.