Divisor-Band Bijections and Totient Shells in the Triangular-Fractional Grid

We define the triangular-fractional grid
\[
    a(n,k)=\frac{n^2-n+k}{n}=n-1+\frac{k}{n},
    \qquad n\geq 1,\quad 1\leq k\leq n,
\]
and study the reduced rational entries generated by this elementary chamber structure.  The main result is an exact divisor-band bijection: a positive rational $r/s$ in lowest terms occurs in chamber $n$ if and only if
\[
    s\mid n,\qquad s(n-1)<r\leq sn,\qquad \gcd(r,s)=1,
\]
and the step is then uniquely $k=n\bigl(r-s(n-1)\bigr)/s$.  This yields a chamber-level decomposition: for each divisor $s\mid n$, exactly $\varphi(s)$ entries in chamber $n$ reduce to denominator $s$, giving a geometric realization of Gauss's identity $\sum_{s\mid n}\varphi(s)=n$.  A distinguished floor shell, obtained by setting the reduced denominator equal to the chamber index, contains exactly $\varphi(m)$ entries in shell $m$; consequently its cumulative square-normalized average satisfies
\[
    \frac{1}{M^2}\sum_{m\leq M}\varphi(m)\longrightarrow\frac{3}{\pi^2}=\frac{1}{2\zeta(2)}
\]
by the classical summatory totient theorem.  We give four mutually reinforcing views of this structure --- arithmetic (residue bands), combinatorial (the chamber decomposition), geometric (a multiplication grid and a square-gap ``collar'' identity $n^2-a(n,n-1)^2=2-1/n^2$), and asymptotic (within-chamber equidistribution and the floor-shell limit) --- and we record a large computational verification ledger, including guaranteed-invalid negative controls.  A short methodological calibration using the Gram-coordinate projection of Riemann zeta zeros is included; it reproduces known normalized-zero behavior and is explicitly \emph{not} presented as evidence of a new zeta-zero resonance.  The contribution is elementary and structural: a closed-form rational grid whose reduced entries are governed exactly by divisor bands and totient shells.