Divisor-Filtered Farey Chambers and Localized Discrepancy

For each integer n >= 1, the elementary chamber grid
a(n,k) = n - 1 + k/n, for 1 <= k <= n,
becomes, after translation to the unit interval and reduction to lowest terms, a divisor-filtered family of reduced fractions. A reduced denominator q appears in chamber n if and only if q divides n, and the number of entries of reduced denominator q is Euler's totient phi(q).

This note studies the discrepancy of such reduced fractions when the allowed denominators are restricted to a divisor window D ⊆ Div(n). For 0 <= x <= 1, define

A_{n,D}(x) = sum_{q in D} #{1 <= a <= qx : gcd(a,q)=1}

and

Phi_D = sum_{q in D} phi(q),

with discrepancy

Delta_{n,D}(x) = A_{n,D}(x) - x Phi_D.

We derive a Möbius-inversion formula for Delta_{n,D}, prove the uniform bound

sup_{0 <= x <= 1} |Delta_{n,D}(x)| <= sum_{q in D} 2^{omega(q)},

and identify several instructive special cases. The full divisor window D = Div(n) collapses exactly to the regular grid, giving

A_{n,Div(n)}(x) = floor(nx).

The exact shell D = {n} recovers the classical reduced-residue discrepancy, while a prime-power shell D = {p^a} has the sharper exact supremum 1 - 1/p.

These results are finite, elementary, and local. They are not global Farey-discrepancy estimates and carry no implication for the Riemann hypothesis.