Möbius Structure and Composite Variance in the Primitive Active-Zone Error

Hardy conjectured that the classical Gauss circle error satises E(x) = O(x1/4+ε) for every ε>0. This paper analyses a related quantity: the uctuation ER(x) = N(x)−x/(2π) in the primitive coprime count N(x) at a single scale. We show that ER admits an exact Möbius decomposition into 2ω(s) fractional-part terms, where ω(s) is the number of distinct prime factors of s= m+ n. For prime s, this reduces to two terms with constant variance. For composite s, we identify the per-s variance formula Var(e|s) = A(s)/12*φ(s)/s+Var(T)/4 , A(s) = 2^ω(s) +1−Prod p|s (1 +1/p^2) , empirically, with no free parameters; a rigorous derivation is open. By the HardyRamanujan theorem (ω(s)∼log log s), this gives σc ∼(ln s)^(ln 2)/2, conrmed numerically to <1% error from x= 10^8 to 10^17. The growth of σc is consistent with an ε-type correction within ER at a single scale, conditional on (i) the empirical formula above and (ii) pairwise decorrelation across s (not proved in this series). No implication for the classical Gauss circle conjecture E(x) = O(x1/4+ε) is established here: the Möbius shell expansion linking ER to E transfers quarter-power bounds only when ε>1/4, and the regime ε→0+ of Hardy's conjecture lies outside this range. The present paper makes no claim toward Hardy's ε.