1/f frequency noise in mechanical resonators scales inversely with volume, not with quality factor

Abstract: Mechanical frequency-shift sensors have never reached their fundamental thermomechanical or quantum limits. Every material, size, and transduction scheme yet examined yields a $1/f$ floor 1-3 orders higher, with a flicker-noise coefficient $h_{-1} \propto 1/V_{\varepsilon}$. Surveying the published record, I find the invariant $$h_{-1}\,V_{\varepsilon} \equiv σ<em>A ^2\, V</em>{\varepsilon}/(2\ln 2)$$ independent of device size across 13 decades; volume-independent noise is excluded by 9$σ$, site-averaging by 3.8$σ$. No $Q$ dependence appears: reported $Q<sup>{-n}$ laws scale equivalently when $Q$ and volume co-vary; only volume scaling survives when they do not. Mass resolution follows as $$|δm| \propto \sqrt{h_{-1}\,V_{\varepsilon} \cdot V}$$.