  We explain how fluctuations of ratios can constrain and falsify the
statistical model of particle production in heavy ion collisions, using $K/\pi$
fluctuations as an example. We define an observable capable of determining
which statistical model, if any, governs freeze-out in ultrarelativistic heavy
ion collisions. We calculate this observable for $K/\pi$ fluctuations, and show
that it should be the same for RHIC and LHC energies, as well as independent of
centrality, if the Grand-Canonical statistical model is an appropriate
description and chemical equilibrium applies. We describe variations of this
scaling for deviations from this scenario, such as light quark chemical
non-equilibrium, strange quark over-saturation and local conservation
(canonical ensemble) for strange quarks. We also introduce a similar observable
capable, together with the published $K^*/K$ measurement, of ascertaining if an
interacting hadron gas phase governs the system between thermal and chemical
freeze-out, and of ascertaining its duration and impact on hadronic chemistry.
