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Fluctuations are not necessarily a threat to stability. Stability does not mean that fluctuations do not exist. Stability means that initial small fluctuation remain small. Indeed, if you write down a periodic wave function for crystal lattice (which can be found in all solid state textbooks) you will see that position uncertainties of atoms are much smaller than the size of the crystal as a whole. This means that quantum fluctuations are small, which corresponds to stability.RockyMarciano said:Are you referring only to classical theory? Because this doesn't seem to be a valid assertion in the quantum realm, at least if we go by its theoretical principles. A solid meter is most likely made up of atoms joined by chemical bonds that act as springs with a ground state energy that fluctuates, the corresponding uncertainty in the length of the spring makes the separation between atoms at each step not well defined so that they shouldn't add up to a fixed and stable expected distance between marks on the meter and therefore it can't justify a robust measure remaining stable independently of how and when it is used as a measuring tool.
Of course in practice these shortcomings are overcome by obtaining a measurement that gives a defined distance that allows to introduce an idealized meter and the atomic fluctuations only produce a minor blurring for the position of each atom(for instance in x-ray scattering). You would have to show from first principles how the meter is stable taking into account the ground state energy fluctuations.