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A finite volume symplectic manifold is said to have "packing stability" ifthe only obstruction to symplectically embedding sufficiently small balls isthe volume obstruction. Packing stability has been shown in a variety of casesand it has been conjectured that it always holds. We give counterexamples tothis conjecture; in fact, we give examples that cannot be fully packed by anydomain with smooth boundary nor by any convex domain. The examples aresymplectomorphic to open and bounded domains in $\mathbb{R}^4$, with thediffeomorphism type of a disc.  The obstruction to packing stability is closely tied to another old question,which asks to what extent an open symplectic manifold has a well-definedboundary; it follows from our results that many examples cannot besymplectomorphic to the interior of a compact symplectic manifold with smoothboundary. Our results can be quantified in terms of the volume decay near theboundary, and we produce, for example, smooth toric domains that are onlysymplectomorphic to the interior of a compact domain if the boundary of thisdomain has inner Minkowski dimension arbitrarily close to $4$.  The growth rate of the subleading asymptotics of the ECH spectrum plays a keyrole in our arguments. We prove a very general "fractal Weyl law", relatingthis growth rate to the Minkowski dimension; this formula is potentially ofindependent interest.

Boundaries of open symplectic manifolds and the failure of packing stability