For any bounded domain, the behavior of optimal eigenvalues over unions of its scaled copies encodes whether Pólya's conjecture holds, yielding a trichotomy and a packing-density lower bound.
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Optimal unions of scaled copies of domains and P\'olya's conjecture
For any bounded domain, the behavior of optimal eigenvalues over unions of its scaled copies encodes whether Pólya's conjecture holds, yielding a trichotomy and a packing-density lower bound.