Self-similar Delone sets and Pisot numbers

What the math gives to ML

The paper identifies an algebraic mechanism controlling whether self-similar Delone patterns remain uniformly discrete: under its stated mild hypotheses, the similarity factor is Pisot exactly when uniform discreteness holds. A Pisot number is an algebraic integer greater than one whose other algebraic conjugates all lie strictly inside the unit disk; these contracting conjugate directions prevent arbitrarily close coincidences across scale levels. The transferable neural-network construction is a Pisot-constrained multiscale feature or codebook generator, where hierarchical representations use powers of a learned algebraic scale and are monitored for minimum separation. This is a speculative but falsifiable route to stable multiscale embeddings, quantization, or memory representations rather than a generic optimizer modification.