Conjecture: Expanding Hilbert spaces in a growing-quasicrystal model of spacetime

Establish whether modeling spacetime as a growing quasicrystal implies that, as the universe grows, there exists a sequence of Hilbert spaces H_n at each growth step n that expands analogously to the expanding Hilbert spaces associated with growing quasicrystals in condensed matter, thereby capturing the set of possible tiling configurations at each step.

Background

The paper proposes that spacetime is a growing quasicrystal and draws an analogy to condensed matter systems where growing quasicrystals are associated with expanding Hilbert spaces. The authors explicitly formulate a conjecture that a similar mechanism operates for the universe, implying a dynamically expanding Hilbert space sequence indexed by growth steps.

They later describe a construction where, at each growth step n, a Hilbert space H_n is spanned by states corresponding to quasicrystalline tiling configurations, suggesting a concrete mathematical structure that could be proven or refuted.