The Arithmetic of Time: the slowest settling clock on at most three letters is the Padovan substitution, unique up to mirror image, and both minimal rules are chiral
Five compared theorems. Among all substitutions on at most three letters whose incidence matrix has a Pisot eigenvalue θ > 1, the least such θ is the plastic number ρ, the real root of x³ = x + 1, attained only by the Padovan rule a ↦ b, b ↦ c, c ↦ ab and its mirror c ↦ ba, up to renaming the letters; both minimal rules are chiral, in opposite directions. Precisely: (1) for a substitution σ on k ≤ 3 letters (images may be empty) and a real eigenvalue θ > 1 of its integer incidence matrix that is a Pisot number, ρ ≤ θ, and θ = ρ exactly when, after renaming the letters, σ is the Padovan rule…
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