{"id":"palomar/PALOMAR-2026-10-18-000019","url":"https://github.com/stalex444/arithmetic-of-time-slowest-clock/tree/49eafbab4af8899b2a66469697bf74683a88ce87","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-18-000019-v1.json","repo":"https://github.com/stalex444/arithmetic-of-time-slowest-clock","tags":["math.CO","math.NT","math.DS","physics.hist-ph","msc:68R15","msc:11R06","msc:37B10","msc:00A79","substitutions","pisot-numbers"],"score":0,"title":"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","trust":"high","domain":"math.CO","source":"palomar","authors":["Stephanie Alexander"],"reviews":0,"subject":"math.CO","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-18","reactions":{},"formalized":true}