{"id":"palomar/PALOMAR-2026-09-01-000008","url":"https://github.com/stalex444/only-two-morphic-numbers/tree/9d3664f4ad3abc3783a9296ad1207a1ef1fee41d","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-09-01-000008-v1.json","repo":"https://github.com/stalex444/only-two-morphic-numbers","tags":["math.NT","math.HO","msc:11R09","msc:11C08","msc:00A67","number-theory","morphic-numbers","lean","golden-ratio"],"score":0,"title":"Only two morphic numbers: the Aarts-Fokkink-Kruijtzer classification of the golden ratio and the plastic number","trust":"high","domain":"math.NT","source":"palomar","authors":["Stephanie Alexander"],"reviews":0,"subject":"math.NT","summary":"Kernel-checked classification of the morphic numbers — the reals p > 1 satisfying both of the golden ratio's defining scale closures: p + 1 = p^k (the sum of consecutive measures of the geometric system is again a measure) and p - 1 = p^(-l) (so is their difference; rendered zpow-free as p^l * (p - 1) = 1). These two equations are the arithmetic of self-similar proportion: a number satisfying both generates a measure system closed under the paper's superposition and juxtaposition, the property pair behind the divine proportion's standing as the plane's design constant. The theorem…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-09-01","reactions":{},"formalized":true,"formalizes":[{"title":"Morphic numbers","identifier":"Nieuw Archief voor Wiskunde (5) 2 (2001), 56-58","relationship":"formalizes"}]}