{"id":"palomar/PALOMAR-2026-09-01-000012","url":"https://github.com/stalex444/golden-family-boundary/tree/fc43a62561b4d7846ca25273229a4f964dff7ae5","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-09-01-000012-v1.json","repo":"https://github.com/stalex444/golden-family-boundary","tags":["math.NT","msc:11R06","msc:11R09","msc:11C08","number-theory","trinomials","lean","pisot-numbers"],"score":0,"title":"The golden family's boundary at n = 3: general-n theorems for x^n = x + 1","trust":"high","domain":"math.NT","source":"palomar","authors":["Stephanie Alexander"],"reviews":0,"subject":"math.NT","summary":"Kernel-checked general-n theorems for the golden family — the ladder of trinomials x^n - x - 1, whose unique real root r_n > 1 descends from the golden ratio (n = 2) through the plastic number (n = 3) toward 1. The family's structural boundary falls at n = 3; this entry proves the morphic half unconditionally at general n and holds the Pisot half as the measure⟺Pisot equivalence, the unconditional n = 4 non-Pisot statement, and the Siegel-conditional strict excess for n >= 4, from the family's own arithmetic, with no per-degree case check above the boundary. Eleven compared statements carry…","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"}]}