{"id":"palomar/PALOMAR-2026-09-02-000001","url":"https://github.com/yuma-mizuno/markoff-modp-generalized/tree/01ed89dd540d942ab1df6a25d88e64b6b951cab1","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-09-02-000001-v2.json","repo":"https://github.com/yuma-mizuno/markoff-modp-generalized","tags":["math.NT","math.DS","msc:11D25","msc:11T06","msc:37P25","number-theory","markoff-surfaces","lean","strong-approximation"],"score":0,"title":"Generalized Markoff strong approximation","trust":"high","domain":"math.NT","source":"palomar","authors":["Yuma Mizuno"],"reviews":0,"subject":"math.NT","summary":"A Lean 4 formalization of the generic branch of eventual strong approximation for generalized Markoff surfaces, specialized to a fixed ordered integral coefficient triple. It proves punctured transitivity for both the full Vieta group and the two-Vieta rotation group, supplies an explicit coefficient-dependent cutoff, and derives surjectivity of coordinatewise reduction onto the full finite-field surface.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-12","reactions":{},"formalized":true,"formalizes":[{"title":"Strong Approximation for the Relative Character Variety of the Four-Times Punctured Sphere","identifier":"arXiv:2603.04096v3","relationship":"adapts"},{"title":"Divisibility by p for Markoff-like Surfaces","identifier":"arXiv:2509.02187v3","relationship":"adapts"}]}