{"id":"hexagon/2610.00309","pdf":"https://hexagonmath.org/pdf/2610.00309v1","url":"https://hexagonmath.org/abs/2610.00309v1","tags":["msc:60K35","msc:92E20","msc:05C80","msc:03D78","msc:68V20","autocatalytic sets","computable analysis","critical window","finite-size effects","low-intensity asymptotics"],"score":0,"title":"Quantitative analysis of critical-window profiles for autocatalytic emergence","domain":"math.PR","origin":"ai-generated","source":"hexagon","authors":["Michael Mislan"],"reviews":0,"subject":"math.PR","summary":"The critical-window law for reflexively autocatalytic and food-generated (RAF) sets in the reversible binary-polymer model states that, when the mean number of reactions catalysed per molecule grows like $\\lambda n$, the RAF probability converges to $S_\\nu(1-\\mathrm{e}^{-\\lambda})$, where $S_\\nu$ is the probability that ordinary reversible closure of the food set is unbounded in an infinite field of independently open reaction channels. The law is known for the ordered split-position catalogue ($\\nu=\\mathrm{sp}$) and for the commuting-factor quotient catalogue ($\\nu=\\mathrm{qt}$). It is an…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-17","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":true,"formalizations":["https://palomar-registry.org/entry?id=PALOMAR-2026-10-04-000007&version=1"],"human_understanding":"some"}