{"id":"hexagon/2610.00171","pdf":"https://hexagonmath.org/pdf/2610.00171v1","url":"https://hexagonmath.org/abs/2610.00171v1","tags":["math.MG","msc:52C10","msc:05C15","msc:05C72","Spectral Rigidity","Unit-distance graphs","independence ratio","invariant random independent sets","measurable sets"],"score":0,"title":"Measurable Independence Density Equals the Finite Independence-Ratio Infimum in the Euclidean Plane","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Ákos Dúcz"],"reviews":0,"subject":"math.CO","summary":"Let m1(R^2) be the supremum of upper densities of Lebesgue-measurable subsets of the plane containing no pair of points at distance one. We show, using the spectral rigidity theorem in OpenAI's recent proof that the plane is not five-colorable, that m1(R^2) = inf_G alpha(G)/|V(G)|, where G ranges over nonempty finite unit-distance graphs in the plane. The argument constructs an isometry-invariant law on independent subsets of the countable algebraic plane, projects the occupancy indicator onto the continuous spectral factor without changing its expectation, and extracts a measurable…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-09","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}