{"id":"openai-math/Finite-cylinder-approximation-of-ruled-sets","pdf":"https://github.com/openai/math/blob/main/preprints/Finite-cylinder-approximation-of-ruled-sets-September-27-2026/main.pdf","url":"https://github.com/openai/math/tree/main/preprints/Finite-cylinder-approximation-of-ruled-sets-September-27-2026","tags":["geometry","ruled-sets","vector-fields"],"score":0,"title":"Finite cylinder approximation of ruled sets","domain":"Convex and metric geometry","family":"100","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.MG","summary":"We approximate compact ruled families of segments by finitely many cylinders with square intercept tiles and perpendicular-base area at most their integral projection cost plus any positive error. The velocity field is C 1 , and its differential has opposite real eigenvalues whose magnitudes are strictly below the inverse segment half-length; both eigenvalues may vanish. The result includes square-zero differentials with unrestricted shear and fields that pass between the two regimes. It holds for arbitrary compact label sets.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-27","reactions":{},"formalized":false,"family_title":"Cylinder coverings below the half-area bound"}