{"id":"openai-math/Positive-scalar-curvature-and-uniform-codimension-two-width","pdf":"https://github.com/openai/math/blob/main/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Positive-scalar-curvature-and-uniform-codimension-two-width-September-23-2026","tags":["differential-geometry","scalar-curvature","topology"],"score":0,"title":"Positive scalar curvature and uniform codimension-two width","domain":"Differential geometry","family":"336","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.DG","summary":"We prove the quantitative continuous form of Gromov's scalar-curvature conjecture in every dimension n ≥ 4. Every complete connected smooth boundaryless n -manifold with scalar curvature at least one admits a continuous map to a simplicial complex of dimension at most $`n-2`$ whose entire fibers have diameter bounded only in terms of n . We also obtain the continuous macroscopic-dimension conclusion for universal covers of closed manifolds with positive scalar curvature in every dimension n ≥ 2.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-09-23","reactions":{},"discussion":[{"url":"https://arxiv.org/abs/2610.28390","date":"2026-10-20","kind":"web","site":"arXiv.org","text":"In this note, we adapt ideas from No. 336 of OpenAI's October 6, 2026 mathematics release to prove the weighted cube inequality in all dimensions.","added":"2026-10-24","title":"A note on the cube inequality","author":"Yuchen Bi","record":"arxiv/2610.28390"}],"formalized":false,"family_title":"Spectral scalar curvature, Urysohn width, and macroscopic dimension"}