{"id":"openai-math/Spectral-scalar-curvature-and-uniform-Urysohn-width","pdf":"https://github.com/openai/math/blob/main/preprints/Spectral-scalar-curvature-and-uniform-Urysohn-width-October-5-2026/main.pdf","url":"https://github.com/openai/math/tree/main/preprints/Spectral-scalar-curvature-and-uniform-Urysohn-width-October-5-2026","tags":["scalar-curvature","riemannian-geometry","urysohn-width","spectral-geometry"],"score":0,"title":"Spectral scalar curvature and uniform Urysohn width","domain":"Differential geometry","family":"336","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.DG","summary":"For every n ≥ 4, a complete connected smooth Riemannian n -manifold without boundary satisfying $`-4\\Delta+\\mathop{\\mathrm{Scal}}\\nolimits \\ge1`$ as a quadratic-form inequality 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 . The bound is measured in the original metric. This extends the uniform Urysohn width theorem from a pointwise scalar-curvature lower bound to a spectral lower bound.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-10-05","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"}