{"id":"openai-math/An-integral-scalar-curvature-bound-for-real-simplicial-volume","pdf":"https://github.com/openai/math/blob/main/preprints/An-integral-scalar-curvature-bound-for-real-simplicial-volume-October-5-2026/v126-proof.pdf","url":"https://github.com/openai/math/tree/main/preprints/An-integral-scalar-curvature-bound-for-real-simplicial-volume-October-5-2026","tags":["differential-geometry","scalar-curvature"],"score":0,"title":"An integral scalar curvature bound for real simplicial volume","domain":"Differential geometry","family":"335","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.DG","summary":"We prove the integral scalar-curvature inequality proposed by Gromov. For every dimension n ≥ 3, there is a constant $`a_n\\gt 0`$, depending only on n , such that $`\\displaystyle \\int_M(\\mathop{\\mathrm{Scal}}\\nolimits _g^-)^{n/2}\\,dV_g\\ge a_n\\|M\\|`$ for every closed connected oriented smooth n -manifold M and every smooth Riemannian metric g . Here $`\\mathop{\\mathrm{Scal}}\\nolimits _g^-:=\\max\\{0,-\\mathop{\\mathrm{Scal}}\\nolimits _g\\}`$, and $`\\|M\\|`$ is real simplicial volume. The proof uses the nonnegative-scalar-curvature vanishing theorem of the companion paper on rational inessentiality.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":1,"published":"2026-10-05","reactions":{},"discussion":[{"url":"https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/#:~:text=This%20trick%20and%20these%20two%20papers%20now%20appear%20in%20OpenAI%E2%80%99s%20deeply%20creative%20solutions%20of%20problems%20207%20and%20335","date":"2026-10-08","kind":"web","site":"Proofs and Prompts","text":"This trick and these two papers now appear in OpenAI’s deeply creative solutions of problems 207 and 335 – problems so different, they could not be farther apart! I am stunned. What do I take away from this night? We cannot defensively narrow our focus to what we currently regard as human…","added":"2026-10-24","title":"100+ reactions to 100+ solutions"}],"formalized":false,"family_title":"Gromov’s integral scalar-curvature bound for simplicial volume"}