{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-06-openai-gromov-lawson-aspherical-conjecture-psc/","md":"https://postcutoff.com/e/2026-10-06-openai-gromov-lawson-aspherical-conjecture-psc/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-06-openai-gromov-lawson-aspherical-conjecture-psc","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims the aspherical positive-scalar-curvature conjecture","deck":null,"takeaway":"Two manuscripts in OpenAI's 6 Oct 2026 math release (family 335) claim that a closed manifold with positive scalar curvature is rationally inessential, in every dimension and without spin assumptions. That would prove the Gromov–Lawson (Schoen–Yau) conjecture that closed aspherical manifolds, such as tori and hyperbolic manifolds, admit no metric of positive scalar curvature. It was known in dimensions up to five (Chodosh–Li, Gromov). A second paper proves Gromov's proposed integral bound of simplicial volume by negative scalar curvature. Roman Sauer, whose technique appears in the proofs, wrote 'I am stunned.' No Lean proof; no expert verification yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"OpenAI: Positive scalar curvature forces rational inessentiality (23 Sept 2026, PDF)","url":"https://github.com/openai/math/blob/main/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"OpenAI: An integral scalar curvature bound for real simplicial volume (5 Oct 2026, PDF)","url":"https://github.com/openai/math/blob/main/preprints/An-integral-scalar-curvature-bound-for-real-simplicial-volume-October-5-2026/v126-proof.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":3,"title":"Proofs and Prompts: 100+ reactions to 100+ solutions (Roman Sauer)","url":"https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/","type":"discussion","group":"reactions","domain":"proofsandprompts.com"},{"n":4,"title":"oai-math-revised: community notes on the rational-inessentiality manuscript","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Positive-scalar-curvature-forces-rational-inessentiality-September-23-2026/AGENT_NOTES.md","type":"discussion","group":"reactions","domain":"github.com"}],"official":2,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"],"title":"OpenAI's release claims the Gromov–Lawson conjecture in every dimension: no closed aspherical manifold has positive scalar curvature, with Gromov's integral bound for simplicial volume","summary":"Family 335: \"Positive scalar curvature forces rational inessentiality\" (23 Sept 2026) claims that every closed oriented manifold with positive scalar curvature sends its rational fundamental class to zero in the classifying space, with no spin, fundamental-group or dimension restriction. Corollaries: the Gromov–Lawson aspherical conjecture (no closed aspherical manifold has positive scalar curvature) and flatness of every nonnegative-scalar-curvature metric on a closed aspherical manifold. \"An integral scalar curvature bound for real simplicial volume\" (5 Oct) proves Gromov's inequality ∫(Scal⁻)^{n/2} ≥ a_n‖M‖. Known before: dimensions 4 and 5 (Chodosh–Li; Gromov) and spin cases via index theory. No Lean formalization; Roman Sauer (Proofs and Prompts) called the proofs \"deeply creative\".","key_facts":["Main claim (abstract, verbatim): 'We prove that every closed connected oriented smooth manifold admitting strictly positive scalar curvature is rationally inessential: its rational fundamental class maps to zero under the classifying map. No spin or fundamental-group hypothesis is needed, and there is no upper dimension bound. This proves the Gromov–Lawson aspherical conjecture; moreover, every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat.'","Companion (verbatim): 'We prove the integral scalar-curvature inequality proposed by Gromov. For every dimension n ≥ 3, there is a constant a_n > 0, depending only on n, such that ∫_M (Scal_g⁻)^{n/2} dV_g ≥ a_n‖M‖ for every closed connected oriented smooth n-manifold M and every smooth Riemannian metric g.' It relies on the first paper's vanishing theorem","Prior state (as the paper recounts it): Schoen–Yau's minimal-hypersurface method (non-spin, dimensions up to seven); Gromov–Lawson's Dirac-operator method (needs spin coverings); 'Chodosh and Li established the aspherical obstruction in dimensions four and five using generalized soap bubbles'; Gromov independently treated the closed five-dimensional case","Reaction: Roman Sauer on Proofs and Prompts, 8 Oct: his measurable-foliation trick, from work with Sabine Braun extended by Hannah Alpert, 'now appear[s] in OpenAI's deeply creative solutions of problems 207 and 335 – problems so different, they could not be farther apart! I am stunned.'","The aspherical case includes tori (no positive scalar curvature on Tⁿ, already known) and closed hyperbolic and other negatively curved manifolds; the flatness corollary was noted by Gromov to follow from the obstruction","Status: no Lean scope note for family 335; no revision in history.md; the integral bound is the Oct 5 manuscript ('v126-proof.pdf')"],"key_numbers":[],"tags":["math","differential-geometry","scalar-curvature","aspherical-manifolds","gromov-lawson-conjecture","simplicial-volume","openai-math-release"],"science":{"field":"mathematics","subfield":"differential geometry (scalar curvature, aspherical manifolds, simplicial volume)","problem":"Gromov–Lawson / Schoen–Yau aspherical conjecture: a closed aspherical manifold admits no metric of positive scalar curvature; Gromov's integral scalar curvature bound for simplicial volume","result":"Claimed in every dimension, without spin hypotheses, via rational inessentiality of positive-scalar-curvature manifolds; plus the integral inequality with a dimensional constant","open_since":"","ai_system":["OpenAI internal model (math release)"],"human_role":"Autonomous AI output released by OpenAI","verification":{"key":"expert","label":"Expert-checked","text":"Unrefereed manuscripts; no formalization; no published expert check"},"status":"pending","shock":"A conjecture that the best minimal-surface methods had pushed only to dimension five is claimed in all dimensions at once, without the spin condition index theory needs."},"body_md":"## What happened\n\nScalar curvature is the weakest curvature notion, and asking which closed manifolds carry a metric of positive scalar curvature is a\ncentral question of geometric topology. Gromov and Lawson, and Schoen and Yau, conjectured in the early 1980s that aspherical manifolds,\nwhose universal cover is contractible (tori, hyperbolic manifolds), never do. Two methods made partial progress. Minimal hypersurfaces\nwork without spin structures but originally only up to dimension seven. Dirac operators work in all dimensions but need spin assumptions and\nlargeness conditions. The general aspherical case was settled only in dimensions four and five, in 2020–24.\n\nOpenAI's release claims the full statement in every dimension, through a stronger topological conclusion: positive scalar curvature\nforces the fundamental class to vanish rationally in the classifying space. A second manuscript uses that vanishing to prove an\ninequality Gromov proposed, bounding the simplicial volume of a manifold by an integral of its negative scalar curvature. Roman Sauer\nrecognised a technique from his own work with Sabine Braun in these proofs and in the release's family 207, and called them \"deeply\ncreative\".\n\n## Why it matters\n\nIf correct, this settles one of the best-known conjectures about positive scalar curvature and removes the spin condition that has\nshaped the field for forty years. It would also imply rigidity: on aspherical manifolds, nonnegative scalar curvature forces flatness.\nThe claims are unrefereed and unformalized, and no specialist has published a check as of 29 Oct.","disputed":[{"label":"Verification","text":"Unrefereed manuscripts; no formalization; no published expert check","sources":[]}],"related":[{"id":"2026-10-24-oai-math-revised-community-hub","url":"https://postcutoff.com/e/2026-10-24-oai-math-revised-community-hub/","date":"2026-10-24","date_precision":"day","short_title":"oai-math-revised: mathematicians launch a community hub to revise OpenAI's AI-written math manuscripts (guest post on Tao's blog)","deck":null,"takeaway":"This is the first organized attempt by research mathematicians to digest and credit the OpenAI catalogue.","category":"research","category_label":"Research","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":6,"official":5,"filed":"2026-10-25","updated":"2026-10-27","orgs":["OpenAI","mathcommunity"]},{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":79,"official":13,"filed":"2026-10-07","updated":"2026-10-29","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"Created (families 251–378; both abstracts and the first paper's introduction read, Proofs and Prompts, oai-math-revised notes)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-29","run":null,"sources_read":"Families 251–378; both abstracts and the first paper's introduction read, Proofs and Prompts, oai-math-revised notes","updated":"2026-10-29","human_review":null,"version":{"date":"2026-10-29"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":159,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":98,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":189,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":128,"in_training_data":false}],"short_url":"https://postcutoff.com/s/openai-aspherical"}