{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-06-openai-yau-nodal-conjecture-surfaces-counterexamples/","md":"https://postcutoff.com/e/2026-10-06-openai-yau-nodal-conjecture-surfaces-counterexamples/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-yau-nodal-conjecture-surfaces-counterexamples","date":"2026-10-06","date_precision":"day","short_title":"OpenAI: Yau's nodal bound holds on surfaces, fails in higher dimensions","deck":null,"takeaway":"Three manuscripts in OpenAI's 6 Oct 2026 math release (family 350, dated 23 Sept) claim to settle Yau's 1982 conjecture on the size of nodal sets of Laplace eigenfunctions for smooth metrics. On every smooth closed surface the nodal length is at most C√λ, which completes the conjecture there. In dimensions 3 and 4, explicit smooth metrics, including some arbitrarily close to the round 3-sphere, violate the upper bound, and on S⁴×S¹ it fails by a power of λ. Lean proofs of all three statements passed an independent Comparator run. Whether the formal statements match the papers has not been checked by specialists.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"OpenAI: Sharp nodal length on smooth surfaces (PDF)","url":"https://github.com/openai/math/blob/main/preprints/Sharp-nodal-length-on-smooth-surfaces-September-23-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"OpenAI: Smooth counterexamples to Yau's nodal upper bound in dimensions three and four (PDF)","url":"https://github.com/openai/math/blob/main/preprints/Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four-September-23-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":3,"title":"OpenAI: Power-law violations of Yau's nodal upper bound (PDF)","url":"https://github.com/openai/math/blob/main/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":4,"title":"OpenAI Lean scope note 350","url":"https://github.com/openai/math/blob/main/lean/docs/350.md","type":"code","group":"primary","domain":"github.com"},{"n":5,"title":"oai-math-revised: Comparator results (25–26 Oct)","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv","type":"code","group":"primary","domain":"github.com"},{"n":6,"title":"Xie, Zhang: Sharp bounds for nodal sets of Steklov eigenfunctions on smooth surfaces (arXiv 2610.28391)","url":"https://arxiv.org/abs/2610.28391","type":"paper","group":"primary","domain":"arxiv.org"}],"official":6,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"],"title":"OpenAI's release claims Yau's nodal-set conjecture splits: the sharp √λ upper bound holds on every smooth surface but fails for smooth metrics in dimensions 3–5, with Lean proofs of both sides","summary":"Family 350: \"Sharp nodal length on smooth surfaces\" claims H¹(nodal set) ≤ C√λ for every eigenfunction on a fixed smooth closed surface, which with the known lower bound proves Yau's conjecture for surfaces (best previous bound: C·λ^{3/4−β}, Logunov–Malinnikova). \"Smooth counterexamples to Yau's nodal upper bound in dimensions three and four\" and \"Power-law violations of Yau's nodal upper bound\" (S⁴×S¹, growth faster than λ^{1/2+ε₀}) claim the upper bound fails for smooth metrics in higher dimensions. It holds for real-analytic metrics (Donnelly–Fefferman, 1988), and Logunov proved the lower bound in all dimensions. All three are formalized: Comparator challenges NodalLength, SmoothYau and YauCounterexample were accepted in oai-math-revised's run of 25–26 Oct. No expert assessment found.","key_facts":["Surfaces (abstract, verbatim): 'We prove that a nonzero real Laplace eigenfunction with eigenvalue λ > 0 on a fixed smooth closed connected Riemannian surface has nodal length at most C√λ. Together with the known lower bound, this proves Yau's conjecture in this setting.'","Dimensions 3–4 (verbatim): 'We construct a smooth metric on the three-sphere, arbitrarily close to the round metric in the smooth topology, and a smooth metric on S²×𝕋² for which sequences of exact real Laplace eigenfunctions have unbounded nodal measure divided by the square root of the eigenvalue… Thus the upper-bound part of Yau's nodal conjecture fails for smooth metrics in dimensions three and four.'","Dimension 5 (verbatim): 'We construct a smooth Riemannian metric on S⁴ × S¹ and a sequence of real Laplace eigenfunctions whose nodal four-volume grows faster than λ^{1/2+ε₀} for one fixed ε₀ > 0. This disproves the smooth upper-bound assertion in Yau's nodal-set conjecture and the proposed bound with an arbitrarily small positive power loss.'","Prior state (surface paper's introduction): Donnelly–Fefferman proved both bounds for real-analytic metrics; Logunov proved the lower bound in every dimension; for smooth surfaces the upper bound was Cλ^{3/4} (Donnelly–Fefferman; Dong), improved to Cλ^{3/4−β} by Logunov and Malinnikova","Lean (scope note lean/docs/350.md): the surface bound with a surface-dependent constant; 'one fixed smooth metric' near round S³ and one on S²×T² 'with an eigenfunction sequence of unbounded nodal-measure ratio'; and one fixed metric on S⁴×S¹ with nodal measure over √λ tending to infinity. oai-math-revised's Comparator run (25–26 Oct, standard axioms only) accepted all three; it checks the proofs against the Lean statements, not the statements against the papers","Follow-up: Xie and Zhang (arXiv 2610.28391, 20 Oct) prove the sharp linear upper bound for interior nodal sets of Steklov eigenfunctions on smooth surfaces, listing OpenAI's preprint in their references"],"key_numbers":[],"tags":["math","spectral-geometry","nodal-sets","eigenfunctions","yau-conjecture","counterexample","lean","openai-math-release"],"science":{"field":"mathematics","subfield":"spectral geometry (nodal sets of Laplace eigenfunctions)","problem":"Yau's conjecture (1982): on a closed smooth Riemannian manifold, c√λ ≤ H^{n−1}(nodal set of an eigenfunction) ≤ C√λ","result":"Claimed: upper bound true on all smooth surfaces (completing the conjecture in dimension 2); upper bound false for smooth metrics in dimensions 3 and 4, and by a power of λ on S⁴×S¹","open_since":"1982","ai_system":["OpenAI internal model (math release)"],"human_role":"Autonomous AI output released by OpenAI, with Lean formalizations","verification":{"key":"peer-review","label":"Peer-reviewed","text":"Lean 4 proofs of the three main statements accepted by an independent Comparator run; fidelity of the formal definitions to the papers not checked by specialists; manuscripts unrefereed"},"status":"pending","shock":"A conjecture usually expected to be true in every dimension is claimed true only on surfaces, with machine-checked counterexamples near the round 3-sphere."},"body_md":"## What happened\n\nWhen a drum vibrates at a high frequency, its \"nodal set\", where the surface does not move, becomes a fine network of curves. Shing-Tung\nYau conjectured in 1982 that its size grows like the square root of the eigenvalue λ, with upper and lower constants depending only\non the manifold. Donnelly and Fefferman proved this for real-analytic metrics in 1988. For smooth metrics, Alexander Logunov proved the\nlower bound in 2018 together with a polynomial upper bound, but the sharp upper bound stayed open even on surfaces.\n\nOpenAI's release claims a split answer. On surfaces the upper bound holds for every smooth metric. In dimensions three and four it\nfails: there are smooth metrics, some as close as one likes to the round 3-sphere, with eigenfunctions whose nodal sets are too large\nfor any constant C. A third paper builds a metric on S⁴×S¹ where the ratio grows like a power of λ, which also rules out the weaker\nform of the bound with a small power loss. All three statements come with Lean proofs, and an independent Comparator run by the\noai-math-revised maintainers accepted them with only the standard axioms.\n\n## Why it matters\n\nYau's nodal-set conjecture is one of the best-known problems in spectral geometry, and Logunov's partial results were among its main\nrecent advances. If the claims hold, the smooth conjecture is true only in dimension two. The Lean proofs make the result unusually\ncheckable. The open question is whether the formal definitions (Hausdorff measure, eigenfunctions, the metrics) say what the papers say.\nThat needs a specialist to read the statement files. We found no public reaction from spectral geometers as of 29 Oct.","disputed":[{"label":"Verification","text":"Lean 4 proofs of the three main statements accepted by an independent Comparator run; fidelity of the formal definitions to the papers not checked by specialists; manuscripts unrefereed","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; three abstracts, the surface paper's introduction, Lean scope note, oai-math-revised Comparator results, arXiv 2610.28391)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-29","run":null,"sources_read":null,"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-yau-nodal-bound"}