{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-23-propeller-conjecture-openai-proof-heilman-simplification/","md":"https://postcutoff.com/e/2026-10-23-propeller-conjecture-openai-proof-heilman-simplification/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":"2026-10-23-propeller-conjecture-openai-proof-heilman-simplification","date":"2026-10-23","date_precision":"day","short_title":"Heilman simplifies OpenAI's propeller conjecture proof","deck":null,"takeaway":"OpenAI's 6 Oct release contained a Lean-formalized proof of the Gaussian propeller conjecture (Khot–Naor 2008) in every dimension, a claim our dataset had missed. On 23 Oct Steven Heilman (USC), co-author of the 2012 three-dimensional case, posted a shorter version that also settles the equality cases and removes the computer search from the 3D step. It is the first expert follow-up to treat the conjecture as proved.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"arXiv 2610.32158: On the Propeller Conjecture (Steven Heilman)","url":"https://arxiv.org/abs/2610.32158","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"OpenAI math catalogue (family 096: The Gaussian propeller conjecture in every dimension)","url":"https://github.com/openai/math/blob/main/CONTENTS.md","type":"paper","group":"primary","domain":"github.com"},{"n":3,"title":"OpenAI Lean comparator challenge: GaussianPropeller","url":"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/GaussianPropeller.lean","type":"code","group":"primary","domain":"github.com"}],"official":3,"filed":"2026-10-26","updated":"2026-10-26","orgs":["OpenAI","University of Southern California"],"title":"Khot–Naor propeller conjecture: Steven Heilman, who proved the 3D case in 2012, simplifies OpenAI's proof and removes its computer assistance","summary":"The propeller conjecture of Subhash Khot and Assaf Naor (2008) says the sum of squared lengths of the Gaussian moments of the parts of any partition of Euclidean space is at most 9/(8π), attained by three 120° sectors. It gives the sharp hardness-of-approximation constant for kernel clustering under the Unique Games Conjecture. OpenAI's 6 Oct 2026 release (family 096) claims it in every dimension with a Lean comparator statement. In arXiv 2610.32158 (23 Oct) Steven Heilman \"simplif[ies] and improve[s] OpenAI's proof\". He shows equality only for the propeller and gives a computer-free proof of the case of four parts, which in 2012 used an ε-net search. GPT-6 helped prepare the manuscript.","key_facts":["Abstract (verbatim): 'We simplify and improve OpenAI’s proof of the propeller conjecture of Khot and Naor from 2008: the sum of the squared lengths of the Gaussian moments of any partition of Euclidean space is at most 9/(8π). We show equality only occurs for three 120 degree sectors in the plane … We also remove computer assistance from the three-dimensional case of the conjecture, which previously used an epsilon net traversal.'","AI statement (verbatim): 'GPT 6 was used to prepare this manuscript, which is based upon the solution of the Propeller Conjecture from [Ope26]. Before the release of [Ope26], we had a somewhat different proof of Theorem 1.1 that was longer and instead relied on the Gaussian isoperimetric inequality and a second variation argument.'","Structure: proves C₄ = 9/(8π) without computer assistance, then, 'as in [Ope26]', shows C_N = C₄ for all N > 4","OpenAI family 096 (catalogue): 'The Gaussian propeller conjecture in every dimension … Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor (8π/9)(1−1/k) for identity-target kernel clustering with fixed k ≥ 3'; manuscript 'The Gaussian propeller bound in every dimension' dated 24 Sept 2026; Lean comparator challenge GaussianPropeller (theorem OAI.GaussianPropeller.all_partitions, standard axioms only)","Prior human work: Khot–Naor (2008, 2013) posed it; Heilman, Jagannath and Naor proved the three-dimensional case (2012) with a computer-assisted ε-net step","Heilman also says that, since the conjecture 'is now proven', the kernel clustering hardness of Khot–Naor needs only the Unique Games assumption, which OpenAI's family 102 also claims to prove"],"key_numbers":[],"tags":["math","computer-science","probability","gaussian-geometry","hardness-of-approximation","openai","openai-math-release","lean","gpt-6","arxiv"],"science":{"field":"mathematics","subfield":"Gaussian geometry / hardness of approximation","problem":"Propeller conjecture (Khot–Naor 2008)","result":"OpenAI's every-dimension proof (Lean comparator statement) simplified by an expert, with equality cases characterised and the 3D computer search removed","open_since":"2008","ai_system":["OpenAI internal model (openai/math release)","GPT-6"],"human_role":"OpenAI model produced the original proof; Heilman (a prior contributor to the problem) rewrote and improved it, using GPT-6 for manuscript preparation","verification":{"key":"lean","label":"Lean-verified","text":"Lean comparator statement in OpenAI's repository; independent human proof posted on arXiv (unrefereed)"},"status":"confirmed","shock":""},"body_md":"## What happened\n\nOpenAI's release of 6 Oct has 372 families, and the propeller conjecture (family 096) was not among the claims we had covered.\nThe conjecture comes from Khot and Naor's study of kernel clustering. Among all ways to cut Gaussian space into pieces, it asks\nwhich maximises the summed squared \"centre of mass\" of the pieces. The answer, 9/(8π), comes from a planar three-bladed propeller.\n\nSteven Heilman of USC proved the three-dimensional case in 2012 with Aukosh Jagannath and Assaf Naor, partly by computer search.\nHis 23 Oct paper takes OpenAI's proof as its starting point, writes it out more simply, adds the characterisation of equality\ncases, and replaces the old computer search with a short argument. He says he had a longer proof of his own before OpenAI's\nrelease.\n\n## Why it matters\n\nIt is one of the clearest cases so far of an expert on a problem endorsing an OpenAI release result by reproving it. Combined\nwith OpenAI's claimed Unique Games proof, it would settle the approximability of kernel clustering.","disputed":[],"related":[{"id":"2026-10-26-ai-assisted-results-october-26-listing","url":"https://postcutoff.com/e/2026-10-26-ai-assisted-results-october-26-listing/","date":"2026-10-26","date_precision":"day","short_title":"AI-assisted results in the 26 Oct 2026 sweep","deck":"Claude-written Lean proof of a frame-indifferent rank-one convex function that is not quasiconvex, a sculptor and Claude Code prove Takemura's conjecture, Claude's contribution to deterministic root-finding, and Zenodo records","takeaway":"Monday's arXiv announcement (papers submitted 22–23 Oct) brought four results with their own entries: OpenAI's propeller conjecture reproved by Heilman, Tokareva's conjecture disproved by MiniMax agents, the triangle lower tail with GPT-6 Astra and Claude, and graph-connectivity communication complexity with ChatGPT. This page lists the next tier, from Lean proofs written entirely by Claude to papers where the AI only checked the text.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":22,"official":22,"filed":"2026-10-26","updated":"2026-10-26","orgs":["Anthropic","OpenAI","various researchers"]},{"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"]},{"id":"2026-10-06-unique-games-conjecture-openai","url":"https://postcutoff.com/e/2026-10-06-unique-games-conjecture-openai/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI model claims a proof of Khot's Unique Games Conjecture, with Lean formalization","deck":null,"takeaway":"If OpenAI's claimed proof holds, it settles the Unique Games Conjecture, a central open problem in hardness of approximation since 2002. It comes with a Lean formalization, but experts have not yet checked it.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":12,"official":5,"filed":"2026-10-07","updated":"2026-10-29","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-26","type":"filed","text":"Created (arXiv PDF and OpenAI catalogue read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-26","run":null,"sources_read":"ArXiv PDF and OpenAI catalogue read","updated":"2026-10-26","human_review":null,"version":{"date":"2026-10-27"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":176,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":115,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":206,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":145,"in_training_data":false}],"short_url":null}