{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-06-openai-unit-distance-upper-bound-power-saving-claim/","md":"https://postcutoff.com/e/2026-10-06-openai-unit-distance-upper-bound-power-saving-claim/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-unit-distance-upper-bound-power-saving-claim","date":"2026-10-06","date_precision":"day","short_title":"Unit-distance bound below n^(4/3) claimed (OpenAI, Lean)","deck":null,"takeaway":"A 53-page manuscript credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claims that n points in the plane determine at most O(n^(4/3−δ)) pairs at distance one, for some unspecified δ > 0. The bound n^(4/3) of Spencer, Szemerédi and Trotter had not been improved since 1984. A companion proves the weak pinned Erdős distance conjecture. Both are formalized in Lean and passed a community comparator run, and Terence Tao's optimization-constants repository now lists the new exponent. On 25 Oct a commenter posting as WillSawin explained the argument on erdosproblems.com and said he sees how to make its ineffective step effective. Since OpenAI's May counterexample gave a lower bound of n^(1+c), the exponent now lies strictly between 1 and 4/3.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"OpenAI (PDF): A power saving for planar unit distances (23 Sept 2026)","url":"https://github.com/openai/math/blob/main/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"OpenAI (PDF): The weak pinned planar distance theorem (23 Sept 2026)","url":"https://github.com/openai/math/blob/main/preprints/The-weak-pinned-planar-distance-theorem-September-23-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":3,"title":"OpenAI Lean scope note for family 167","url":"https://github.com/openai/math/blob/main/lean/docs/167.md","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"teorth/optimizationproblems PR #204: unit distance bounds, upper bound 4/3 − ε (merged 18 Oct)","url":"https://github.com/teorth/optimizationproblems/pull/204","type":"code","group":"primary","domain":"github.com"},{"n":5,"title":"oai-math-revised comparator results (PlanarUnitDistances, PinnedDistances accepted)","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":"Thomas Bloom: Erdős problems in the OpenAI proofs (problems #90, #604) and WillSawin's comment of 25 Oct","url":"https://www.erdosproblems.com/forum/thread/blog:10","type":"discussion","group":"reactions","domain":"erdosproblems.com"},{"n":7,"title":"Gil Kalai: Updates: Sharing AI progress on mathematics (amazing!) (7 Oct)","url":"https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/","type":"discussion","group":"reactions","domain":"gilkalai.wordpress.com"},{"n":8,"title":"oai-math-revised AGENT_NOTES for the unit-distance manuscript","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-power-saving-for-planar-unit-distances-September-23-2026/AGENT_NOTES.md","type":"discussion","group":"reactions","domain":"github.com"}],"official":5,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"],"title":"OpenAI's math release claims the first improvement on the Spencer–Szemerédi–Trotter bound of 1984: n points in the plane span O(n^(4/3−δ)) unit distances; also the weak pinned distinct-distance conjecture; both in Lean, and Will Sawin has explained a route to an effective proof","summary":"\"A power saving for planar unit distances\" (OpenAI, 23 Sept 2026, 53 pages; family 167 of the 6 Oct release): \"We prove a power saving for the planar unit-distance problem: for some absolute β < 4/3, every set of n points in the Euclidean plane determines O(n^β) unordered pairs at unit distance.\" The companion \"The weak pinned planar distance theorem\" (27 pages) proves that, for every ε > 0, all but o(n) points of an n-point planar set determine at least n^(1−ε) distinct distances. Both are formalized in Lean (PlanarUnitDistances, PinnedDistances) and were accepted in the oai-math-revised comparator run. Thomas Bloom: the PDF \"gives no indication as to the value of c obtained\" and combines the point–circle incidence view with \"some kind of entropy-based argument together with input from algebraic number theory\". teorth/optimizationproblems merged the 4/3 − ε upper bound on 18 Oct.","key_facts":["Manuscript (abstract, verbatim): 'We prove a power saving for the planar unit-distance problem: for some absolute β < 4/3, every set of n points in the Euclidean plane determines O(n^β) unordered pairs at unit distance.' Theorem 1.1: absolute constants 0 < C < ∞ and 1 ≤ β < 4/3 with u(n) ≤ C·n^β for every n. 53 pages, dated 23 Sept 2026","Background (paper): Erdős posed the problem in 1946 with the lattice lower bound n^(1+c/log log n); Spencer, Szemerédi and Trotter proved u(n) = O(n^(4/3)) in 1984, reproved by Székely's crossing-number method; Pach, Raz and Solymosi obtained a logarithmic improvement conditional on a rigidity conjecture. In May 2026 an OpenAI construction disproved Erdős's conjecture u(n) = n^(1+o(1)) (see 2026-05-20-ai-disproves-erdos-unit-distance-conjecture); Sawin sharpened it to n^1.014114, and later work pushed the lower exponent to about 1.043","Companion (abstract, verbatim): 'We prove the weak pinned Erdős distinct-distance conjecture. For every fixed ε > 0, all but o(n) points of any n-point planar set determine at least n^(1−ε) distinct nonzero distances.' 27 pages; Bloom lists it under Erdős problem #604","Lean (OpenAI scope note lean/docs/167.md): 'there are absolute constants C>0 and 1≤β<4/3 such that every finite planar point set of size n determines at most Cn^β unordered unit-distance pairs. The same constants work for every n'; the pinned-distance statement is a separate challenge. Both 'accepted' in the oai-math-revised comparator run of 25–26 Oct","Thomas Bloom (erdosproblems.com, problem #90, 8 Oct): 'The best lower bound for this problem, obtained by OpenAI in May, is n^{1+c} for some constant c>0. The best upper bound previously, due to Spencer, Szemerédi, and Trotter [SST84], was n^{4/3}. The PDF gives no indication as to the value of c obtained.' The proof 'appears to begin with a point-circle incidence interpretation (which was used in [SST84]), and couples that with some kind of entropy-based argument together with input from algebraic number theory'","Expert reading (forum comment by WillSawin, 25 Oct): 'The biggest progress is that I see how to remove the argument with ultrafilters from section 8, which was one of the things making the argument ineffective, and replace it with in-principle effective algebraic geometry reasoning.' He describes the strategy as extracting structure from near-extremal configurations of the n^(4/3) argument and deriving a contradiction 'in a somewhat surprising way', with a key role for the Weil height of ratios on unit circles, and says of the entropy step: 'I don't understand this'","Recognition of the bound: teorth/optimizationproblems PR #204 (merged by teorth on 18 Oct) adds 'OpenAI, A power saving for planar unit distances' as the source of the upper exponent 4/3 − ε with ε not explicit, and corrects 'The comment saying the exponent 4/3 had not been improved'. Gil Kalai's 7 Oct post lists the improved planar unit-distance bound among the release's results"],"key_numbers":[],"tags":["math","discrete-geometry","combinatorics","unit-distance-problem","incidence-geometry","erdos-problems","openai-math-release","lean","formal-verification"],"science":{"field":"mathematics","subfield":"discrete geometry (incidence geometry, Erdős distance problems)","problem":"Erdős unit distance problem: the maximum number u(n) of unit distances among n points in the plane; the upper bound O(n^(4/3)) (Spencer–Szemerédi–Trotter, 1984) had not been improved. Also the weak pinned distinct-distance conjecture","result":"Claimed u(n) = O(n^(4/3−δ)) for an absolute, unspecified δ > 0, and the weak pinned distance theorem; both formalized in Lean","open_since":"1946","ai_system":["OpenAI internal model (math release)"],"human_role":"Released as model output under OpenAI's fixed procedure, with no human author named; detailed public reading by an outside mathematician (forum comment), no written check","verification":{"key":"peer-review","label":"Peer-reviewed","text":"Lean formalizations accepted by an independent comparator run; unrefereed manuscripts; an outside expert's partial reading proposes an effective replacement for one step"},"status":"pending","shock":"The 4/3 exponent, unchanged since 1984, gives way to an argument that brings in heights from algebraic number theory, the field that also produced the May 2026 lower-bound construction."},"body_md":"## What happened\n\nErdős asked in 1946 how many pairs among n points in the plane can be at distance exactly one. Spencer, Szemerédi and Trotter proved\nthe upper bound O(n^(4/3)) in 1984 from incidences between points and unit circles. Later work improved only the constant. In\nMay 2026 an OpenAI model refuted Erdős's conjecture that the truth is n^(1+o(1)), with a number-field construction giving n^(1+c).\n\nOpenAI's 6 Oct release now claims the matching improvement from above: O(n^(4/3−δ)) for an absolute δ that the paper does not compute.\nA companion paper proves the weak pinned distance conjecture. Both are formalized in Lean, and a community comparator run accepted them.\nOn 25 Oct a commenter posting as WillSawin, the name of the mathematician who sharpened the May construction, posted a long reading on\nerdosproblems.com. He traces the argument to near-extremal incidence structure and Weil heights of ratios of points on unit circles,\nand says he can replace its ultrafilter step with effective algebraic geometry.\n\n## Why it matters\n\nIf correct, the claim breaks a 42-year-old barrier in incidence geometry. Combined with May's lower bound, the unit-distance exponent\nlies strictly between 1 and 4/3, and both moves came from OpenAI models using algebraic number theory. A detailed outside reading and\nthe formal proof make this one of the better-checked claims in the release, but δ is still unknown and the proof is not refereed.","disputed":[{"label":"Verification","text":"Lean formalizations accepted by an independent comparator run; unrefereed manuscripts; an outside expert's partial reading proposes an effective replacement for one step","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"]},{"id":"2026-10-06-openai-erdos-arithmetic-progressions-conjecture-claim","url":"https://postcutoff.com/e/2026-10-06-openai-erdos-arithmetic-progressions-conjecture-claim/","date":"2026-10-06","date_precision":"day","short_title":"Erdős's arithmetic-progression conjecture claimed (OpenAI, Lean)","deck":null,"takeaway":"A 198-page manuscript credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claims Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arbitrarily long arithmetic progressions, the problem for which Erdős offered $5000. It does so by claiming quasipolynomial bounds in Szemerédi's theorem for every progression length. A Lean proof of the qualitative conjecture was accepted in a community comparator run. Thomas Bloom of erdosproblems.com reads the Lean code as proving a weaker bound than the paper claims, and describes the proof as a combination of existing techniques. No expert has published a check of the 198 pages.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":10,"official":7,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-05-20-ai-disproves-erdos-unit-distance-conjecture","url":"https://postcutoff.com/e/2026-05-20-ai-disproves-erdos-unit-distance-conjecture/","date":"2026-05-20","date_precision":"day","short_title":"OpenAI model disproves Erdős's 80-year-old unit distance conjecture","deck":null,"takeaway":"This is the moment AI crossed from solving competition problems to settling a famous open research conjecture, reshaping debate about AI's role in mathematics.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":9,"official":3,"filed":"2026-09-29","updated":"2026-10-29","orgs":["OpenAI"]}],"people":[{"id":"thomas-bloom","name":"Thomas Bloom","url":"https://postcutoff.com/person/thomas-bloom/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"Created (CONTENTS entry, both PDFs (53 and 27 pp), Lean scope note, comparator results, Bloom's post and the WillSawin comment, teorth PR #204, Kalai)"}],"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/unit-distance-bound-below-n"}