---
id: "2026-10-06-openai-borsuk-conjecture-dimension-nine-counterexample"
url: "https://postcutoff.com/e/2026-10-06-openai-borsuk-conjecture-dimension-nine-counterexample/"
as_of: "2026-10-30T13:42:00+01:00"
date: "2026-10-06"
date_precision: day
category: science
importance: 5
confidence: medium
status: [Awaiting review]
verification: Peer-reviewed
sources: 18
editor: human editor
human_review: null
version: "2026-10-29"
---

As of: 2026-10-30 13:42 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-06-openai-borsuk-conjecture-dimension-nine-counterexample/

# Borsuk's conjecture fails in dimension 9 (OpenAI claim, Lean-checked)

Full title: OpenAI's release claims Borsuk's conjecture fails in dimension 9, down from the 63-dimensional record; the Lean proof is accepted and Gil Kalai calls it major progress

"A nine-dimensional counterexample to Borsuk's covering assertion" (OpenAI, 23 Sept 2026, released 6 Oct, family 156) claims that X = {uuᵀ : u ∈ ℝ⁴, ‖u‖ = 1}, the rank-one orthogonal projectors on ℝ⁴ with the Frobenius metric, lies in the 9-dimensional space of trace-one symmetric matrices, has diameter √2 and cannot be covered by ten sets of smaller diameter. Borsuk's conjecture would then fail in every dimension d ≥ 9; the smallest known failing dimension had been 63 (Grinsztajn, May 2026). The proof is topological and ends in a finite combinatorial obstruction. The main theorem is formalized in Lean, and the formalization was accepted in the community Comparator run of 25–26 Oct. No human referee report has been published. Kalai called the result major progress and stressed that it still has to be checked. The construction is infinite and says nothing new about finite sets.

## Key facts
- Claim (abstract, verbatim): 'The compact set of rank-one orthogonal projectors on ℝ⁴, with the Frobenius metric, cannot be covered by ten sets of strictly smaller diameter. It therefore gives a counterexample to Borsuk's conjecture in dimension nine.'
- Scope (paper, §1): 'The theorem does not determine the smallest dimension in which it fails or the exact value of b(X).' Corollary 7.1 adds points at distance √2 to get compact counterexamples in every dimension d ≥ 9. 'Covers and partitions give the same number', so the result is the partition form of Borsuk's question
- Geometry: ‖Pₓ − P_y‖² = 2 − 2⟨u, v⟩², so 'two projectors are at this distance exactly when their lines are orthogonal'. A cover by ten smaller sets would give ten nonnegative functions on ℝP³, summing to one, with orthogonal lines never sharing a positive label. An odd map on symmetric matrices, mod-two degree, 'triangle systems on six labels' and a combinatorial block argument on ten labels then give the contradiction
- Record before: 63 (Max Grinsztajn with GPT-5.5 Pro, May 2026, 321 points), 64 (Jenrich–Brouwer, 2014), 65 (Bondarenko, 2014). The construction is classical: Kalai's 2015 survey (§2.3) lists 'the image of all unit vectors' under x ↦ x ⊗ x, and says (§2.4) that the double-cap conjecture, if true, would make this set a Borsuk counterexample 'for d > 70 or so'. Kalai on 21 Oct: 'The construction itself is one of the constructions Jeff and I studied'. 'What is new is the proof that this set cannot be covered by ten sets of smaller diameter'
- Gil Kalai (blog, 21 Oct): 'One of the 700 or so papers released by Open AI on October 6, 2026 asserts that f(9)>10. This is a major progress on the Borsuk's problem. (Of course, the proof needs to be carefully checked. The main theorem has also been formalized in Lean.)' And: 'The topology here looks more complicated and the proof looks very beautiful!'
- Finite sets: Gábor Damásdi, commenting on Kalai's post (21 Oct), reports that ChatGPT showed him 'that every finite subset of its non-10-colorable example is actually 8-colorable!' He partitions S³ by the coordinate hyperplanes into 16 regions, giving opposite regions the same colour (Kalai: 'a very nice argument'). Kalai's 'Test Your Human Intuition 60' (11 Oct) asks whether the finite and infinite Borsuk numbers agree; on 21 Oct he conjectured that g(d) < f(d) 'for every large d'. Wikipedia's article (checked 29 Oct) still gives 63 and says the infinite construction would not disprove the finite version in ℝ⁹
- Lean: comparator challenge BorsukNine.lean states main_theorem: the projector set is compact, lies in the trace-one symmetric matrices, has Metric.diam = √2, and has no cover by ten subsets of diameter < √2. OpenAI's Main.lean also proves an ℝ⁹ form (euclidean_nine_counterexample). Accepted in oai-math-revised's Comparator run of 25–26 Oct (414 of 416 challenges accepted, none rejected); accepted means the proof matches the challenge statement with standard axioms, not that a person has checked that the statement says what the paper claims
- Independent replay: Erdős Problem a Day, a tracker built by Patrick White with Claude, reports (#505, 20 Oct) that it built OpenAI's 269 Lean modules (78,964 lines) with 0 errors and that leanchecker --fresh exited 0. Its referee seats, all served by Claude Opus 5.5, read the printed argument and found it 'HOLDS AS READ'. It also gives an exact certificate that eleven pieces suffice, so b(X) = 11 if Theorem 1.1 holds (the acprk/openai-math-followups notes had reported this 'numerically' on 10 Oct). Its verdict: 'Nothing we found disputes the claim'
- Trackers: google-deepmind/formal-conjectures PR #6898 (merged 11 Oct) marks Borsuk refuted at n = 9 (borsuk_conjecture.not_nine); PR #7312 (merged 23 Oct) adds the every-n ≥ 9 statement of Corollary 7.1 without a proof. Tao's optimization-constants table (constant 28a, checked 29 Oct) lists 9 with an asterisk 'pending a second party', calling it 'a large jump from the previous record of 63'
- Separately, on the upper-bound side: an AI-generated Hexagon note curated by Andriy Prymak (hexagon 2610.00163, 8 Oct) claims b(n) ≤ (1.22304666… + o(1))ⁿ, slightly improving the base √(3/2) ≈ 1.22474 of Schramm (1988) and Bourgain–Lindenstrauss (1991). 'The main proof arguments were primarily generated using ChatGPT-6 Astra with discussion and prompts by Andrii Arman, Andriy Bondarenko, Andriy Prymak and Danylo Radchenko'

## What happened

In 1933 Borsuk asked whether every bounded set in d-dimensional space can be cut into d + 1 pieces of smaller diameter.
Kahn and Kalai showed in 1993 that the answer is no in very high dimensions, and the smallest known failing dimension then came
down step by step (946, 561, 298, 65, 64, …) to 63 in May 2026. Every one of these examples was a finite point set.

OpenAI's 6 Oct release contains an 18-page manuscript that jumps to dimension 9 with an infinite set: the projectors uuᵀ for unit
vectors u in ℝ⁴. Two of them are at the maximal distance √2 exactly when their lines are orthogonal. The claim is that this copy of
real projective 3-space cannot be covered by ten pieces of diameter strictly below √2. The proof turns a hypothetical cover into a
map with a parity (mod-two degree) obstruction and reduces the last step to a finite combinatorial problem on ten labels. The
construction itself was known: Kalai had proposed this set in his 2015 survey, but his argument needed the unproved double-cap
conjecture and would only have worked above about 70 dimensions. The new part is the covering obstruction.

The main theorem has a Lean 4 proof, which oai-math-revised's Comparator run accepted. Erdős Problem a Day, a tracker run with Claude,
also rebuilt the Lean proof on its own machine and reports an exact certificate that eleven pieces suffice for this set. Gil Kalai,
who studied this construction with Jeff Kahn, wrote on 21 Oct that the paper is "a major progress on the Borsuk's problem", with the
caveat that "the proof needs to be carefully checked". In the comments, Gábor Damásdi showed that every finite subset of the set
can be split into eight pieces. So the result does not lower the finite record of 63. Kalai expects the finite and infinite Borsuk
numbers to differ in large dimensions.

## Why it matters

If the proof is right, it closes most of the gap in one of the best-known problems of discrete geometry: Borsuk's conjecture holds in
dimensions up to 3 and would fail from 9 on, leaving 4 to 8 open. It is also a rare case where a release result has a short, readable
statement, a machine-checked proof and a named expert who finds the argument beautiful. What it does not do is change the finite
question. And the formal proof certifies the Lean statement, not the paper's printed route; as of 29 Oct no human topologist or
combinatorialist had published a full reading.

## What is disputed or not yet verified

- Verification: Lean 4 proof of the main theorem, accepted by the community Comparator run (25–26 Oct) and replayed by an AI-run tracker; manuscript unrefereed; no published full check by a human expert (as of 29 Oct)

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 159 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 98 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 189 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 128 days after its cutoff

## Sources

1. [OpenAI: A nine-dimensional counterexample to Borsuk's covering assertion (23 Sept 2026, PDF)](https://github.com/openai/math/blob/main/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/paper.pdf) (github.com, paper)
2. [openai/math: Lean scope note for family 156](https://github.com/openai/math/blob/main/lean/docs/156.md) (github.com, code)
3. [openai/math: comparator challenge BorsukNine.lean](https://github.com/openai/math/blob/main/lean/ComparatorChallenges/BorsukNine.lean) (github.com, code)
4. [openai/math: OAI/Geometry/Borsuk/Main.lean (ℝ⁹ form)](https://github.com/openai/math/blob/main/lean/OAI/Geometry/Borsuk/Main.lean) (github.com, code)
5. [oai-math-revised: Comparator run of 25–26 Oct (results.tsv)](https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv) (github.com, code)
6. [acprk/openai-math-followups: Borsuk numbers of projective-space projector sets (10 Oct 2026)](https://github.com/acprk/openai-math-followups/blob/main/results/03-borsuk.md) (github.com, code)
7. [Formal Conjectures PR #6898: mark problems solved in the OpenAI math release (merged 11 Oct)](https://github.com/google-deepmind/formal-conjectures/pull/6898) (github.com, code)
8. [Tao et al., optimization constants: constant 28a (Borsuk)](https://teorth.github.io/optimizationproblems/constants/28a.html) (teorth.github.io, docs)
9. [Kalai (2015): Some old and new problems in combinatorial geometry I: Around Borsuk's problem](https://arxiv.org/abs/1505.04952) (arxiv.org, paper)
10. [Hexagon 2610.00163: An exponential improvement for Borsuk's problem (AI-generated, curated by Andriy Prymak)](https://hexagonmath.org/abs/2610.00163) (hexagonmath.org, paper)
11. [oai-math-revised: community notes on the Borsuk manuscript](https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026/AGENT_NOTES.md) (github.com, discussion)
12. [Gil Kalai: Progress on Borsuk's Problem and Answer to TYI 60 (21 Oct 2026)](https://gilkalai.wordpress.com/2026/10/21/progress-on-borsuks-problem-and-answer-to-tyi-60/) (gilkalai.wordpress.com, discussion)
13. [Gábor Damásdi's comment: every finite subset is 8-colourable (21 Oct 2026)](https://gilkalai.wordpress.com/2026/10/21/progress-on-borsuks-problem-and-answer-to-tyi-60/#comment-100326) (gilkalai.wordpress.com, discussion)
14. [Gil Kalai: Test Your Human Intuition 60: The Finite and Infinite Borsuk's Problem (11 Oct 2026)](https://gilkalai.wordpress.com/2026/10/11/test-your-intuition-60-the-finite-and-infinite-borsuks-conjecture/) (gilkalai.wordpress.com, discussion)
15. [Gil Kalai: Updates: Sharing AI progress on mathematics (7 Oct 2026)](https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/) (gilkalai.wordpress.com, discussion)
16. [Erdős Problem a Day #505: Lean replay and an eleven-piece certificate (20 Oct 2026, AI-run)](https://erdosproblemaday.com/day/505-openai-borsuk-fails-in-dimension-nine-refereed) (erdosproblemaday.com, discussion)
17. [Wikipedia: Borsuk's conjecture](https://en.wikipedia.org/wiki/Borsuk%27s_conjecture) (en.wikipedia.org, discussion)
18. [Latent Seminar: Rank-one projectors and Borsuk's covering problem (exposition)](https://lowrank.github.io/latent-seminar/OpenAI-Math/156-borsuk-nine) (lowrank.github.io, discussion)

## Changes

- 2026-10-29 (filed): Created (manuscript PDF, BorsukNine.lean, formalization.yaml, oai-math-revised AGENT_NOTES, Kalai's posts of 7, 11 and 21 Oct and their comments via the WordPress API, Erdős Problem a Day #505 and its About page, acprk note, Formal Conjectures PR #6898, Tao's table 28a, Wikipedia wikitext, Kalai's 2015 survey, Hexagon 2610.00163 and its PDF; HN and web searches found no other expert comment)

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