Source authenticated

The smallest dimension of a counterexample to Borsuk's conjecture: failure in dimension nine

Theorem 1.1: the compact set of rank-one orthogonal projectors $uu^T$ , $u\in\mathbb R^4$ a unit vector, with the Frobenius metric, lies in the nine-dimensional affine space of trace-one symmetric $4\times4$ matrices, has diameter $\sqrt2$ and cannot be covered by ten sets of smaller diameter, so Borsuk's assertion fails in dimension 9 (the record was 63). A corollary extends this to compact counterexamples in every dimension $d\ge9$ . Not shown: the smallest failing dimension (dimensions 4 to 8 remain open) or the exact Borsuk number of the set. The witness is a continuum, not a finite point set.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The smallest dimension of a counterexample to Borsuk's conjecture: failure in dimension nine · Borsuk's conjecture fails in dimension 9

Confidence: Not scored

Registry verification: lean checked · announcement · partial

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: The smallest dimension of a counterexample to Borsuk's conjecture: failure in dimension nine

specialist registry · passed

No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against Borsuk's question. Lean-checked on the release's Comparator challenge BorsukNine (declaration OAI.BorsukNine.main_theorem, listed in lean/formalization.yaml). Its statement was read here: the image of the unit sphere of R^4 under $u\mapsto uu^T$ , inside the Euclidean space of 4x4 matrices, is compact, lies in the trace-one symmetric matrices, has diameter $\sqrt2$ , and admits no cover by ten subsets each of diameter less than $\sqrt2$ . That is the headline; that trace-one symmetric matrices form a nine-dimensional affine space is elementary and not itself stated. Not rebuilt here. The paper says it does not determine the smallest failing dimension.

Source ↗

The smallest dimension of a counterexample to Borsuk's conjecture: failure in dimension nine

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Lean Comparator challenge BorsukNine (OpenAI math release)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Borsuk 1933, Drei Satze uber die n-dimensionale euklidische Sphare

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

Source ↗

Artifacts and verifiers

Lean Comparator challenge BorsukNine (OpenAI math release)

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

Borsuk 1933, Drei Satze uber die n-dimensionale euklidische Sphare evidence for this event

This event attributed to Unreleased internal OpenAI model

The smallest dimension of a counterexample to Borsuk's conjecture: failure in dimension nine evidence for this event

VibeMathed record: The smallest dimension of a counterexample to Borsuk's conjecture: failure in dimension nine evidence for this event

Lean Comparator challenge BorsukNine (OpenAI math release) evidence for this event

The smallest dimension of a counterexample to Borsuk's conjecture: failure in dimension nine parent of this event

Act on this frontier

Verify, challenge, or extend the result.

Help verify Open canonical problem Explore evidence graph