Ziegler's Cross-Polytope Conjecture (simplicial 0/1-polytopes)
Ziegler proved every simplicial -dimensional 0/1-polytope has at most vertices, and asked whether attaining vertices forces central symmetry (i.e. a 0/1 cross-polytope). Known true for ; open since ~2000.
- Result
- Disproved
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Combinatorics, Discrete Geometry
- Posed by
- Günter M. Ziegler
- Year posed
- 2000
- Years open
- 26y
- Solved
- 2026-06-30
- Model
- DeepSeek V4 Flash, GLM 5.2
- Vendor
- DeepSeek / Zhipu AI
- Collaborators
- Volker Kaibel, Sebastian Pokutta
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 15 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
An agentic research framework (locally-deployed open-weights DeepSeek V4 Flash + GLM 5.2, augmented with reflection prompts) first produced a flawed proof that the conjecture holds, then attempted a Lean 4 formalization as verification. The formalization failed, and from that failure the agent extracted the combinatorial condition that yielded an explicit counterexample: 14 vertices in whose convex hull is simplicial but not centrally symmetric.
Verification
arXiv preprint 2606.31640 (30 Jun 2026) by Volker Kaibel and Sebastian Pokutta. The counterexample is explicit and computer-checkable (exhaustive enumeration finds exactly five such non-centrally-symmetric polytopes in dimension 7, of two combinatorial types); a domain-expert preprint, not yet peer-reviewed. Notably found with locally-run open-weights models, not closed frontier LLMs.
Source
- PaperarXiv:2606.31640