Reflexive polytopes conjecture disproved, GPT-6.1 Sol
Awaiting review
The takeaway
Johannes Hofscheier and Benjamin Nill (arXiv 2611.08819) give counterexamples to Nill’s own 2004 conjecture that a d-dimensional reflexive polytope has at most 6^{d/2} vertices, in every dimension d ≥ 8, and show the count can be doubly exponential in d.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- 3 of 5
- Last verified
- 9 November 2026
Your AI and this story
- GPT-6 Astra193 days after its cutoff
- Claude Opus 5.5132 days after its cutoff
- Gemini 4 Argon223 days after its cutoff
- Grok 4.7162 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 132 days before it.
Key facts
- Conjecture (Nill, 2004): at most 6^{d/2} vertices; false for every d ≥ 8
- Counterexamples in dimensions 8 and 9, built from an explicit 3-dimensional lattice polytope with 21 points; checkable with standard software
- Large d: vertex (equivalently facet) count can be doubly exponential, as large as theoretically possible; the number-theoretic step originally used OpenAI’s solution of Erdős’s short Egyptian fractions conjecture, now replaced by Bloom and van Doorn’s human-written Hexagon paper
- AI statement: ‘The example, the lemmata and the general result were found in middle of October 2026 using GPT-6.1 Sol based upon prompts with ideas by the first author originally aimed at proving rather than disproving Conjecture 1.1’
What happened
In arXiv 2611.088191 (9 Nov 2026), Hofscheier and Nill report that GPT-6.1 Sol, prompted with an approach meant to prove the conjecture, produced counterexamples in dimensions 8 and 9 and a general construction. The authors write that the doubly exponential growth is “rather surprising”. Reflexive polytopes correspond to Gorenstein toric Fano varieties.
Why it matters
The conjecture’s own author confirms the disproof, and the low-dimensional counterexamples are explicit, which makes the claim easy to check. Unrefereed.
What is disputed or not yet verified
| Verification | Preprint; explicit examples checkable by computer |
|---|
Sources
1 source from 1 site. Numbers match the chips in the text.
1 source: 1 primary
Primary
Changes
- Filed