---
id: "2026-11-09-nill-reflexive-polytopes-conjecture-disproved-gpt-6-1-sol"
url: "https://postcutoff.com/e/2026-11-09-nill-reflexive-polytopes-conjecture-disproved-gpt-6-1-sol/"
as_of: "2026-11-09T21:13:00+01:00"
date: "2026-11-09"
date_precision: day
category: science
importance: 3
confidence: medium
status: [Awaiting review]
verification: Preprint only
sources: 1
editor: human editor
human_review: null
version: "2026-11-09"
---

As of: 2026-11-09 21:13 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/2026-11-09-nill-reflexive-polytopes-conjecture-disproved-gpt-6-1-sol/

# Reflexive polytopes conjecture disproved, GPT-6.1 Sol

Full title: Nill's 2004 conjecture on vertices of reflexive polytopes disproved with GPT-6.1 Sol: doubly exponentially many vertices possible

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. "These results were found using GPT-6.1 Sol." The examples can be checked with standard polyhedral software.

## 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 2611.08819](https://arxiv.org/abs/2611.08819) (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

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 193 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 132 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 223 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 162 days after its cutoff

## Sources

1. [Hofscheier, Nill: Reflexive polytopes can have doubly exponentially many vertices (arXiv 2611.08819)](https://arxiv.org/abs/2611.08819) (arxiv.org, paper)

## Changes

- 2026-11-09 (filed): Created

## Related

- 2026-11-09: [AI-assisted results, 9 Nov: arXiv and Hexagon](https://postcutoff.com/e/2026-11-09-ai-assisted-results-november-9-listing/index.md)