---
id: "2026-07-17-cycle-double-cover-conjecture-proved"
url: "https://postcutoff.com/e/2026-07-17-cycle-double-cover-conjecture-proved/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-07-17"
date_precision: day
category: science
importance: 5
confidence: medium
status: [Result confirmed]
verification: Expert-checked
sources: 4
editor: human editor
human_review: null
version: "2026-10-19"
---

As of: 2026-11-13 13:49 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/2026-07-17-cycle-double-cover-conjecture-proved/

# GPT-5.6 Sol Ultra proves the 50-year-old cycle double cover conjecture

In mid-July 2026 OpenAI released a preprint crediting GPT-5.6 Sol Ultra, 'in less than an hour', with a proof of the cycle double cover conjecture (Szekeres 1973, Seymour 1979): every bridgeless graph has a collection of cycles covering each edge exactly twice. Independent expositions by graph theorists Sang-il Oum and Jim Geelen followed.

## Key facts
- OpenAI released the proof as a PDF (cdn.openai.com); arXiv 2607.15399 is Jim Geelen's 3-page notes 'OpenAI's proof of the Cycle Double Cover Theorem' ('my attempt to clarify, to myself, the OpenAI proof … This write-up offers no new insights'); Oum's exposition is arXiv 2607.16356 (17 Jul 2026)
- Proof attributed entirely to GPT-5.6 Sol Ultra; the write-up was done with Codex
- Independent checks and expositions by Sang-il Oum and Jim Geelen; a public Lean formalisation is reported but not verified here

## What happened

OpenAI published a proof of the cycle double cover conjecture that it attributed wholly to its top model. Leading graph theorists independently re-expounded and checked the argument within days.

## Why it matters

Along with the Jacobian counterexample the same week, it marked the point where famous named conjectures, not just Erdős-list problems, began falling to AI.

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 78 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 17 days after its cutoff
- Gemini 4 Hydrogen (training cutoff March 2026): 108 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 47 days after its cutoff

## Sources

1. [OpenAI: cycle double cover proof (PDF)](https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_proof.pdf) (cdn.openai.com, paper)
2. [Jim Geelen: OpenAI's proof of the Cycle Double Cover Theorem (arXiv 2607.15399)](https://arxiv.org/abs/2607.15399) (arxiv.org, paper)
3. [Sang-il Oum: exposition of the proof (arXiv 2607.16356)](https://arxiv.org/abs/2607.16356) (arxiv.org, paper)
4. [AI Weekly: OpenAI attributes cycle double cover proof to GPT-5.6 Sol Ultra](https://aiweekly.co/alerts/openai-attributes-cycle-double-cover-proof-to-gpt-56-sol-ultra) (aiweekly.co, press)

## Changes

- 2026-10-19: Fixed a misattribution: arXiv 2607.15399 is Jim Geelen's notes on OpenAI's proof, not an OpenAI preprint (key fact and link title corrected from the arXiv metadata); verification names the independent checkers
- 2026-09-29 (filed): Created

## Related

- 2026-08-01: [OpenAI's unreleased 'Astra' model claims ten advances in maths and theoretical CS, with Lean proofs](https://postcutoff.com/e/2026-08-01-openai-astra-ten-advances/index.md)
- 2026-07-09: [OpenAI broadly releases GPT-5.6 after government-gated preview](https://postcutoff.com/e/2026-07-09-gpt-5-6-sol-terra-luna/index.md)