What is the Cycle Double Cover Conjecture?

The Cycle Double Cover Conjecture asks whether every bridgeless graph has a set of cycles such that each edge lies in exactly two of them. The question dates to Tutte, Szekeres, and Seymour in the 1970s and is central to topological graph theory. GPT-5.6 Sol Ultra's proof answers it in the positive.

How did GPT-5.6 Sol Ultra produce the proof?

GPT-5.6 Sol Ultra spawned 64 subagents that were trained to cooperate and share context during a single task, coordinating for just under one hour. The proof itself came from Sol Ultra; Codex with GPT-5.6 Sol handled the writeup. OpenAI released both the proof PDF and the prompt used to elicit it.

Has the proof been verified by mathematicians?

The Cycle Double Cover proof has not yet been formally verified. It is not written in Lean, and mathematicians on Hacker News and X are cautiously reading it but have not signed off. OpenAI released the prompt alongside the proof so the community can attempt to reproduce and check the argument.

How does this compare to OpenAI's Erdős result from May?

In May 2026 an OpenAI reasoning model disproved a 1946 Erdős unit-distance conjecture. The Cycle Double Cover proof is the first public showcase of the Sol Ultra subagent architecture, which made GA one day earlier, and tackles a much older graph-theory conjecture using 64 cooperating agents instead of a single reasoning trace.

OpenAI · 2026-07-10 · major

GPT-5.6 Sol Ultra proves Cycle Double Cover Conjecture — 64 subagents, one hour

OpenAI released a proof of the 50-year-old Cycle Double Cover Conjecture written by GPT-5.6 Sol Ultra. The Ultra tier ran 64 cooperating subagents for just under an hour to produce the proof.

OpenAI says GPT-5.6 Sol Ultra proved the 50-year-old Cycle Double Cover Conjecture using 64 cooperating subagents.

Quick facts

Maker	OpenAI
Model	GPT-5.6 Sol Ultra
Problem	Cycle Double Cover Conjecture (open since the 1970s)
Method	64 cooperating subagents, one Ultra run
Time	Under one hour
Writeup	GPT-5.6 Sol via Codex
Verification	Not yet formalized in Lean; awaiting peer review

What is it?

OpenAI published a proof of the Cycle Double Cover Conjecture, a 1970s open problem in graph theory that asks whether every bridgeless graph has a collection of cycles covering every edge exactly twice. The proof was produced entirely by GPT-5.6 Sol Ultra and typeset by Codex with GPT-5.6 Sol.

How does it work?

The Sol Ultra tier, GA'd on July 9, spawns subagents that are trained to cooperate and communicate during a task instead of running as independent parallel searches. On this problem the model ran 64 subagents for just under an hour, delegating sub-lemmas and synthesizing them into a single argument. OpenAI released the exact prompt alongside the PDF so third parties can try to reproduce the run.

Why does it matter?

The Cycle Double Cover result is the first public showcase of Sol Ultra's cooperating-subagent architecture on a named open problem. If the proof holds up, it turns AI-driven mathematics into a second data point after May's Erdős unit-distance disproof rather than a one-off. Mathematicians are cautiously interested but note the proof is not yet in Lean; verification is the next step.

Who is it for?

graph theorists, math researchers, AI capability watchers

Frequently asked questions

What is the Cycle Double Cover Conjecture?: The Cycle Double Cover Conjecture asks whether every bridgeless graph has a set of cycles such that each edge lies in exactly two of them. The question dates to Tutte, Szekeres, and Seymour in the 1970s and is central to topological graph theory. GPT-5.6 Sol Ultra's proof answers it in the positive.
How did GPT-5.6 Sol Ultra produce the proof?: GPT-5.6 Sol Ultra spawned 64 subagents that were trained to cooperate and share context during a single task, coordinating for just under one hour. The proof itself came from Sol Ultra; Codex with GPT-5.6 Sol handled the writeup. OpenAI released both the proof PDF and the prompt used to elicit it.
Has the proof been verified by mathematicians?: The Cycle Double Cover proof has not yet been formally verified. It is not written in Lean, and mathematicians on Hacker News and X are cautiously reading it but have not signed off. OpenAI released the prompt alongside the proof so the community can attempt to reproduce and check the argument.
How does this compare to OpenAI's Erdős result from May?: In May 2026 an OpenAI reasoning model disproved a 1946 Erdős unit-distance conjecture. The Cycle Double Cover proof is the first public showcase of the Sol Ultra subagent architecture, which made GA one day earlier, and tackles a much older graph-theory conjecture using 64 cooperating agents instead of a single reasoning trace.

Try it

https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_proof.pdf