Report: GPT-5.6 Sol Ultra produces proof of the Cycle Double Cover Conjecture [pdf]

{ "title": "OpenAI's GPT-5.6 Sol Proves 50-Year-Old Math Conjecture", "slug": "openai-gpt-5-6-sol-cycle-double-cover-conjecture-proof", "summary": "OpenAI's GPT-5.6 Sol Ultra has produced a proof of the Cycle Double Cover Conjecture, a 50-year-old problem in graph theory. The model, launching July 9, 2026, also sets new coding benchmarks and introduces ChatGPT Work. This marks a significant milestone in AI-driven scientific discovery, demonstrating frontier reasoning capabilities alongside practical enterprise tools.", "meta_description": "OpenAI's GPT-5.6 Sol Ultra proves the Cycle Double Cover Conjecture, a 50-year-old math problem. The model also sets new coding benchmarks and launches ChatGPT Work.", "content": "

AI Achieves a Mathematical Milestone

OpenAI's GPT-5.6 Sol Ultra has produced a formal proof of the Cycle Double Cover Conjecture, a problem in graph theory that has stood for over 50 years. The proof, published as a PDF by OpenAI, demonstrates the model's ability to perform complex, multi-step mathematical reasoning. This achievement places GPT-5.6 at the forefront of AI-driven scientific discovery, a capability that has long been a holy grail for the industry.

The conjecture, independently posed by William Tutte, Alon Itai, and others, asserts that every bridgeless undirected graph contains a collection of cycles that covers each edge exactly twice. The proof, attributed entirely to GPT-5.6 Sol Ultra and written with Codex, leverages advanced concepts in graph theory and linear algebra, including the 8-flow theorem and a novel labeling argument.

How the Proof Works

The proof, detailed in a 3-page PDF released by OpenAI, follows a standard reduction to cubic graphs. It then uses a nowhere-zero 8-flow, a known result, to assign elements of the group F₂³ to edges. The key innovation is a lemma that converts this flow into a labeling of edges with two-element subsets of the group, satisfying a local parity condition.

This labeling is then shown to directly yield a cycle double cover. The final step involves solving a system of linear equations over F₂, which the model proves always has a solution using a duality argument. The entire proof is attributed to GPT-5.6 Sol Ultra, with the writeup assisted by Codex.

GPT-5.6 Family: Sol, Terra, and Luna

OpenAI launched the GPT-5.6 family on July 9, 2026, with three tiers: Sol (flagship), Terra (balanced), and Luna (lightweight). The flagship Sol model is priced at $5 per million input tokens and $30 per million output tokens, making it significantly more cost-effective than Anthropic's Claude Fable 5, which costs roughly one-third more per task.

Early benchmarks show Sol scoring 80 on the Coding Agent Index, 2.8 points above Fable 5, while using less than half the output tokens and taking less than half the time. On the Artificial Analysis Intelligence Index, Sol scores 59, just 0.9 points behind Fable 5's 59.9. CEO Sam Altman stated that Sol is 54% more token efficient on agentic coding tasks.

New Reasoning Modes and Desktop Consolidation

OpenAI introduced two new reasoning modes for Sol: Max mode, which allocates additional compute for complex problems, and Ultra mode, which coordinates four agents in parallel for demanding workflows. These modes are designed to handle the kind of multi-step reasoning required for tasks like the Cycle Double Cover proof.

In a strategic move, OpenAI merged the Codex app into a revamped ChatGPT desktop application, creating a unified " , "superapp"