Cycle Double Cover Conjecture — GPT 5.6 Sol Ultra


Lab	OpenAI
Model	GPT 5.6 Sol Ultra
Field	Mathematics — graph theory
Result (yield)	Complete proof of the Cycle Double Cover Conjecture, a 1973-era open problem (Szekeres 1973 / Seymour 1979). Verified by adversarial multi-agent audit.
Date	2026
Source (PDF)	https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_prompt.pdf

Why it's battle-tested

The prompt produced a full, unconditional proof of the Cycle Double Cover Conjecture for all finite bridgeless loopless multigraphs. The document above is OpenAI's verbatim release of the prompt as given.

Key prompt techniques

Precise, self-contained problem statement — all definitions (multigraph, bridge, cycle, cycle double cover) are given explicitly; no terms left to interpretation.
Explicit completion criteria — partial progress, special graph classes, bounded-length variants, reductions to other unproved conjectures, and fixed-size computational verification are named and excluded as insufficient.
Anti-circularity guards — forbids "proving" the conjecture via equivalent statements.
Aggressive dynamic multi-agent orchestration — up to 64 concurrent agents, portfolio diversity, independence preservation, approach-family registry, adversarial verification.
Anti-persuasion-by-optimism — status reports and "routine" hand-waves are rejected; concrete lemmas, constructions, equations or counterexamples are demanded.
Minimum time budget — "Spend at least 8 hours on this before even thinking of returning."
Scoped web use — background literature only; no searching for the solution itself, and no declaring the problem open.

Hidden user prompts (unpublished)

The task prompt above is complete as the agent message. However, these user-authored inputs to the run were never published and are therefore missing here:

The harness system prompt of OpenAI's internal agent runtime.
The "multiagent v2" orchestration configuration (64-agent setup, sub-agent briefs, tool schemas).

A reproduction must supply equivalents; yield is not attributable to the task prompt alone.

Files

prompt.md — the full prompt, nothing else. Machine-extracted from the source PDF and stripped of the PDF's title/abstract framing; the file begins at the prompt's own "Current task statement" heading and ends at the prompt's final line. Copy the whole file and send it as one message. (The PDF's typography is authoritative for exact glyphs.)