GPT-5.6 Sol Ultra Proves Cycle Double Cover Conjecture
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- OpenAI attributes the proof of the Cycle Double Cover Conjecture entirely to its GPT-5.6 'Sol Ultra' model, with Codex (using GPT-5.6 Sol) credited for the writeup
- The conjecture, posed independently by Tutte, Itai and Rodeh, Szekeres, and Seymour, asserts that every finite bridgeless undirected graph admits a collection of cycles covering every edge exactly twice
- The proof reduces to loopless cubic graphs, applies the nowhere-zero 8-flow result (Kilpatrick and Jaeger) together with Tutte's group-flow theorem, and resolves the remaining combinatorial obstruction via a linear-algebra argument over Γ = F³₂
- Prior partial results had covered planar graphs (Jaeger), 3-edge-colourable cubic graphs (Szekeres), and bridgeless graphs with no Petersen subdivision (Alspach, Goddyn, and Zhang), leaving snarks as the remaining obstacle
- The key technical step converts a nowhere-zero Γ-flow into an edge-labeling by two-element subsets of Γ such that each element appears zero or two times at every vertex — a relaxed variant of a proper 3-edge-colouring
Why it matters: An open graph-theory conjecture that resisted decades of partial progress — including named results by Jaeger, Szekeres, and Alspach–Goddyn–Zhang — now carries an alleged AI-authored proof credited entirely to GPT-5.6 Sol Ultra. If the argument survives peer scrutiny, working graph theorists lose a central open target and OpenAI gains a hard-math benchmark well beyond product demos and contest problems.
