What was already known
The planar and K₅-minor-free cases were known; the general finite multigraph case remained open.
AI involvement
Ulyanov credits GPT-5.6 Pro with essentially the proof after two prompts and a few human hints. Ulyanov wrote the explanation and formalized the circuit conclusion.
The author explicitly assigns essentially the proof to GPT-5.6 Pro, while acknowledging limited human hints.
Importance
Resolves the general compatibility conjecture for Eulerian multigraphs.
Limitations
The public Lean endpoint certifies the compatible circuit partition. The paper’s stronger four-colour statement is explained in prose but is not identified as that public endpoint.
Technical verification
The author’s Lean record reports a successful build, standard-axiom audit and comparator check of the ordinary-circuit endpoint under matching multigraph and degree hypotheses. The final circuit-decomposition claim is formally checked; the four-colour intermediate has separate Lean components, and the exact prose strengthening is not identified as the public endpoint.
Independent checking
pinned build, axiom audit and Comparator statement check
- Checker
- Lean kernel and Comparator through Nikolay Ulyanov’s sabidussi-lean project
- Scope
- compatible partition into ordinary connected 2-regular circuits for finite endpoint multigraphs with Euler tour and minimum degree four
- Basis
- Reported by the linked checker
Human understanding
Ulyanov’s §§2–6 explain the parity encoding, local assignments, counting obstruction and extraction of circuits; the author also prepared the formal statement.
Correction and publication history
The assessed conclusion is the compatible circuit partition.
Read this published revision