{"schema":"vela.problem-packet.v0.1","problem":617,"statement":"Let $r\\geq 3$. If the edges of $K_{r^2+1}$ are $r$-coloured then there exist $r+1$ vertices with at least one colour missing on the edges of the induced $K_{r+1}$.","status":"open","seam":"raw","closureRoutes":[{"type":"witness","verifierKind":"unsat_cert","note":"LRAT certificate that no balanced 5-coloring of K_26 exists. The frozen checker now supports RUP and RAT steps (deletion lines still unsupported); symmetry-breaking clauses must be included in the CNF and justified as RAT additions."},{"type":"counterexample_witness","verifierKind":"balanced_coloring","note":"A balanced 5-coloring of K_26 itself: every 6-subset sees all 5 colors. Frozen verifier checks all C(26,6)=230,230 subsets directly. SHIPPED in vela-verify (golden: the pentagon 2-coloring of K_5; mutations in corpus/invalid)."},{"type":"formal_proof","verifierKind":"lean","note":"Lean patch building clean under the math CI profile (no sorry, no new axioms)"},{"type":"obstruction_report","verifierKind":"review","note":"precise, artifact-backed reason a route cannot work"}],"obligations":[{"findingId":"vf_da8d51a6cae5ec11","banked":"K25 balanced-coloring construction known; the K26 finite case is the first open case and reduces to a SAT instance","open":"produce the LRAT/RAT certificate for the K26 SAT encoding and re-check it with `vela reproduce` (unsat_cert); the symmetry-breaking step needs RAT, beyond the current RUP checker.","dependents":1,"lease":null}],"attestations":[],"attempts":[],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}