{"schema":"https://assignee.net/schemas/math-result-v1","schema_version":"1.0","contract_version":"math-result-v1.0","schema_documentation":"https://assignee.net/schemas","changelog_url":"https://assignee.net/changelog","publisher":{"name":"Assignee Research","url":"https://assignee.net"},"result":{"id":"843d975d77414a55a0e5bc5c9e03a4ce","problem_id":"6e0ee417-7bc2-4398-b325-71c51b274fee","problem_name":"Ramsey R(5,5) — upper bound improvement","domain":"Ramsey","statement":"In any 2-coloring of the edges of K_43 that contains no monochromatic K_5, there exists no vertex v such that the red degree of v is exactly 21 AND the red neighborhood of v induces a subgraph containing a red triangle. Equivalently, if a vertex has red degree 21 in an extremal R(5,5) coloring on 43 vertices, its red neighborhood must be triangle-free (and thus an independent set of size 21 is impossible, but specifically the induced subgraph has no K_3).","status":"falsified","url":"https://assignee.net/math#result-843d975d77414a55a0e5bc5c9e03a4ce","doi":null},"verification":{"state":"FALSIFIED","label":"Falsified","proof_claim":false,"method":"python_computation","result":"falsified","n_cases":0,"counterexample_available":true,"cpu_seconds":0.08,"lean4_source_public":false},"artifact_set":[{"type":"manifest","label":"Math result manifest","url":"https://assignee.net/math/843d975d77414a55a0e5bc5c9e03a4ce/manifest.json","format":"application/json"},{"type":"report","label":"Public report PDF","url":"https://assignee.net/math/843d975d77414a55a0e5bc5c9e03a4ce/paper.pdf","format":"application/pdf"}],"interpretation":"Computational evidence is not a formal proof. Formal verification is claimed only when public Lean4 source is attached.","limitations":["Python check code, local file paths, and private execution logs are not exposed in public manifests.","Computational evidence reports bounded search only and can be invalidated by later counterexamples.","Formal proof verification requires public Lean4 source; otherwise the record remains a proof attempt or report."]}