{"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":"695e319f6af9452e965e393042bf9b61","problem_id":"3ac6796c-0870-445d-b7cd-474654ab6e29","problem_name":"Primes of form n^2+1 — density conjecture","domain":"Number Theory","statement":"For any integer n >= 2, let P_n be the set of primes of the form k^2+1 where 1 <= k <= n. Let G_n be the maximum gap between consecutive elements in the sorted sequence P_n (defining the first gap as p_1 - 1). Then G_n is strictly less than 2.5 * sqrt(p_max) * ln(p_max), where p_max is the largest prime in P_n. If P_n has fewer than 2 elements, the conjecture holds trivially for that n.","status":"falsified","url":"https://assignee.net/math#result-695e319f6af9452e965e393042bf9b61","doi":null},"verification":{"state":"FALSIFIED","label":"Falsified","proof_claim":false,"method":"python_computation","result":"falsified","n_cases":0,"counterexample_available":true,"cpu_seconds":1.47,"lean4_source_public":false},"artifact_set":[{"type":"manifest","label":"Math result manifest","url":"https://assignee.net/math/695e319f6af9452e965e393042bf9b61/manifest.json","format":"application/json"},{"type":"report","label":"Public report PDF","url":"https://assignee.net/math/695e319f6af9452e965e393042bf9b61/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."]}