{"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":"2329816d326f4d73ab059a048631fe9e","problem_id":"d571909f-3a0e-4875-b12f-1a529fb1546d","problem_name":"Primes of form n^2+1 — density and distribution","domain":"Number Theory","statement":"Conjecture: For the sequence of primes of the form n^2+1, let P_k = n_k^2+1 be the k-th such prime. The difference between consecutive roots, d_k = n_{k+1} - n_k, satisfies d_k <= floor(sqrt(2)*sqrt(n_k)) for all k >= 2. This refines the expected Cramer-type gap behavior for this sparse subsequence of primes, suggesting the gaps grow no faster than the square root of the index parameter scaled by sqrt(2).","status":"falsified","url":"https://assignee.net/math#result-2329816d326f4d73ab059a048631fe9e","doi":null},"verification":{"state":"FALSIFIED","label":"Falsified","proof_claim":false,"method":"python_computation","result":"falsified","n_cases":0,"counterexample_available":true,"cpu_seconds":1.41,"lean4_source_public":false},"artifact_set":[{"type":"manifest","label":"Math result manifest","url":"https://assignee.net/math/2329816d326f4d73ab059a048631fe9e/manifest.json","format":"application/json"},{"type":"report","label":"Public report PDF","url":"https://assignee.net/math/2329816d326f4d73ab059a048631fe9e/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."]}