{"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":"a7f5bd3a2fcc4f22a428809c6bd91945","problem_id":"d571909f-3a0e-4875-b12f-1a529fb1546d","problem_name":"Primes of form n^2+1 — density and distribution","domain":"Number Theory","statement":"For the sequence of primes of the form p = n^2 + 1, let S(x) be the set of such primes less than or equal to x. Define the 'quadratic gap ratio' for a prime p = n^2 + 1 (where n > 1) as R(p) = (p_next - p) / (2n), where p_next is the next prime in the sequence S. The conjecture states that the limit inferior of R(p) as p approaches infinity is strictly less than 0.5. Specifically, there exist infinitely many indices n such that the gap to the next prime of the form m^2+1 is smaller than n (i.e.,","status":"falsified","url":"https://assignee.net/math#result-a7f5bd3a2fcc4f22a428809c6bd91945","doi":null},"verification":{"state":"FALSIFIED","label":"Falsified","proof_claim":false,"method":"python_computation","result":"falsified","n_cases":0,"counterexample_available":true,"cpu_seconds":1.95,"lean4_source_public":false},"artifact_set":[{"type":"manifest","label":"Math result manifest","url":"https://assignee.net/math/a7f5bd3a2fcc4f22a428809c6bd91945/manifest.json","format":"application/json"},{"type":"report","label":"Public report PDF","url":"https://assignee.net/math/a7f5bd3a2fcc4f22a428809c6bd91945/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."]}