{"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":"9f7d6c51e37b4088becf9e35d33cc33c","problem_id":"ebb83cdc-3fec-4974-84a9-c5dccb942b46","problem_name":"Twin prime density — Hardy-Littlewood conjecture verification","domain":"Number Theory","statement":"For all x >= 1000, the actual count of twin prime pairs up to x strictly exceeds the standard Hardy-Littlewood prediction (2*C2*x/ln(x)^2) but remains bounded above by the prediction augmented with a specific second-order correction term: 2*C2*x/(ln(x)^2 - 1.5*ln(x)). This implies the convergence to the asymptotic limit is approached from below with a predictable logarithmic lag.","status":"falsified","url":"https://assignee.net/math#result-9f7d6c51e37b4088becf9e35d33cc33c","doi":"10.5281/zenodo.20573415"},"verification":{"state":"FALSIFIED","label":"Falsified","proof_claim":false,"method":"python_computation","result":"falsified","n_cases":0,"counterexample_available":true,"cpu_seconds":0.06,"lean4_source_public":false},"artifact_set":[{"type":"manifest","label":"Math result manifest","url":"https://assignee.net/math/9f7d6c51e37b4088becf9e35d33cc33c/manifest.json","format":"application/json"},{"type":"report","label":"Public report PDF","url":"https://assignee.net/math/9f7d6c51e37b4088becf9e35d33cc33c/paper.pdf","format":"application/pdf"},{"type":"external_record","label":"DOI","url":"https://doi.org/10.5281/zenodo.20573415","format":"text/html"}],"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."]}