{"id":"W4416099375","doi":"10.48550/arxiv.2503.22004","title":"On Opial's Lemma","year":2025,"lang":"en","type":"preprint","venue":"ArXiv.org","topic":"Optimization and Variational Analysis","field":"Computer Science","cited_by":0,"is_retracted":false,"has_abstract":true,"ca_institutions":"","funders":"Natural Sciences and Engineering Research Council of Canada","keywords":"Lemma (botany); Monotone polygon; Convergence (economics); Hilbert space; Limit (mathematics); Weak convergence; Projection (relational algebra)","routes":{"ca_aff":false,"ca_fund":true,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":true},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"codex-gemma-dda1882f352a","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.0001587755,0.0001794978,0.0002249439,0.0002255349,0.0001089143,0.0001429427,0.001082914,0.0001569215,0.0001788659],"category_scores_gemma":[0.0001041744,0.0001689788,0.0001750504,0.0003863633,0.00001506279,0.0001051774,0.001012406,0.0003054353,0.0004003819],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00006516548,"about_ca_system_score_gemma":0.0002125771,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.00006856106,"about_ca_topic_score_gemma":0.000009550911,"domain_scores_codex":[0.9986849,0.00007304643,0.0002544588,0.0005705128,0.0002615679,0.0001554624],"domain_scores_gemma":[0.9986981,0.0001122422,0.0001404197,0.0008287008,0.000156664,0.00006384256],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"simulation_or_modeling","study_design_scores_codex":[0.000008977087,0.0002291087,0.03930999,0.00007000326,0.0002999573,0.00001340967,0.0003447715,0.1681497,0.00001373451,0.7747517,0.0128851,0.003923609],"study_design_scores_gemma":[0.0007207027,0.00007393039,0.1558991,0.0002917323,0.000135955,0.000001791379,0.0000102978,0.7584415,0.0005290211,0.04898788,0.03382141,0.001086676],"study_design_candidate":"simulation_or_modeling","study_design_consensus":null,"genre_codex":"methods","genre_gemma":"empirical","genre_scores_codex":[0.01917122,0.00006308841,0.9298739,0.01271021,0.001872105,0.0002008704,0.00002405264,0.0003257181,0.03575886],"genre_scores_gemma":[0.9187124,0.0001195236,0.03637316,0.007959381,0.0003934046,0.00006597485,0.0001491658,0.00001488003,0.03621215],"genre_candidate":"methods","genre_consensus":null,"teacher_disagreement_score":0.8995411,"threshold_uncertainty_score":0.6890756,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.03972902393439452,"score_gpt":0.2813416872812596,"score_spread":0.2416126633468651,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}