{"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":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.003354953,0.0006007742,0.000865993,0.0020084,0.001552078,0.001876959,0.001328795,0.0008980446,0.007075842],"category_scores_gemma":[0.01220379,0.0003527448,0.00108545,0.001442899,0.003776328,0.00590084,0.004693937,0.00435319,0.001444823],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.001161914,"about_ca_system_score_gemma":0.0008511202,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.001105346,"about_ca_topic_score_gemma":0.0006030906,"domain_scores_codex":[0.9981869,0.0005174385,0.00009713586,0.0003632232,0.0006313554,0.0002039439],"domain_scores_gemma":[0.9958913,0.00222713,0.0002228516,0.0003996842,0.001018485,0.0002404759],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.00002856903,0.00001097431,0.0002863095,0.00004977787,0.000006609484,0.00005764397,0.0001288753,0.003028075,0.0007896414,0.9861558,0.001301673,0.008156181],"study_design_scores_gemma":[0.00001393776,0.00006288589,0.000560342,0.00005294397,0.00001068342,0.0001798785,0.0001053378,0.04318753,0.001472225,0.9422646,0.01206256,0.00002705076],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"methods","genre_gemma":"empirical","genre_scores_codex":[0.09223166,0.003217783,0.7763734,0.004178453,0.0005584389,0.0001406565,0.0007382037,0.0003129032,0.1222485],"genre_scores_gemma":[0.806823,0.004162342,0.1457758,0.002283643,0.000888733,0.0005251426,0.0006646609,0.0004604195,0.0384163],"genre_candidate":"empirical","genre_consensus":null,"teacher_disagreement_score":0.007075842,"threshold_uncertainty_score":0.02367109,"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."}}