{"id":"W4406279647","doi":"10.1007/s00220-024-05166-5","title":"Magnetic Flatness and E. Hopf’s Theorem for Magnetic Systems","year":2025,"lang":"en","type":"article","venue":"Communications in Mathematical Physics","topic":"Geometry and complex manifolds","field":"Mathematics","cited_by":5,"is_retracted":false,"has_abstract":true,"ca_institutions":"","funders":"Natural Sciences and Engineering Research Council of Canada; Fundação de Amparo à Pesquisa do Estado de São Paulo; Deutsche Forschungsgemeinschaft; Instituto Serrapilheira; National Science Foundation","keywords":"Flatness (cosmology); Holomorphic function; Curvature; Mathematics; Mathematical analysis; Sectional curvature; Physics; Pure mathematics; Geometry; Scalar curvature; Quantum mechanics","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":[0.0005837907,0.0001922638,0.0004089876,0.0001145676,0.0002011929,0.0001011641,0.0008893413,0.00009685181,0.00002516053],"category_scores_gemma":[0.0005394666,0.000180168,0.00007662964,0.0005055007,0.0002930685,0.00008093486,0.0004829366,0.0002176988,0.00001941828],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00004159405,"about_ca_system_score_gemma":0.00003647543,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.000004019164,"about_ca_topic_score_gemma":0.000005797008,"domain_scores_codex":[0.9987116,0.0001647324,0.0005124792,0.0002226245,0.0001295616,0.000258992],"domain_scores_gemma":[0.9946205,0.00325628,0.00008615427,0.001876367,0.0001120797,0.00004861719],"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":[0.000006999949,0.0003699306,0.00006825609,0.0007667268,0.00001387665,2.48743e-7,0.0002692315,0.00000744345,0.00005852882,0.9919733,0.0004997499,0.005965673],"study_design_scores_gemma":[0.0004699977,0.00005274941,0.0001663478,0.0003105865,0.00007475633,0.000003507723,0.0005007098,0.04420289,0.00002873551,0.9526661,0.001356722,0.0001668761],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"methods","genre_gemma":"empirical","genre_scores_codex":[0.05562267,0.008441855,0.7468724,0.003305489,0.0003356198,0.004869024,0.00005918755,0.0004311616,0.1800625],"genre_scores_gemma":[0.9035113,0.0001152388,0.09290307,0.0000842567,0.00003528441,0.0006945619,0.0000112398,0.00003264782,0.002612443],"genre_candidate":"empirical","genre_consensus":null,"teacher_disagreement_score":0.8478886,"threshold_uncertainty_score":0.7347039,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.0776415015370549,"score_gpt":0.3530329496240395,"score_spread":0.2753914480869846,"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."}}