{"id":"W4410447212","doi":"10.1017/fms.2025.39","title":"Free summands of stably free modules","year":2025,"lang":"en","type":"article","venue":"Forum of Mathematics Sigma","topic":"Rings, Modules, and Algebras","field":"Mathematics","cited_by":0,"is_retracted":false,"has_abstract":true,"ca_institutions":"University of British Columbia","funders":"Natural Sciences and Engineering Research Council of Canada","keywords":"Mathematics; Free product; Pure mathematics; Arithmetic; Chemistry; Group (periodic table)","routes":{"ca_aff":true,"ca_fund":true,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.0007198279,0.0006231976,0.0005597582,0.002562434,0.001499695,0.002516575,0.0006320574,0.0004250649,0.008500474],"category_scores_gemma":[0.001580603,0.0004053933,0.0006181231,0.0009469544,0.002520564,0.003068716,0.002272525,0.0009587946,0.001128335],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00144165,"about_ca_system_score_gemma":0.0004859118,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.0009965051,"about_ca_topic_score_gemma":0.0013171,"domain_scores_codex":[0.9990626,0.0001109864,0.00006189799,0.0001401964,0.0004088506,0.0002153957],"domain_scores_gemma":[0.9988899,0.0002943753,0.0001762804,0.00006449636,0.0002258152,0.0003492231],"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":[0.0001664058,0.0001496845,0.004738505,0.0002104949,0.00006562446,0.001951164,0.002037357,0.00245529,0.02137305,0.9490675,0.002671061,0.01511392],"study_design_scores_gemma":[0.00005254431,0.0002237775,0.009803913,0.0001019746,0.00007849115,0.001487563,0.0009301746,0.011948,0.02947595,0.9279159,0.01788032,0.0001014298],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.8884113,0.001233415,0.03062128,0.0004731458,0.0001488862,0.00005258323,0.0003328009,0.0005346254,0.07819197],"genre_scores_gemma":[0.9775374,0.0002218757,0.003128766,0.00006541566,0.0001581963,0.00003011898,0.0002545768,0.00005802436,0.01854554],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.008500474,"threshold_uncertainty_score":0.02843696,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.02368585795954809,"score_gpt":0.2687525261550934,"score_spread":0.2450666681955453,"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."}}