{"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":"codex-gemma-dda1882f352a","candidate_categories":["metaepi_narrow"],"consensus_categories":[],"category_scores_codex":[0.0005957261,0.0002864472,0.0008300634,0.0002763572,0.00008088731,0.00002483039,0.00115387,0.0001676267,0.0001338567],"category_scores_gemma":[0.001864815,0.0002523393,0.0002736579,0.0003060083,0.0001934372,0.0001385419,0.0005418709,0.0001633772,0.000006439911],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00003533322,"about_ca_system_score_gemma":0.00007439211,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.0000342438,"about_ca_topic_score_gemma":0.0001052521,"domain_scores_codex":[0.9977233,0.0000352337,0.00105493,0.0002708121,0.0005113005,0.0004043868],"domain_scores_gemma":[0.9964793,0.0008046654,0.0005493116,0.001878265,0.0002202938,0.00006819922],"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":[0.00001810979,0.0007176638,0.0006506609,0.002880374,0.0001909321,0.000002353105,0.0009719915,0.00001542733,0.002602737,0.9409736,0.05027933,0.0006968136],"study_design_scores_gemma":[0.001008292,0.00009506468,0.0001719358,0.000715087,0.0001204314,0.000002593356,0.001418295,0.00104354,0.06359557,0.9308033,0.0008184121,0.0002074674],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.8571978,0.000638911,0.08020481,0.001038347,0.0003975664,0.0007467981,0.000331256,0.0001930695,0.05925143],"genre_scores_gemma":[0.8593113,0.00006244486,0.136009,0.00006267443,0.00005702695,0.00003252731,0.000009276098,0.00006824797,0.004387533],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.06099283,"threshold_uncertainty_score":0.9999929,"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."}}