{"id":"W4414163255","doi":"10.1080/00029890.2025.2545706","title":"A Lebesgue Integral Variant of the Additive Square Problem","year":2025,"lang":"en","type":"article","venue":"American Mathematical Monthly","topic":"semigroups and automata theory","field":"Computer Science","cited_by":0,"is_retracted":false,"has_abstract":true,"ca_institutions":"","funders":"Natural Sciences and Engineering Research Council of Canada","keywords":"Lebesgue integration; Square (algebra); Riemann integral; Lebesgue–Stieltjes integration; Lebesgue measure; Mean square","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":[0.0002495692,0.0001508566,0.000331126,0.00006640918,0.00007812224,0.00005321595,0.001197858,0.00003187103,0.00003590946],"category_scores_gemma":[0.0002169785,0.00009002468,0.0001567338,0.0008222562,0.0004425859,0.0001295242,0.00045842,0.000164226,0.00002687577],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00003390263,"about_ca_system_score_gemma":0.00009418098,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.00009998332,"about_ca_topic_score_gemma":0.000007393548,"domain_scores_codex":[0.9987665,0.0001293318,0.0003364749,0.0002758494,0.0002407358,0.0002510666],"domain_scores_gemma":[0.9984185,0.0005122473,0.000177744,0.0007669857,0.00007106565,0.00005347304],"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":[0.000004504101,0.0001370156,0.00003727394,0.00003974156,0.00003450569,0.000004940215,0.0007885576,0.00001162673,0.0001249777,0.9786966,0.001452677,0.01866762],"study_design_scores_gemma":[0.0001962388,0.0001218569,0.002225103,0.0004069227,0.00002628391,0.000008348562,0.0008175371,0.03971376,0.002046834,0.9530154,0.001253223,0.0001684205],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"methods","genre_gemma":"empirical","genre_scores_codex":[0.01574134,0.00009133489,0.6936024,0.00582723,0.0002154699,0.0007187251,0.00004741205,0.0002866618,0.2834694],"genre_scores_gemma":[0.9359956,0.000001852801,0.06260742,0.0007964675,0.00001555661,0.00005639929,0.000001057084,0.000008314246,0.0005172872],"genre_candidate":"empirical","genre_consensus":null,"teacher_disagreement_score":0.9202543,"threshold_uncertainty_score":0.36711,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.00493977295177866,"score_gpt":0.2273676459635397,"score_spread":0.222427873011761,"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."}}