{"id":"W7117420770","doi":"10.61091/um125-10","title":"A normal distribution of Raab direction in binomial coefficients triangle","year":2025,"lang":"","type":"article","venue":"Utilitas Mathematica","topic":"Mathematical Dynamics and Fractals","field":"Mathematics","cited_by":0,"is_retracted":false,"has_abstract":true,"ca_institutions":"","funders":"","keywords":"Gaussian binomial coefficient; Binomial coefficient; Binomial distribution; Binomial (polynomial); Distribution (mathematics); Sequence (biology)","routes":{"ca_aff":false,"ca_fund":false,"ca_venue":true,"about_ca":false,"invisible_to_affiliation_only":true},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"codex-gemma-dda1882f352a","candidate_categories":["metaepi_narrow","insufficient_payload"],"consensus_categories":[],"category_scores_codex":[0.002954331,0.0006575271,0.002013666,0.000583293,0.0001804445,0.0001071121,0.0006196555,0.0005829178,0.00109697],"category_scores_gemma":[0.005978058,0.0006313868,0.000634482,0.001803,0.000437162,0.0002346035,0.0003829818,0.0005588145,0.0001418286],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0004437867,"about_ca_system_score_gemma":0.0003010581,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.00006431862,"about_ca_topic_score_gemma":0.00002752464,"domain_scores_codex":[0.9938688,0.0004032344,0.003324241,0.0007280457,0.000761301,0.0009143311],"domain_scores_gemma":[0.9944992,0.002897503,0.0008743978,0.001227033,0.0003114759,0.000190363],"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":[0.000507162,0.008735809,0.0004440823,0.01156547,0.0003570595,0.00001666835,0.001998221,0.00004833314,0.0005727115,0.9638501,0.00237709,0.009527226],"study_design_scores_gemma":[0.006001499,0.0003719315,0.004110818,0.007013984,0.0008850357,0.0000176901,0.001601143,0.355829,0.001768777,0.6190519,0.002385008,0.0009632187],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.6389105,0.0007555233,0.2714298,0.0006869973,0.001389297,0.003933455,0.0009195594,0.000160582,0.08181427],"genre_scores_gemma":[0.9922389,0.00006644332,0.004030891,0.00002996327,0.0000517232,0.0001275085,0.00009095679,0.0000541472,0.003309414],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.3557807,"threshold_uncertainty_score":0.9998162,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.01748514848315048,"score_gpt":0.2972859598466086,"score_spread":0.2798008113634581,"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."}}