{"id":"W4416097310","doi":"10.48550/arxiv.2505.04882","title":"The Degree Polynomial","year":2025,"lang":"en","type":"preprint","venue":"ArXiv.org","topic":"Graph theory and applications","field":"Mathematics","cited_by":0,"is_retracted":false,"has_abstract":true,"ca_institutions":"","funders":"Natural Sciences and Engineering Research Council of Canada","keywords":"Degree (music); Multigraph; Polynomial; Order (exchange); Degree of a polynomial; Reciprocal polynomial","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":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.0004189882,0.0003647345,0.0004286898,0.001381732,0.0006930864,0.002250225,0.000640104,0.0004956066,0.01009722],"category_scores_gemma":[0.0035306,0.0002523375,0.0003842544,0.001256711,0.001736168,0.003464546,0.001452278,0.00143463,0.002028317],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.001332861,"about_ca_system_score_gemma":0.0004609186,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.000840597,"about_ca_topic_score_gemma":0.0006104084,"domain_scores_codex":[0.9994287,0.00005984365,0.00001627747,0.0002281318,0.0001340248,0.0001329405],"domain_scores_gemma":[0.998679,0.0006174704,0.000181218,0.0001934513,0.000167296,0.000161635],"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":[0.0001086858,0.00002728456,0.002071143,0.0001472989,0.00002344914,0.00008002618,0.0002663861,0.005739933,0.007790444,0.9250343,0.006968773,0.0517423],"study_design_scores_gemma":[0.00002400985,0.00006535945,0.004963356,0.00006890699,0.0000304249,0.0009037296,0.0001560204,0.02441613,0.006279004,0.9097417,0.05329651,0.00005483527],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.5308248,0.004995128,0.2729848,0.005636488,0.0005982905,0.00009461195,0.004025223,0.0006422486,0.1801985],"genre_scores_gemma":[0.9576745,0.001856531,0.01441136,0.0002734461,0.0003631756,0.00005316553,0.0006439422,0.000177199,0.02454673],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.01009722,"threshold_uncertainty_score":0.03377861,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.1779776798518906,"score_gpt":0.3481721697215069,"score_spread":0.1701944898696162,"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."}}