{"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":"codex-gemma-dda1882f352a","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.0004122997,0.0002059691,0.0002205389,0.00004597359,0.0004033183,0.00005138775,0.000774087,0.0002206557,0.0000533989],"category_scores_gemma":[0.0002889315,0.0001487717,0.0002131728,0.0001155715,0.000132411,0.0000188001,0.000662161,0.0006145429,0.0001633284],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0000314377,"about_ca_system_score_gemma":0.0001142603,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.00002079278,"about_ca_topic_score_gemma":0.00005672544,"domain_scores_codex":[0.9989411,0.00009309154,0.0003088606,0.0003141456,0.0001129735,0.000229787],"domain_scores_gemma":[0.9976445,0.0008468505,0.0001706316,0.001225263,0.00006312031,0.00004957442],"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":[0.00003007576,0.0001285131,0.01530761,0.0001636902,0.0002215804,0.000002460973,0.0003843113,0.000005565375,0.0001824514,0.9244213,0.05477575,0.004376676],"study_design_scores_gemma":[0.0001778841,0.000008492649,0.01238078,0.0001131031,0.0001485763,0.000001751051,0.0001139043,0.00002484334,0.001401393,0.8974363,0.08793247,0.0002605243],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.9460015,0.0002930907,0.0006977936,0.002740939,0.0007982073,0.0006118168,0.00007197926,0.0002116997,0.04857295],"genre_scores_gemma":[0.9770519,0.0001093075,0.001006059,0.0002204891,0.0003558474,0.0003534175,0.00001872275,0.00002427046,0.02086006],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.03315672,"threshold_uncertainty_score":0.6066735,"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."}}