{"paper":{"title":"Exact $3$-Adic Periods of Symmetric Dellac Numbers and Their Polynomial Extensions","arxiv_id":"2611.05962","primary_cat":"math.CO","submitted_at":"2026-11-04T04:59:44+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":0.99,"id":"s0","own":false,"role":"background","text":"The odd and even symmetric Dellac numbers are the specializations $a_n=f_n(0)$ and $e_n=f_n(1)$ of the Randrianarivony--Zeng family $f_n(s)=2^{-n}D_n(s)$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.2,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.39,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.99,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":1,"p":0.88,"id":"s1","own":true,"role":"result","text":"Using a finite Stieltjes-fraction truncation, we establish the sharp congruence $v_3(a_n)=1$ for $n\\ge2$ and obtain explicit congruences modulo $81$ for both sequences.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.97,"main_claim_p":0.36,"quantitative":true,"quantitative_p":0.96,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.88,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.12,"interpretation":0.0}},{"i":2,"p":0.9,"id":"s2","own":true,"role":"result","text":"We also establish universal eventual periodicity modulo arbitrary positive integers.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.2,"quantitative":false,"quantitative_p":0.14,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.9,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.1,"interpretation":0.0}},{"i":3,"p":0.76,"id":"s3","own":true,"role":"contribution","text":"For every integer $s\\ge0$ and $k\\ge1$, we give and analyse formulas for the \\emph{least} eventual period of $f_n(s)$ modulo $3^k$ and its \\emph{least starting index}.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.39,"quantitative":false,"quantitative_p":0.25,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.24,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.76,"interpretation":0.0}},{"i":4,"p":0.94,"id":"s4","own":true,"role":"result","text":"The period is governed by five linear factors in $s$; the onset follows a nine-class formula modulo $9$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.91,"main_claim_p":0.18,"quantitative":true,"quantitative_p":0.76,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.94,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.05}},{"i":5,"p":1.0,"id":"s5","own":true,"role":"result","text":"In particular, for $k\\ge3$ the even symmetric Dellac period is three times the odd period and begins one index earlier.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.92,"main_claim_p":0.2,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":6,"p":1.0,"id":"s6","own":true,"role":"method","text":"The proofs combine a finite Stieltjes-fraction certificate, a graded $3$-adic pole expansion, and an explicit finite top-degree certificate over $\\F_3$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.92,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.17,"role_probabilities":{"gap":0.0,"method":1.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":7,"p":1.0,"id":"s7","own":true,"role":"method","text":"Finite integer-arithmetic checks supplement the proofs.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.89,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":1.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":8,"p":0.49,"id":"s8","own":true,"role":"contribution","text":"We also record the connection with the type-$B$ homogenized Linial arrangement.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.71,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.33,"background":0.09,"limitation":0.0,"future_work":0.0,"contribution":0.49,"interpretation":0.09}}]},"compiled":{"main_claim":{"p":0.89,"text":"For every integer $s\\ge0$ and $k\\ge1$, we give and analyse formulas for the \\emph{least} eventual period of $f_n(s)$ modulo $3^k$ and its \\emph{least starting index}.","location":"abstract s3","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"TypeSafe API","model":"classifier-1.13.0","endpoint":"https://api.typesafe.ai/v1/systemone","provider":"TypeSafe"},"packing":"one request per paper","requests":1,"input_tokens":6981,"cost_usd_reported":0.0},"compiled_at":"2026-11-05","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.04,"states_limitation":0.13,"human_participants":0.01,"controlled_comparison":0.02,"randomized_assignment":0.01,"states_quantitative_result":0.97,"compares_with_prior_results":0.1,"mentions_code_or_data_availability":0.04},"evidential_basis":{"p":1.0,"value":"proof or derivation","probabilities":{"argument":0.0,"literature":0.0,"proof_or_derivation":1.0,"analysis_of_existing_data":0.0,"observation_or_experiment":0.0,"simulation_or_computation":0.0}},"contribution_type":{"p":1.0,"value":"theoretical result","probabilities":{"review":0.0,"method_or_tool":0.0,"empirical_study":0.0,"simulation_study":0.0,"theoretical_result":1.0,"dataset_or_benchmark":0.0,"position_or_commentary":0.0}},"contribution_statement":{"p":0.93,"text":"For every integer $s\\ge0$ and $k\\ge1$, we give and analyse formulas for the \\emph{least} eventual period of $f_n(s)$ modulo $3^k$ and its \\emph{least starting index}.","location":"abstract s3"}},"coverage":{"tier":"abstract","levels_run":["P abstract"],"not_examined":["full text","figures and tables","references and citing sentences","relations to other papers"],"sources_read":["arXiv listing abstract v1"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":9,"sentences_in_abstract":9},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; 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