{"paper":{"title":"Havel--Hakimi Residues of Common-Divisor Graphs: Complete Asymptotics and Prime-Counting Structure","arxiv_id":"2608.04040","primary_cat":"math.CO","submitted_at":"2026-08-03T21:09:37+00:00","version_compiled":"v2"},"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 v2","sentences":[{"i":0,"p":0.99,"id":"s0","own":false,"role":"background","text":"Let $G_n$ be the graph on $\\{2,\\ldots,n\\}$ in which two integers are adjacent when they have a common divisor greater than one.","hedge":{"p":0.9,"label":"asserted"},"own_p":0.08,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.17,"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":0.59,"id":"s1","own":true,"role":"result","text":"We determine the complete asymptotic expansion of its Havel--Hakimi residue $\\R(G_n)$, confirming a leading-constant prediction of Staton recorded in Fajtlowicz's \\emph{Written on the Wall}.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.54,"quantitative":false,"quantitative_p":0.1,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.59,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.41,"interpretation":0.0}},{"i":2,"p":1.0,"id":"s2","own":true,"role":"result","text":"If $A=\\sum_{k=2}^{\\infty}(\\log k)/(k^2(k-1))$, then the first two terms are $\\R(G_n)=(\\zeta(2)-1)n/\\log n+(\\zeta(2)-1-A)n/\\log^2n +O(n/\\log^3n)$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.9,"main_claim_p":0.57,"quantitative":true,"quantitative_p":0.95,"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":3,"p":0.91,"id":"s3","own":true,"role":"result","text":"More precisely, the difference between $\\R(G_n)$ and the prime-vertex contribution to the Caro--Wei sum is $O_\\beta(n\\exp\\{-(\\log n)^\\beta\\})$ for every fixed $0<\\beta<1/2$; this estimate yields every coefficient in the expansion.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.89,"main_claim_p":0.22,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.03,"result":0.91,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.05,"interpretation":0.01}},{"i":4,"p":0.99,"id":"s4","own":true,"role":"method","text":"The upper bound follows from a degree-preserving realization in which almost all relevant prime vertices are partitioned into cliques.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.86,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.16,"role_probabilities":{"gap":0.0,"method":0.99,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":5,"p":0.9,"id":"s5","own":true,"role":"result","text":"We also prove that the unlabeled graph determines $\\pi(n)$ through its simplicial true-twin classes.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.89,"main_claim_p":0.11,"quantitative":false,"quantitative_p":0.11,"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":6,"p":0.85,"id":"s6","own":true,"role":"result","text":"Stable inverses for weighted sums of the resulting degree-class counts give criteria equivalent to the Riemann hypothesis, including one involving only the Caro--Wei sum.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.88,"main_claim_p":0.11,"quantitative":false,"quantitative_p":0.18,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.85,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.1,"interpretation":0.05}},{"i":7,"p":0.59,"id":"s7","own":true,"role":"result","text":"An exact local-defect identity additionally reduces the conjectured sharp $+2$ residue bound to explicit prefix estimates.","hedge":{"p":0.89,"label":"asserted"},"own_p":0.84,"main_claim_p":0.07,"quantitative":true,"quantitative_p":0.97,"role_probabilities":{"gap":0.0,"method":0.05,"result":0.59,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.31,"interpretation":0.05}}]},"compiled":{"main_claim":{"p":0.95,"text":"We determine the complete asymptotic expansion of its Havel--Hakimi residue $\\R(G_n)$, confirming a leading-constant prediction of Staton recorded in Fajtlowicz's \\emph{Written on the Wall}.","location":"abstract s1","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":6544,"cost_usd_reported":0.0},"compiled_at":"2026-09-23","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.07,"states_limitation":0.16,"human_participants":0.01,"controlled_comparison":0.06,"randomized_assignment":0.01,"states_quantitative_result":0.98,"compares_with_prior_results":0.89,"mentions_code_or_data_availability":0.02},"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":1.0,"text":"We determine the complete asymptotic expansion of its Havel--Hakimi residue $\\R(G_n)$, confirming a leading-constant prediction of Staton recorded in Fajtlowicz's \\emph{Written on the Wall}.","location":"abstract s1"}},"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 v2"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":8,"sentences_in_abstract":8},"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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