{"paper":{"title":"A first-exit proof of Cusick's sum-of-digits conjecture","arxiv_id":"2606.23398","primary_cat":"math.NT","submitted_at":"2026-06-22T14:21:11+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":0.89,"id":"s0","own":true,"role":"contribution","text":"We prove Cusick's conjecture on the binary sum-of-digits function.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.97,"main_claim_p":0.89,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.11,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.89,"interpretation":0.0}},{"i":1,"p":0.99,"id":"s1","own":true,"role":"result","text":"More precisely, for every integer \\(t\\ge 1\\) we show that \\[ c_t:=\\lim_{N\\to\\infty}\\frac{1}{N} \\#\\{0\\le n<N:\\ s_2(n+t)\\ge s_2(n)\\}>\\frac{1}{2}, \\] and in fact obtain the explicit bound \\[ c_t\\ge \\frac{1}{2}+2^{-2s_2(t)-1}, \\] where \\(s_2(m)\\) denotes the number of ones in the binary expansion of \\(m\\).","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.86,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.99,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":2,"p":1.0,"id":"s2","own":true,"role":"method","text":"The proof is based on an exact deconvolution which replaces the distribution of \\(s_2(n+t)-s_2(n)\\) by a finite stopped random-walk law.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.06,"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":3,"p":0.99,"id":"s3","own":true,"role":"method","text":"The required bias is then proved through first-exit medians for principal subsequence ideals.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.87,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.04,"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}}]},"compiled":{"main_claim":{"p":0.65,"text":"We prove Cusick's conjecture on the binary sum-of-digits function.","location":"abstract s0","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":3822,"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":0.21,"states_limitation":0.06,"human_participants":0.01,"controlled_comparison":0.06,"randomized_assignment":0.01,"states_quantitative_result":0.98,"compares_with_prior_results":0.11,"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":0.53,"text":"We prove Cusick's conjecture on the binary sum-of-digits function.","location":"abstract s0"}},"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":4,"sentences_in_abstract":4},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; the figure is how often the published label matches their consensus","sentence_role_agreement":0.92,"main_claim_agreement":0.908,"typed_citation_agreement":0.933,"typed_citation_agreement_95_interval":[0.917,0.945],"review_comments_accurate":0.758,"review_comments_accurate_95_interval":[0.684,0.82],"review_caught_admitted_error_found_or_related":0.448}}