{"paper":{"title":"A tyro's approach to the tilted sieve: beyond the Erd\\H{o}s--Rankin bound","arxiv_id":"2609.28253","primary_cat":"math.NT","submitted_at":"2026-09-23T15:15:59+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":0.55,"id":"s0","own":true,"role":"contribution","text":"We give an elementary exposition of the tilted sieve introduced by GPT-5.6~Sol \\cite{GPT2026}, showing that one residue class modulo each prime $p \\le x$ can cover an interval of length \\begin{equation*} \\gg \\frac{x\\log x}{(\\log_{2} x)\\log_{3} x}.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.66,"main_claim_p":0.79,"quantitative":true,"quantitative_p":0.97,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.45,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.55,"interpretation":0.0}},{"i":1,"p":0.79,"id":"s1","own":true,"role":"result","text":"\\end{equation*} This improves the classical Erd\\H{o}s--Rankin bound by a factor of $(\\log_{2} x)/(\\log_{3} x)^{2}$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.74,"main_claim_p":0.52,"quantitative":true,"quantitative_p":0.97,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.79,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.19}},{"i":2,"p":0.66,"id":"s2","own":true,"role":"limitation","text":"Although weaker than the strongest known bounds, it shows what the tilt and its associated covering of composite survivors achieve without Maynard sieve weights or a hypergraph covering theorem.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.61,"main_claim_p":0.09,"quantitative":true,"quantitative_p":0.59,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.0,"limitation":0.66,"future_work":0.0,"contribution":0.01,"interpretation":0.33}},{"i":3,"p":1.0,"id":"s3","own":true,"role":"method","text":"The proof uses the prime number theorem, Mertens' reciprocal-prime formula, and elementary probability.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.71,"main_claim_p":0.02,"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":4,"p":0.76,"id":"s4","own":true,"role":"contribution","text":"The appendices provide a historical survey and self-contained proofs of the classical Erd\\H{o}s--Rankin bound.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.56,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.04,"result":0.0,"background":0.2,"limitation":0.0,"future_work":0.0,"contribution":0.76,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":0.61,"text":"We give an elementary exposition of the tilted sieve introduced by GPT-5.6~Sol \\cite{GPT2026}, showing that one residue class modulo each prime $p \\le x$ can cover an interval of length \\begin{equation*} \\gg \\frac{x\\log x}{(\\log_{2} x)\\log_{3} x}.","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":4414,"cost_usd_reported":0.0},"compiled_at":"2026-09-24","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.05,"states_limitation":0.96,"human_participants":0.02,"controlled_comparison":0.16,"randomized_assignment":0.01,"states_quantitative_result":0.98,"compares_with_prior_results":0.98,"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":0.97,"value":"theoretical result","probabilities":{"review":0.0,"method_or_tool":0.03,"empirical_study":0.0,"simulation_study":0.0,"theoretical_result":0.97,"dataset_or_benchmark":0.0,"position_or_commentary":0.0}},"contribution_statement":{"p":0.93,"text":"We give an elementary exposition of the tilted sieve introduced by GPT-5.6~Sol \\cite{GPT2026}, showing that one residue class modulo each prime $p \\le x$ can cover an interval of length \\begin{equation*} \\gg \\frac{x\\log x}{(\\log_{2} x)\\log_{3} x}.","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 v1"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":5,"sentences_in_abstract":5},"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}}