{"paper":{"title":"Unconditional Bounds for Two, Three and Four Consecutive Moderate Gaps between Zeros of the Riemann Zeta Function","arxiv_id":"2610.22927","primary_cat":"math.NT","submitted_at":"2026-10-16T04:43:59+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":"result","text":"We prove unconditional positive-proportion results for consecutive moderate gaps between zeros of the Riemann zeta function.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.97,"main_claim_p":0.92,"quantitative":false,"quantitative_p":0.13,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.55,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.45,"interpretation":0.0}},{"i":1,"p":1.0,"id":"s1","own":true,"role":"result","text":"More precisely, for every fixed \\(0<c\\leq 1/4\\), the lower asymptotic densities, normalized by \\(N(T)\\), of occurrences of two, three, and four consecutive gaps, each of length at least \\(2\\pi c/\\log T\\), are at least \\(0.3450014\\ldots\\), \\(0.1812517\\ldots\\), and \\(0.0175021\\ldots\\), respectively.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.76,"quantitative":true,"quantitative_p":0.99,"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":2,"p":0.71,"id":"s2","own":false,"role":"background","text":"Here zeros are ordered by ordinate and counted with multiplicity, and \\(N(T)\\) denotes the number of nontrivial zeros with ordinates in \\((0,T]\\).","hedge":{"p":0.9,"label":"asserted"},"own_p":0.32,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.11,"role_probabilities":{"gap":0.0,"method":0.05,"result":0.06,"background":0.71,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.17}},{"i":3,"p":0.7,"id":"s3","own":true,"role":"interpretation","text":"This answers unconditionally and quantitatively a question studied by Gonek and Sahay under the Riemann hypothesis and the pair correlation conjecture.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.74,"main_claim_p":0.14,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.01,"method":0.0,"result":0.02,"background":0.11,"limitation":0.0,"future_work":0.0,"contribution":0.16,"interpretation":0.7}},{"i":4,"p":0.99,"id":"s4","own":false,"role":"gap","text":"In particular, at the time of their work, it was not known unconditionally that even two consecutive moderate gaps occur infinitely often for any fixed \\(c>0\\).","hedge":{"p":0.99,"label":"asserted"},"own_p":0.06,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.18,"role_probabilities":{"gap":0.99,"method":0.0,"result":0.0,"background":0.01,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":5,"p":1.0,"id":"s5","own":true,"role":"method","text":"The proof combines Lamzouri's Hilbert-space formulation of the Montgomery--Taylor second-moment method with local spectral estimates for simple critical-line zeros and a cluster-counting argument that accounts for multiple zeros and zeros off the critical line.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.83,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.04,"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}}]},"compiled":{"main_claim":{"p":0.91,"text":"We prove unconditional positive-proportion results for consecutive moderate gaps between zeros of the Riemann zeta 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":5266,"cost_usd_reported":0.0},"compiled_at":"2026-10-20","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.24,"states_limitation":0.17,"human_participants":0.01,"controlled_comparison":0.02,"randomized_assignment":0.01,"states_quantitative_result":0.99,"compares_with_prior_results":0.92,"mentions_code_or_data_availability":0.01},"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":"We prove unconditional positive-proportion results for consecutive moderate gaps between zeros of the Riemann zeta 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 v1"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":6,"sentences_in_abstract":6},"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}}