{"paper":{"title":"Primes in short intervals assuming the Riemann hypothesis","arxiv_id":"2610.27830","primary_cat":"math.NT","submitted_at":"2026-10-20T08:19:43+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":1.0,"id":"s0","own":false,"role":"background","text":"In 2001 Baker, Harman, and Pintz proved, for $x$ sufficiently large, the existence of at least one prime in the sub-linear interval $[x,x+x^{\\frac{1}{2} +\\frac{1}{40}}]$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.02,"main_claim_p":0.05,"quantitative":true,"quantitative_p":0.97,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":0.99,"id":"s1","own":false,"role":"gap","text":"But just how large is sufficiently large?","hedge":{"p":0.92,"label":"unclear"},"own_p":0.07,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.05,"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":2,"p":0.56,"id":"s2","own":true,"role":"contribution","text":"In the current note I shall make the Baker--Harman--Pintz theorem effective, but at an extremely high price.","hedge":{"p":0.97,"label":"asserted"},"own_p":0.59,"main_claim_p":0.06,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.03,"method":0.0,"result":0.0,"background":0.0,"limitation":0.4,"future_work":0.0,"contribution":0.56,"interpretation":0.0}},{"i":3,"p":0.91,"id":"s3","own":true,"role":"result","text":"Contingent upon assuming validity of the Riemann hypothesis (RH), I shall show that the Baker--Harman--Pintz theorem holds for $x \\geq 3.5\\times 10^{90}$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.47,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.91,"background":0.0,"limitation":0.02,"future_work":0.0,"contribution":0.07,"interpretation":0.0}},{"i":4,"p":0.97,"id":"s4","own":true,"role":"result","text":"Furthermore, again assuming RH, I shall also show that for any $0< m\\leq15.831$ and $x \\geq 1$ there is at least one prime in the interval $[x,x+x^{\\frac{1}{2} +\\frac{1}{m}}]$, and that for $M\\geq 2.29$ and $y \\geq 1$ there is at least one prime in the interval $[y^M, (y+1)^M]$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.92,"main_claim_p":0.38,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.97,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.03,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":0.78,"text":"Contingent upon assuming validity of the Riemann hypothesis (RH), I shall show that the Baker--Harman--Pintz theorem holds for $x \\geq 3.5\\times 10^{90}$.","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":4577,"cost_usd_reported":0.0},"compiled_at":"2026-10-21","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.06,"states_limitation":0.94,"human_participants":0.01,"controlled_comparison":0.02,"randomized_assignment":0.01,"states_quantitative_result":0.98,"compares_with_prior_results":0.92,"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":0.44,"text":"Contingent upon assuming validity of the Riemann hypothesis (RH), I shall show that the Baker--Harman--Pintz theorem holds for $x \\geq 3.5\\times 10^{90}$.","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":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}}