{"paper":{"title":"Primes between 8th powers?","arxiv_id":"2610.20058","primary_cat":"math.NT","submitted_at":"2026-10-14T11:59:21+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":"OpenAI (using a non-public \"internal frontier model'') has recently (6 October 2026) made a number of remarkable claims, including some claims regarding the quasi-Riemann hypothesis (QRH).","hedge":{"p":1.0,"label":"asserted"},"own_p":0.04,"main_claim_p":0.02,"quantitative":true,"quantitative_p":0.77,"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":0.76,"id":"s1","own":false,"role":"background","text":"While there are good reasons to be cautious regarding these claims, one can nevertheless ask what (if true) their implications would be.","hedge":{"p":0.62,"label":"hedged_possibility"},"own_p":0.06,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.02,"method":0.0,"result":0.0,"background":0.76,"limitation":0.17,"future_work":0.01,"contribution":0.02,"interpretation":0.02}},{"i":2,"p":0.63,"id":"s2","own":false,"role":"background","text":"While there is widespread agreement that any version of the QRH will provide various improved bounds on the location of the prime numbers, I feel that there is some value in making these improvements explicit.","hedge":{"p":0.58,"label":"asserted"},"own_p":0.09,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.1,"role_probabilities":{"gap":0.22,"method":0.0,"result":0.0,"background":0.63,"limitation":0.0,"future_work":0.0,"contribution":0.14,"interpretation":0.01}},{"i":3,"p":0.7,"id":"s3","own":true,"role":"result","text":"Specifically, if we assume validity of the claimed 7/8 version of the QRH then I shall demonstrate (for all $n\\in \\mathbb{Z^+}$): 1) the existence of primes between eighth powers, in the interval $(n^8,(n+1)^8)$; 2) the existence of primes in the shorter interval $(n^8,n^8+8n^7)$; 3) the existence of primes in the even shorter interval $(n^8,n^8+n^7).$ Similarly for reals greater than unity $x>1$ one can demonstrate: 4) the existence of primes in the Hoheisel-style interval $(x,x+x^{7/8})$.","hedge":{"p":0.89,"label":"asserted"},"own_p":0.82,"main_claim_p":0.91,"quantitative":true,"quantitative_p":0.97,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.7,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.3,"interpretation":0.0}},{"i":4,"p":1.0,"id":"s4","own":false,"role":"limitation","text":"All of this is contingent on the validity of the claimed 7/8 version of the QRH.","hedge":{"p":0.63,"label":"asserted"},"own_p":0.12,"main_claim_p":0.06,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.0,"limitation":1.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":1.0,"text":"Specifically, if we assume validity of the claimed 7/8 version of the QRH then I shall demonstrate (for all $n\\in \\mathbb{Z^+}$): 1) the existence of primes between eighth powers, in the interval $(n^8,(n+1)^8)$; 2) the existence of primes in the shorter interval $(n^8,n^8+8n^7)$; 3) the existence of primes in the even shorter interval $(n^8,n^8+n^7).$ Similarly for reals greater than unity $x>1$ one can demonstrate: 4) the existence of primes in the Hoheisel-style interval $(x,x+x^{7/8})$.","location":"abstract s3","hedge_strength":{"p":0.89,"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":4542,"cost_usd_reported":0.0},"compiled_at":"2026-10-15","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.04,"states_limitation":0.96,"human_participants":0.02,"controlled_comparison":0.02,"randomized_assignment":0.01,"states_quantitative_result":0.84,"compares_with_prior_results":0.12,"mentions_code_or_data_availability":0.02},"evidential_basis":{"p":0.93,"value":"proof or derivation","probabilities":{"argument":0.06,"literature":0.01,"proof_or_derivation":0.93,"analysis_of_existing_data":0.0,"observation_or_experiment":0.0,"simulation_or_computation":0.0}},"contribution_type":{"p":0.93,"value":"theoretical result","probabilities":{"review":0.0,"method_or_tool":0.0,"empirical_study":0.0,"simulation_study":0.0,"theoretical_result":0.93,"dataset_or_benchmark":0.0,"position_or_commentary":0.07}},"contribution_statement":{"p":0.99,"text":"Specifically, if we assume validity of the claimed 7/8 version of the QRH then I shall demonstrate (for all $n\\in \\mathbb{Z^+}$): 1) the existence of primes between eighth powers, in the interval $(n^8,(n+1)^8)$; 2) the existence of primes in the shorter interval $(n^8,n^8+8n^7)$; 3) the existence of primes in the even shorter interval $(n^8,n^8+n^7).$ Similarly for reals greater than unity $x>1$ one can demonstrate: 4) the existence of primes in the Hoheisel-style interval $(x,x+x^{7/8})$.","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}}