{"paper":{"title":"The Hardy--Littlewood asymptotic for primes $p$ and $p+4$ from a uniform zeta-ratios conjecture","arxiv_id":"2610.32672","primary_cat":"math.NT","submitted_at":"2026-10-23T14:14:23+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. 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$U^{1/4}$.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.66,"main_claim_p":0.05,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.15,"result":0.63,"background":0.01,"limitation":0.02,"future_work":0.0,"contribution":0.19,"interpretation":0.0}},{"i":2,"p":0.97,"id":"s2","own":true,"role":"result","text":"For a fixed real smooth weight supported in $(1,101/100)$, we obtain an error $O_\\varepsilon(x^{-1/16+\\varepsilon})$ in the correlation sum with weight $W(n/x)^2/n$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.06,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.01,"result":0.97,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.0}},{"i":3,"p":0.96,"id":"s3","own":true,"role":"result","text":"Adding the Riemann hypothesis improves this error to $O_\\varepsilon(x^{-3/8+\\varepsilon})$.","hedge":{"p":0.98,"label":"asserted"},"own_p":0.77,"main_claim_p":0.06,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.96,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.03}},{"i":4,"p":0.99,"id":"s4","own":true,"role":"method","text":"The proof retains the crossed zero residues in Mellin inversion and estimates their mixed term with the logarithmic-derivative integral.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.91,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.06,"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}},{"i":5,"p":0.58,"id":"s5","own":true,"role":"result","text":"Two nonnegative time weights have the same diagonal term and opposite coefficients for the difference four.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.67,"main_claim_p":0.03,"quantitative":true,"quantitative_p":0.59,"role_probabilities":{"gap":0.0,"method":0.28,"result":0.58,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.12,"interpretation":0.02}},{"i":6,"p":0.98,"id":"s6","own":true,"role":"result","text":"We deduce the logarithmic asymptotic for the weighted prime-pair sum $B_4$ and an asymptotic for the second finite difference of integrals of squared increments of the truncated Chebyshev function.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.19,"quantitative":false,"quantitative_p":0.13,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.98,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.0}},{"i":7,"p":0.72,"id":"s7","own":true,"role":"interpretation","text":"The latter is connected to a logarithmically weighted M\\\"obius approximant in the Nyman--Beurling criterion by finite 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