{"paper":{"title":"Probing automorphic induction through zero correlations and secondary arithmetic terms","arxiv_id":"2611.02058","primary_cat":"math.NT","submitted_at":"2026-11-01T17:03:24+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":0.88,"id":"s0","own":false,"role":"background","text":"Zero correlations offer a way to test the cuspidal structure of an automorphic L-function, but their leading self statistics retain only coarse degree and multiplicity data.","hedge":{"p":0.94,"label":"asserted"},"own_p":0.1,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.12,"role_probabilities":{"gap":0.09,"method":0.0,"result":0.0,"background":0.88,"limitation":0.03,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":0.5,"id":"s1","own":true,"role":"method","text":"We therefore adjoin a specified cuspidal L-function as a probe and study the resulting raw mixed correlations.","hedge":{"p":0.88,"label":"asserted"},"own_p":0.88,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.5,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.5,"interpretation":0.0}},{"i":2,"p":0.97,"id":"s2","own":true,"role":"result","text":"Assuming a factorization into cuspidal L-functions over Q, we show that a mixed pair statistic detects the probe's occurrence and recovers its multiplicity.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.92,"main_claim_p":0.48,"quantitative":false,"quantitative_p":0.1,"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}},{"i":3,"p":0.87,"id":"s3","own":true,"role":"method","text":"A suitable Gevrey-order-two test eliminates the smooth-density terms and gives a subexponential error.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.72,"main_claim_p":0.07,"quantitative":true,"quantitative_p":0.77,"role_probabilities":{"gap":0.0,"method":0.87,"result":0.09,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.04,"interpretation":0.0}},{"i":4,"p":0.73,"id":"s4","own":false,"role":"result","text":"For arbitrary smooth plateau tests, the prime number theorem of Harcos and Thorner gives an error smaller than every fixed inverse power of log T, without a self-contragredience assumption.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.15,"main_claim_p":0.05,"quantitative":true,"quantitative_p":0.94,"role_probabilities":{"gap":0.0,"method":0.04,"result":0.73,"background":0.17,"limitation":0.0,"future_work":0.0,"contribution":0.06,"interpretation":0.0}},{"i":5,"p":0.54,"id":"s5","own":false,"role":"background","text":"The stronger prime number theorem of Liu and Ye gives a subexponential remainder under its stated structural assumptions, with the possible norm-twist terms made explicit.","hedge":{"p":0.95,"label":"asserted"},"own_p":0.16,"main_claim_p":0.06,"quantitative":true,"quantitative_p":0.65,"role_probabilities":{"gap":0.0,"method":0.05,"result":0.33,"background":0.54,"limitation":0.0,"future_work":0.0,"contribution":0.08,"interpretation":0.0}},{"i":6,"p":0.58,"id":"s6","own":true,"role":"method","text":"To obtain arithmetic information beyond this overlap, we compute the correlation expansion at every fixed level and inverse-logarithmic order within the original strict small-support range.","hedge":{"p":0.98,"label":"asserted"},"own_p":0.89,"main_claim_p":0.08,"quantitative":false,"quantitative_p":0.1,"role_probabilities":{"gap":0.02,"method":0.58,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.4,"interpretation":0.0}},{"i":7,"p":0.96,"id":"s7","own":true,"role":"method","text":"We treat the pair and higher correlations separately, using Jiang's rational-field fixed-data estimates to justify the required prime-power convergence.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.68,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.0,"method":0.96,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.04,"interpretation":0.0}},{"i":8,"p":0.9,"id":"s8","own":true,"role":"result","text":"The pair calculation identifies the mixed arithmetic finite parts and gives detailed error estimates, including the contribution of norm twists.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.91,"main_claim_p":0.12,"quantitative":false,"quantitative_p":0.11,"role_probabilities":{"gap":0.0,"method":0.01,"result":0.9,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.09,"interpretation":0.0}},{"i":9,"p":0.93,"id":"s9","own":true,"role":"result","text":"The higher-level calculation retains connected prime-power invariants, whose first cubic contribution appears at order (log T)^{-3}.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.88,"main_claim_p":0.08,"quantitative":true,"quantitative_p":0.93,"role_probabilities":{"gap":0.0,"method":0.01,"result":0.93,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.06,"interpretation":0.0}},{"i":10,"p":0.73,"id":"s10","own":true,"role":"contribution","text":"Explicit third- and fourth-level formulas and examples explain the arithmetic content of these coefficients and the limitations of self-correlations.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.88,"main_claim_p":0.11,"quantitative":false,"quantitative_p":0.35,"role_probabilities":{"gap":0.0,"method":0.05,"result":0.17,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.73,"interpretation":0.05}},{"i":11,"p":0.94,"id":"s11","own":true,"role":"method","text":"Varying powers of the probe separates their mixed components.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.81,"main_claim_p":0.08,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.0,"method":0.94,"result":0.01,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.05,"interpretation":0.0}},{"i":12,"p":0.5,"id":"s12","own":true,"role":"future_work","text":"These formulas, together with the distinct-index sieve, provide a collection of arithmetic probe responses toward the further problem of recovering automorphic induction structure.","hedge":{"p":0.9,"label":"asserted"},"own_p":0.86,"main_claim_p":0.35,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.02,"result":0.03,"background":0.0,"limitation":0.0,"future_work":0.5,"contribution":0.24,"interpretation":0.21}}]},"compiled":{"main_claim":{"p":0.66,"text":"These formulas, together with the distinct-index sieve, provide a collection of arithmetic probe responses toward the further problem of recovering 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