{"paper":{"title":"Contour Hankel dynamics and indicator fields for the Riemann $\\Xi$-function","arxiv_id":"2608.11520","primary_cat":"math.NA","submitted_at":"2026-08-12T00:17:16+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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zeros.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.88,"main_claim_p":0.09,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.0,"method":0.67,"result":0.21,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.12,"interpretation":0.0}},{"i":2,"p":0.96,"id":"s2","own":true,"role":"result","text":"This representation yields exact zero-free and rank criteria and, for conjugation-compatible contour-coordinate pairs, an inertia formula: once the matrix order is at least the number of distinct coordinate nodes, the negative index equals the number of distinct nonreal conjugate pairs.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.89,"main_claim_p":0.29,"quantitative":true,"quantitative_p":0.86,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.96,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.03,"interpretation":0.01}},{"i":3,"p":0.54,"id":"s3","own":true,"role":"interpretation","text":"Consequently, the Riemann hypothesis admits a local finite-dimensional Hankel-positivity formulation, although establishing this positivity independently of the zero set remains unresolved.","hedge":{"p":0.82,"label":"asserted"},"own_p":0.62,"main_claim_p":0.48,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.04,"background":0.0,"limitation":0.42,"future_work":0.0,"contribution":0.0,"interpretation":0.54}},{"i":4,"p":0.82,"id":"s4","own":true,"role":"result","text":"As the contour moves, the Hankel matrix evolves by a continuous congruence flow between zero crossings and undergoes finite-rank jumps at crossing events.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.85,"main_claim_p":0.08,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.0,"method":0.09,"result":0.82,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.08,"interpretation":0.01}},{"i":5,"p":0.93,"id":"s5","own":true,"role":"result","text":"An isolated zero produces a signed rank-one jump, whereas a nonreal conjugate pair produces a rank-two indefinite event in a real-axis circular scan that meets the pair.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.82,"main_claim_p":0.06,"quantitative":true,"quantitative_p":0.86,"role_probabilities":{"gap":0.0,"method":0.02,"result":0.93,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.04}},{"i":6,"p":0.73,"id":"s6","own":true,"role":"method","text":"Removing the continuous coordinate drift yields a piecewise-constant matrix process from which crossing coordinates, multiplicities, and zero locations can be recovered.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.86,"main_claim_p":0.09,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.73,"result":0.13,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.14,"interpretation":0.0}},{"i":7,"p":0.8,"id":"s7","own":true,"role":"result","text":"Numerical experiments 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