{"paper":{"title":"Collective Contraction in the de Bruijn--Newman Heat Flow: Off-Zero Logarithmic Derivatives and Certified Barrier Refinements","arxiv_id":"2609.37164","primary_cat":"math.NT","submitted_at":"2026-09-29T09:56:08+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":0.78,"id":"s0","own":false,"role":"background","text":"We study the de Bruijn--Newman family $H_\\tau(z)$, with heat-deformation parameter $\\tau$ and $H_0(z)=\\tfrac18,\\xi(\\tfrac12+\\tfrac{iz}{2})$.","hedge":{"p":0.86,"label":"asserted"},"own_p":0.33,"main_claim_p":0.02,"quantitative":true,"quantitative_p":0.56,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.78,"limitation":0.0,"future_work":0.0,"contribution":0.22,"interpretation":0.0}},{"i":1,"p":0.63,"id":"s1","own":false,"role":"background","text":"The moving zeros considered here are therefore zeros of $H_\\tau$, not directly of $\\zeta(s)$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.25,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.09,"role_probabilities":{"gap":0.0,"method":0.05,"result":0.0,"background":0.63,"limitation":0.15,"future_work":0.0,"contribution":0.05,"interpretation":0.12}},{"i":2,"p":0.9,"id":"s2","own":true,"role":"result","text":"Retaining the collective interaction term in the zero dynamics, we prove that for a simple nonreal zero $z=x+iy$ of maximal imaginary height, [ \\frac{d}{d\\tau}y^2\\le -2-4y^2\\mathcal G_\\tau(z), ] where $\\mathcal G_\\tau\\ge0$ is a projected interaction sum.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.15,"quantitative":true,"quantitative_p":0.97,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.9,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.1,"interpretation":0.0}},{"i":3,"p":0.48,"id":"s3","own":true,"role":"contribution","text":"A second exact inequality bounds $\\mathcal G_\\tau$ below by an off-zero logarithmic derivative $-\\operatorname{Im}(H_\\tau'/H_\\tau)(x+i\\eta)$, providing a direct interface with the effective Polymath approximation $H_\\tau=B_\\tau F_\\tau$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.86,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.34,"role_probabilities":{"gap":0.0,"method":0.26,"result":0.25,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.48,"interpretation":0.01}},{"i":4,"p":0.97,"id":"s4","own":true,"role":"result","text":"At the frontier $X=6000000185827$, $\\tau_0=129/800$, and $y_0^2=87677/2500000$, a directed Cauchy certificate gives $\\mathcal G_\\tau>3/2$ for $\\tau_0\\le\\tau\\le0.178$, $x\\ge X$, and $|y|\\le y_0$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.9,"main_claim_p":0.06,"quantitative":true,"quantitative_p":0.99,"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":5,"p":0.86,"id":"s5","own":true,"role":"result","text":"Combining this contraction with certified upper and lower zero-free barriers, a two-envelope argument reduces the maximal nonreal height to $0.08$ by time $0.1747532546428610755\\ldots$; the classical de Bruijn contraction then completes the landing.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.91,"main_claim_p":0.39,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.12,"result":0.86,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.0}},{"i":6,"p":0.91,"id":"s6","own":true,"role":"result","text":"Relative to the publicly replayable but not yet peer-reviewed $0.1787854$ base certificate, this yields the audit-relative bound [ \\Lambda\\le0.1779532546428610755\\ldots . ] The collective contraction, logarithmic-derivative bridge, and new high-$x$ certificates are independent contributions.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.88,"main_claim_p":0.58,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.91,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.07,"interpretation":0.02}}]},"compiled":{"main_claim":{"p":0.97,"text":"Relative to the publicly replayable but not yet peer-reviewed $0.1787854$ base certificate, this yields the audit-relative bound [ \\Lambda\\le0.1779532546428610755\\ldots . ] The collective contraction, logarithmic-derivative bridge, and new high-$x$ certificates are independent contributions.","location":"abstract s6","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":6422,"cost_usd_reported":0.0},"compiled_at":"2026-09-30","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.05,"states_limitation":0.64,"human_participants":0.01,"controlled_comparison":0.13,"randomized_assignment":0.01,"states_quantitative_result":0.99,"compares_with_prior_results":0.93,"mentions_code_or_data_availability":0.29},"evidential_basis":{"p":0.98,"value":"proof or derivation","probabilities":{"argument":0.0,"literature":0.0,"proof_or_derivation":0.98,"analysis_of_existing_data":0.0,"observation_or_experiment":0.0,"simulation_or_computation":0.02}},"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":0.49,"text":"Retaining the collective interaction term in the zero dynamics, we prove that for a simple nonreal zero $z=x+iy$ of maximal imaginary height, [ \\frac{d}{d\\tau}y^2\\le -2-4y^2\\mathcal G_\\tau(z), ] where $\\mathcal G_\\tau\\ge0$ is a projected interaction sum.","location":"abstract s2"}},"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":7,"sentences_in_abstract":7},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; 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