{"paper":{"title":"Prime--Zero Duality: Fractal Geometry, Renormalization-Group Flow, and an Information-Ontological Framework for Number Theory","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"A duality measure between primes and zeta zeros converges to a fixed point of 4, structurally supporting the critical line at Re(s) = 1/2.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhengqiang Li","submitted_at":"2026-04-16T03:59:18Z","abstract_excerpt":"The prime numbers and the non-trivial zeros of the Riemann zeta function are globally linked by the explicit formula of analytic number theory. Whether they share a hidden, scale-by-scale geometric symmetry has remained unexplored. We address this by measuring the joint fractal structure of a prime residue class (p=1,5,9,13 mod 16) and the zero distribution of zeta(s). Our central finding is that the duality measure K = 1/d_P + 1/zeta_R is remarkably stable, varying by only 17% across scales L=100--2000, captured by a finite-size scaling law K(L) = K_IR + a*L^{-b}. After geometric normalizatio"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"After geometric normalization, the data converge to a universal infrared fixed point K_IR = 4 with critical exponent b ~ 0.51, robust across two random-matrix symmetry classes; a structural argument for the Riemann Hypothesis emerges: the generator kappa with kappa^2 = ijk = -1 enforces, via exchange symmetry I_P <-> I_Z, the fixed point I_P* = I_Z* = 2, encoding the critical line Re(s) = 1/2.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the numerically fitted duality measure K can be interpreted as a conserved information current whose scaling reflects a renormalization-group flow derived from a variational information action S[I_P, I_Z], and that the ad-hoc generator kappa enforces the fixed point corresponding to the critical line.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Numerical measurements show a prime-zero duality measure K stabilizing to an infrared fixed point of 4 under finite-size scaling with exponent near 0.5, interpreted as renormalization-group flow that structurally supports the Riemann hypothesis via an information action and quaternion-like generator","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A duality measure between primes and zeta zeros converges to a fixed point of 4, structurally supporting the critical line at Re(s) = 1/2.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f6a88c128edbda1b46c1126a5e3176ba95f44932471169a2dd401862f86a4a63"},"source":{"id":"2604.14596","kind":"arxiv","version":1},"verdict":{"id":"d65e22c8-5f85-407c-885f-c1da576a10f4","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T10:12:01.114574Z","strongest_claim":"After geometric normalization, the data converge to a universal infrared fixed point K_IR = 4 with critical exponent b ~ 0.51, robust across two random-matrix symmetry classes; a structural argument for the Riemann Hypothesis emerges: the generator kappa with kappa^2 = ijk = -1 enforces, via exchange symmetry I_P <-> I_Z, the fixed point I_P* = I_Z* = 2, encoding the critical line Re(s) = 1/2.","one_line_summary":"Numerical measurements show a prime-zero duality measure K stabilizing to an infrared fixed point of 4 under finite-size scaling with exponent near 0.5, interpreted as renormalization-group flow that structurally supports the Riemann hypothesis via an information action and quaternion-like generator","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the numerically fitted duality measure K can be interpreted as a conserved information current whose scaling reflects a renormalization-group flow derived from a variational information action S[I_P, I_Z], and that the ad-hoc generator kappa enforces the fixed point corresponding to the critical line.","pith_extraction_headline":"A duality measure between primes and zeta zeros converges to a fixed point of 4, structurally supporting the critical line at Re(s) = 1/2."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.14596/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":27,"sample":[{"doi":"","year":1973,"title":"H. L. Montgomery, The pair correlation of zeros of the zeta function, InAnalytic Number Theory(Proc. Sympos. Pure Math., Vol. XXIV), pp. 181–193. Amer. Math. Soc., 1973","work_id":"ea9eb974-7f79-4787-8bdb-5fe1b7c1c443","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1987,"title":"A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp.48(177), 273–308, 1987","work_id":"3639f3e5-9eb0-48c3-90b3-8db0855cd686","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"A. M. Odlyzko, Tables of zeros of the Riemann zeta function,https://www-users. cse.umn.edu/~odlyzko/zeta_tables/(accessed April 2026)","work_id":"62a98456-e7d3-4136-bd3c-b4a3937d548b","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2014,"title":"Falconer,Fractal Geometry: Mathematical Foundations and Applications, 3rd ed","work_id":"70fc93c2-7285-43af-b91a-e8379740125e","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2004,"title":"M. L. Mehta,Random Matrices, 3rd ed. Academic Press, 2004. 101","work_id":"edf36983-33c1-425b-ab67-12226bc48183","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":27,"snapshot_sha256":"db43fbb2df7d6ceefa2167e6dba66fc4ca145ae2a59868e042bd51c5756aae48","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}