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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":true,"formal_links_present":false},"canonical_record":{"source":{"id":"2604.14596","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2026-04-16T03:59:18Z","cross_cats_sorted":[],"title_canon_sha256":"9b21939884457be428f580b9222a84b9a35ade52e8b4513acd000df1be446454","abstract_canon_sha256":"a6625443707ecbb573e89a4296c559981122b03f2c903f26e0960cb70474cfff"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T23:04:17.608065Z","signature_b64":"Ya0lcBWUyAIxiFHEHzFMgutMHHkmb0Mo+3JqRTmiseBE6NDQamwGrly/D1CUmqm2k1SE6KWZp2vMQQmtlYiXDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"105b11a0f62e2475208dddcbe0be2506bf5713a40c5447aea2e43acf5f758522","last_reissued_at":"2026-05-26T23:04:17.607102Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T23:04:17.607102Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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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. 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