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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","authors_text":"Zhengqiang Li","cross_cats":[],"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.","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2026-04-16T03:59:18Z","title":"Prime--Zero Duality: Fractal Geometry, Renormalization-Group Flow, and an Information-Ontological Framework for Number Theory"},"references":{"count":27,"internal_anchors":0,"resolved_work":27,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"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","year":1973},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"A. M. 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