Prime--Zero Duality: Fractal Geometry, Renormalization-Group Flow, and an Information-Ontological Framework for Number Theory

Abstract

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 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 (beta=2,4), echoing Montgomery--Odlyzko universality. We interpret K as a conserved information current between the arithmetic and spectral domains, with the scaling law reflecting a renormalization-group flow from an ultraviolet fixed point K_UV = 11 (Hurwitz&#39;s theorem on normed division algebras) to K_IR = 4. The exponent b ~ 1/2 is derived from a variational information action S[I_P, I_Z]. 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. Upgrading this to a rigorous proof is the central open problem. We also explore, in a speculative spirit, whether (K_IR, b, kappa) resonate with quantities in quantum gravity and learning theory, including the Bekenstein--Hawking entropy formula. These analogies define open problems at the interface of number theory, physics, and information science.