Self-approximation for Riemann Hypothesis

We present a rigorous formulation of that idea through shifts generated by the continuous Gram function. We study the set of parameters τ for which ζ(s+it_τ) uniformly approximates ζ(s) on compact subsets of D with connected complements. The method is based on Gram-shift universality, weak convergence of probability measures on the space of analytic functions and the support theorem for the random Euler product. The support is precisely the analytic functions that vanish nowhere in D and the identically zero function. This structure leads to two precise equivalences: The Riemann hypothesis is true if and only if the Gram-shift self-approximation holds with positive lower density for each admissible compact set and each approximation radius; and equivalently, when the limiting density exists and it is positive except at most countably many radii.