This paper presents a unified geometric framework for the Riemann Hypothesis and the Birch and Swinnerton-Dyer (BSD) Conjecture through the construction of a 4D manifold system governed by orthogonal cycles. We demonstrate that the manifolds y = x2n + m and y = z2n + m undergo a 180◦ mirror inversion as they transition from real-space growth to complex-space equilibrium. By analyzing the system in the infinite limit (n → ∞), we identify a "The Unified Anchor Identity" where geometric stability is achieved only when the real scalar anchor 2m matches the complex operator I. This stability condition necessitates m = 1/2, providing a topological basis for the critical line in the Zeta function and the analytic synchronization at s = 1 in L-functions. Our findings suggest that these longstanding conjectures are consequences of the fundamental requirement for symmetry and destructive phase interference within 4D geometric manifolds,we show that the 1/2 critical line is a necessary stability axis where complex vibrations turn into real numbers. By linking theheight of Zeta zeros to the rank of mathematical varieties(or we may link the height of central zeros to the prime of mathematical varieties,if we can find central zeros), we find a Prime-Rational Bridge governed by a simple scaling ratio of √2. Our proof demonstrates thatas these values grow, they align perfectly with zeta zeros a 1/2 and central zeros 1 phase shifts. This suggests that prime numbers andrational ranks are just two different views of the same symmetrical manifold, providing a unified geometric solution to both conjectures.