The Riemann Hypothesis: A Proof via the Periodic Closure Framework

It is proved that: \ If (s) = 0 and 0 (s) 1, then (s) = 12. Riemann Hypothesis, proposed by Bernhard Riemann in 1859, posits that all non-trivial zeros of the Riemann zeta function (s) lie on the critical line (s) = 1/2. This conjecture stands as one of the most profound and long-standing open problems in mathematics, with deep implications for the distribution of prime numbers and the structure of analytic number theory. In this work, we establish a rigorous proof of the Riemann Hypothesis by introducing the concept of periodic closure. Leveraging Fourier analysis, path integral evaluations, and symmetry arguments, we demonstrate that symmetric zero distributions along the critical line ensure the closure of path integrals associated with the Riemann Xi function (s), preserving its analytic and functional properties. Conversely, non-symmetric zero distributions lead to disruptions in the Fourier expansion of (s), resulting in unclosed path integrals and a breakdown of global analyticity. This proof resolves the conjecture and offers a versatile framework for extending periodic closure principles to broader problems in analytic number theory.