A Proof of the Riemann Hypothesis Based on Euler-Maclaurin Analysis and Topological Phase Synchronization

This paper presents a proof of the Riemann Hypothesis by analyzing
the topological phase synchronization of nontrivial zeros. Using a
finite-N analogue of the Riemann zeta function derived via the
Euler-Maclaurin formula, we formulate the trigonometric arguments at
the zeros as continuous phase flows lifted to the universal covering
space of the circle. We demonstrate that the phase trajectories of
symmetric zero pairs, dictated by the functional equation, are strictly
locked. This synchronization is protected by topological rigidity,
rendering residual error terms inconsequential in the asymptotic limit.
Based on this topological constraint, we apply a proof by contradiction
assuming the existence of a zero off the critical line (a = 1/2). We
show that the real-part disparities strictly violate the phase
synchronization condition, leading to a mathematical contradiction.
Consequently, we conclude that all nontrivial zeros must lie exactly on
the critical line a = 1/2.