A Study of the Riemann Hypothesis Based on Stationary‑Point Analysis of Quadruple Coherent Clusters

The Riemann hypothesis is a long-standing core proposition in analytic number theory, asserting that all non-trivial zeros of the Riemann ζ-function have real part equal to $(1/2)$. Existing investigations mainly proceed from zero-density estimates, remainder estimates for explicit formulas, and numerical verification, yet no universally accepted complete proof has been obtained. Based on the functional equation of the ζ-function and the conjugate symmetry of its zeros, this paper establishes a research framework taking the quadruple set ρ,ρ̄,1-ρ,1-ρ̄ as the minimal analysis unit. First, we perform trigonometric decomposition for the contribution terms corresponding to each quadruple set. In the asymptotic region of large imaginary part γ 1, we distinguish sine-dominant oscillatory terms from cosine background oscillatory terms. We construct phase windows and derive a uniform logarithmic upper bound for zero counts inside each window, ruling out extreme configurations of local infinite zero accumulation. Further, we carry out intra-cluster coherent superposition and inter-cluster phase-cancellation analysis. Without assuming the Riemann hypothesis in advance, we unconditionally prove the uniform boundedness of the oscillatory series [T(u)=∑_{γ>γ*}sin(γ u)/γ,] which provides underlying support for the steady-state constraints of coherent clusters. This paper introduces an amplitude-envelope functional and transforms the critical resonant configurations of coherent clusters into a minimax stationary-point problem of envelope functions. Combined with complex-vector superposition rules for the pair (ρ,1-ρ) and mean-value inequalities for vector norms, we characterize the constraint boundaries of resonant superposition. We also establish a homology theorem linking Newton-iteration zero-finding and stationary-point problems, which bridges the intrinsic connections among the Perron-integral kernel, the logarithmic derivative of ζ-function, and stationary-point conditions. We then apply proof by contradiction: if exceptional zeros whose real parts deviate from $(1/2)$ exist inside the critical strip, the vector-superposition conditions of their corresponding quadruple sets will structurally conflict with the global steady-state constraints of coherent clusters. This contradiction implies that no such exceptional zeros exist, so the Riemann hypothesis holds. Our argument does not rely on constructing contradictions via traditional strong magnitude scaling, and thus avoids the risk of circular reasoning inherent in magnitude-based arguments. Prospective extensions of the framework include rigorous functional formalization of coherent constraints and migration toward generalized L-functions and the generalized Riemann hypothesis.