On the Polynomial Divergence of the Riemann Zeta Function and the Asymptotic Instability of the Critical Line: A Rigorous Analytic Treatment

This paper provides a rigorous analytic proof for the polynomial divergence of the Rie-mann zeta function ζ(s) in the neighborhood of the critical line ℜ(s) = 1/2. By meticulouslyderiving the Stirling expansion of the functional equation and applying Riemann-Stieltjes in-tegration to weighted second-moment estimates, we establish that any symmetric deviationδ from the critical line leads to a non-convergent energy flux. We prove that the functionalJθ (T) diverges as O(T1+2δ ), which strictly exceeds the classical Tlog T bound for all δ>0.These results demonstrate a structural instability of the non-trivial zero distribution un-der infinitesimal perturbations, implying that all zeros are topologically locked within theσ= 1/2 manifold.