Starting from the classical integral representation of the $\zeta(s)$ function introduced by Riemann in 1859, this paper reexamines its analytic symmetry structure. By performing a geometric decomposition of the integral representation, we demonstrate that on the critical line $\Re(s)=\frac{ 1}{ 2}$, the value of $\xi(s)$ corresponds strictly to the \textbf{ real-part projection} of a specific analytic component. This discovery equivalently transforms the problem of complex zeros into a problem of \textbf{ sign evolution} along the real axis. Based on this geometric framework, we \textbf{ construct} an analytic mechanism of \textbf{ "Two-End Anchoring, Interval Counting"}: the global argument increment on the region boundary \textbf{ anchors} the initial value of the phase function, while the geometric decomposition structure on the critical line \textbf{ locks} its final value. This mechanism reveals an \textbf{ intrinsic coherence} between global topological constraints and local sign oscillations. Unlike traditional methods that rely on asymptotic estimates (such as the Big $O$ error term), the analysis in this paper is \textbf{ grounded in} exact identities. It unveils the \textbf{ geometric determinism} underlying the zero-counting formula, offering a novel perspective for analytic number theory independent of asymptotic analysis.
Categories: math.NT