Constraint Structure and Zero Counting in the Integral Representation of the Zeta Function

Abstract

Starting from the classical integral representation of the $\hat{I}\P (s)$ 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 $\hat{I}¾(s)$ 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.