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

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 $\mathrm{\Re }(s)=\frac{1}{2}$, the value of $\xi (s)$ corresponds strictly to the $\mathbf{\text{real-part}}\mathbf{\text{projection}}$ of a specific analytic component. This discovery equivalently transforms the problem of complex zeros into a problem of $\mathbf{\text{sign}}\mathbf{\text{evolution}}$ along the real axis. Based on this geometric framework, we $\mathbf{\text{construct}}$ an analytic mechanism of $\mathbf{\text{"Two-End}}\mathbf{\text{Anchoring,}}\mathbf{\text{Interval}}\mathbf{\text{Counting"}}$: the global argument increment on the region boundary $\mathbf{\text{anchors}}$ the initial value of the phase function, while the geometric decomposition structure on the critical line $\mathbf{\text{locks}}$ its final value. This mechanism reveals an $\mathbf{\text{intrinsic}}\mathbf{\text{coherence}}$ 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 $\mathbf{\text{grounded}}\mathbf{\text{in}}$ exact identities. It unveils the $\mathbf{\text{geometric}}\mathbf{\text{determinism}}$ underlying the zero-counting formula, offering a novel perspective for analytic number theory independent of asymptotic analysis.