The fluid antenna system (FAS) concept is an emerging paradigm that promotes the utilization of the feature of shape and position reconfigurability in antennas to broaden the design of wireless communication systems. This also means that spatial diversity can be exploited in an unconventional way. However, a rigorous framework for error probability analysis of FAS under realistic spatially correlated channels has been lacking. In this paper, we fill this gap by deriving a tight, closed-form asymptotic expression for the symbol error rate (SER) that establishes the fundamental scaling law linking the system's SER to the channel's spatial correlation structure. A key insight of our analysis is that the achievable diversity gain is governed not by the number of antenna ports, but by the channel's effective rank. To find this critical parameter, we propose a novel dual-pronged approach. First of all, we develop a geometry-based algorithm that extracts distinct performance thresholds from the channel's eigenvalue spectrum. Second, we theoretically prove that the effective rank converges to a fundamental limit dictated solely by the antenna's normalized aperture width. We further establish the equivalence between the threshold identified by the geometric algorithm and the derived theoretical limit, providing rigorous validation for the proposed method. Our effective rank model achieves higher accuracy than existing approaches in the literature. Building on this framework, we offer a complete characterization of diversity and coding gains. The analysis leads to a definitive design insight: FAS performance improvements are fundamentally driven by enlarging the antenna's explorable aperture, which increases the effective channel rank, whereas increasing port density within a fixed aperture yields diminishing returns.