Dissipative partial differential equations, such as the Kuramoto-Sivashinsky equation (KSE), feature infinite-dimensional state spaces that contract onto a finite-dimensional inertial manifold [1]. While analytical bounds exist, they are often loose [2]. Precise estimates utilizing Covariant Lyapunov Vectors (CLVs) [3] or Unstable Periodic Orbits [4] reveal extensive scaling but are computationally prohibitive for large spatial domains. Furthermore, existing data-driven model reduction methods [5] often rely on heuristic error thresholds, lacking a rigorous mechanism to identify the minimal intrinsic dimension. To address these challenges, we introduce a Spectral Autoencoder that identifies the inertial manifold dimension ($d_M$) via a novel ``horizon-induced scaling bifurcation." We reveal that a topological phase transition in the reconstruction error landscape emerges strictly when the observation time horizon exceeds the system's mixing scale. This bifurcation serves as a robust indicator of $d_M$. Our results confirm that the manifold dimension scales linearly with system size ($d_M \approx 0.364L$). Additionally, we demonstrate the holographic nature of the attractor: the manifold topology is robustly encoded in the low-frequency core and remains invariant under high-frequency spectral truncation. This framework bridges deep learning and dynamical systems theory, offering a scalable tool for characterizing turbulence [6].