Entropy-Topology Correspondence in Quantum Chaotic Systems: A Metric-Preserving Learning Approach to the SYK Model

The reconciliation of unitary quantum mechanics with the smooth geometry of General Relativity remains a central challenge in quantum gravity, epitomized by the Firewall Paradox. In this work, we investigate the emergent geometry of the black hole information manifold using a metric-preserving machine learning framework. We simulate a 1D quantum black hole using the Sachdev-Ye-Kitaev (SYK) model and reconstruct its bulk geometry via an autoencoder constrained by the Quantum Bures Metric. Our analysis reveals a strong negative correlation between the thermodynamic entropy and the box-counting dimension of the emergent manifold, with finite size scaling consistent with perfect anti-correlation in the thermodynamic limit. Crucially, control experiments reveal that this correspondence is chaos-dependent: integrable free fermion systems show dramatically weaker correlations, while spatially local Heisenberg chains exhibit intermediate behavior. This establishes chaos as a key requirement for strong entropy-topology correspondence, distinguishing maximally chaotic from integrable thermal quantum states. We observe no topological phase transitions across the thermal crossover, supporting the hypothesis that high-entropy chaotic thermalization drives geometric smoothing, consistent with the ER=EPR conjecture.