Nonlinear Stability of Emergent Hyperbolic Causal Structure from Bures Gradient Flows

The Informational Geometry of Emergent Spacetime (IGES) framework posits that spacetime geometry and Lorentzian causality emerge from the relentless minimization of quantum information degradation, described as a gradient flow on the Bures manifold of density operators. While previous work established the emergence of a symmetric hyperbolic hydrodynamic system in the linearized regime, the stability of this structure under full nonlinear dynamics remained a critical open question. In this work, we provide strong numerical evidence for the Nonlinear Emergent Causal Structure Scenario. By simulating a 1D lattice of coupled qubits under the exact Information Open-systems Flow (IOF) on a 15-dimensional Bures manifold, we demonstrate that the effective hyperbolic causal cone acts as a robust, dynamically stable attractor. Key findings include: 1. The global persistence of fast-slow mode separation far from equilibrium. 2. The continuous maintenance of a strictly positive spectral gap ($\Delta(t) > 0$). 3. The preservation of Onsager symmetry within the effective reduced generator to machine precision. 4. The exponential decay of cone deviation ($\delta(t)$) at a rate matching the spectral gap, with the effective speed of sound ($c_s$) converging to the theoretical attractor value $\sqrt{\lambda\gamma}$. These results suggest that the arrow of time and causal structure are not linear artifacts but fundamental properties of the nonlinear information-geometric flow. We discuss the implications of this stability for the information-geometric origin of gravity and draw phenomenological parallels with the rapid hydrodynamic attractor observed in Quark-Gluon Plasma (QGP) experiments at CERN.