Nonlinear Emergent Causal Structure from Bures Gradient Flows: Numerical Evidence for a Stable Hyperbolic Attractor

The Emergent Spacetime program (Informational Geometry of Emergent Spacetime – IGES) has shown that the linearized Information Open-systems Flow (IOF) on the Bures manifold of density operators projects to a symmetric hyperbolic hydrodynamic system whose characteristic cone is exactly Lorentzian. This established that spacetime causal structure can emerge from quantum information degradation, but only in the tangent-space approximation. In this work, we address the principal remaining gap: we provide strong numerical evidence for a Nonlinear Emergent Causal Structure Scenario, supporting the conclusion that an effective hyperbolic causal cone acts as a dynamically stable attractor of the full, nonlinear IOF dynamics. We implement a complete numerical simulation of two coupled qubits with dephasing on a 1D lattice, evolving the exact IOF on the 15-dimensional Bures manifold. At each time step we extract the Bures Hessian of the information degradation functional, construct the dynamic spectral gap ∆(t), compute the nonlinear Schur complement Keff(t), and measure the instantaneous cone deviation δ(t) ≡ supk |ω 2 (t, k) − c 2 s(t)k 2 |. We find that: (1) the fast/slow mode separation persists globally; (2) the spectral gap remains strictly positive, ∆(t) ≥ ∆0 > 0; (3) the effective reduced generator Keff(t) remains symmetric in the Bures metric to machine precision; (4) the cone deviation δ(t) decays as e −∆t , falling below 10−4 after a few gap times. These results provide significant evidence that the emergent hyperbolic causal structure is not a linear artifact – the effective causal cone is a robust attractor of the full gradient flow. This nonlinear stability suggests compatibility with Jacobson-type thermodynamic gravity scenarios. The complete simulation code and diagnostic data are provided as supplemental material.