Hypoelliptic Propagation of Gevrey Singularities and the Microlocal Bifurcation of Wavefront Sets for Degenerate Hyperbolic Operators with Ergodic Principal Symbols at the Event Horizon

Mathematical Applications of Science Fiction We present a rigorous microlocal analysis of a class of degenerate hyperbolic operators P (x, D) on a Lorentzianmanifold (M, g) possessing an event horizon H. We investigate the propagation of Gevrey regularity Gs(s ≥ 1) across the characteristic variety Σ ⊂ T ∗M \ 0. The principal symbol pm(x, ξ) is assumed to vanish tosecond order on a symplectic submanifold, exhibiting an ergodic Hamiltonian flow on the reduced phase space.We establish a global hypoellipticity result for operators where the sub-principal symbol satisfies a generalizedMelin inequality twisted by the ergodic measure. Furthermore, we characterize the bifurcation of thewavefront set WFG (u) at the horizon, proving that singularities propagate along branching bicharacteristicsdetermined by the eigenvalues of the fundamental matrix of the linearization of the Hamiltonian vector field.We utilize FBI transforms, Fourier Integral Operators (FIOs) with complex phase, and symplectic reductionto prove that information, encoded as microlocal singularities, is scrambled but conserved in the Gevrey class,suggesting a holographic resolution to the classical information loss paradox in the semiclassical limit.