Horizons, Ordered Green's Functions, and Entropic Geometry: Propagation, Scrambling, and Emergent Einstein Dynamics.

We present a propagation-first synthesis connecting horizon thermodynamics, ordered Green's functions, and entropy-weighted metric fluctuations. Horizons impose a restriction on propagation domains that produces an ordering asymmetry in correlation functions. This asymmetry naturally generates the KMS condition and the effective temperatures associated with Unruh and Hawking radiation. We show that horizon emission and absorption can be expressed in terms of Einstein-like coefficients derived from ordered Green's functions and greybody transmission factors. In an entropy-weighted metric fluctuation framework, diffusion kernels and heat kernels provide a bridge from metric fluctuations to propagators. Gravitational dynamics then arise as a consistency condition between propagation and geometry, yielding an expectation-valued Einstein equation. We further clarify the horizon information problem: information is not destroyed but scrambled into high-order correlations through redshift and phase mixing near horizons.