Krylov Subspace Dynamics as Near-Horizon AdS₂ Holography

We establish a holographic gravitational dual for the fundamental dynamical equations governing operator growth in Krylov subspace. Specifically, we show that the deep interior of the Krylov subspace maps directly to the near-horizon regime of AdS₂ gravity. We demonstrate that, in the continuum limit, the discrete evolution on the Krylov chain transforms into the dynamics of a continuous field, which is isomorphic to the Klein-Gordon equation for a scalar field in the AdS₂ throat. This correspondence identifies the linear growth rate of Lanczos coefficients with the Hawking temperature, α=πT, thereby recovering the saturation of the maximal chaos bound. Notably, the Breitenlohner-Freedman bound, a fundamental stability criterion in AdS gravity, emerges as a necessary consistency requirement for the dual description of Krylov subspace dynamics. Our results advance a Krylov-based holographic dictionary in a unified SL\(2, mathbb{R}\) representation, revealing that the emergent geometry of Krylov subspace is a reflection of the near-horizon AdS spacetime.