We give arguments for the idea that a suitable discretization of the near horizon geometry acommodates several of the required ingredients for a departure from the semiclassical and local quantum field theory description of black hole horizons, which have led to serious conflicts between the equivalence principle and quantum mechanics. As a first step, we demonstrate how this could be done in the simplest case of the AdS2=SL(2,R)/SO(1,1,R) factor of near horizon geometries of extremal black holes. We employ modular discretization, i.e. we replace the set of real numbers R by the set of integers modulo N, for any positive integer N,under which AdS2 is reduced to the finite geometry, AdS2[N]=SL(2,Z/NZ)/SO(1,1,Z/NZ) . One of the important reasons for such a drastic departure from the continuous geometry is that we can model the discrete time dynamics of stringbit-like microscopic degrees of freedom, living on the AdS2[N] space-time, by generalized Arnold cat maps, A in SL(2,N/NZ), at the classical and quantum level. These maps are known to exhibit, very important and well studied, properties of strong arithmetic chaos, producing dynamical entropy,nonlocality and factorization in the cutoff discretization N,necessary for fast quantum information processing. Finally we propose a new kind of unitary and holographic correspondence, for AdS2[N]/CFT1[N], via coherent states of the bulk and boundary coset geometries.