Arithmetic Discrete Geometry and the Langlands Program: A Limit Approximation Approach in Characteristic p

This paper systematically constructs a discretized theory of the Langlands program for function fields of characteristic p > 0. By introducing equipped graphs as combinatorial approximations of algebraic curves, we propose an "arithmetic discretization" paradigm: Local Theory: Define discrete valuation rings Oₓ^ and discrete local fields Fₓ^, rigorously establishing the closure decomposition S^_{} = S^_ of the discrete affine Grassmannian Gr₆, ₗ^ and discrete geometric Satake equivalence (verified for G = SL (2) ) ; Global Framework: On equipped single-loop graphs (b₁ () =1), construct the discrete moduli stack BunG^ () of principal G-bundles and the discrete L G-local system stack LocSysL G^ (), proposing the categorical equivalence conjecture D℈ₓ (BunG^) -₄₈₆₄₍ QCoh (LocSysL G^) ; Computability: Approximate Frobenius traces of classical sheaves via inverse systems, providing an algorithmic foundation for efficient L-function computation (Weak Comparison Conjecture). This work offers a discrete realization path for the characteristic p Langlands correspondence, combining rigor with computability.