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 \mathcal{O}_x^{\delta} and discrete local fields F_x^{\delta}, rigorously establishing the closure decomposition \overline{S^{\delta}_{\lambda}} = \bigsqcup_{\mu \leq \lambda} S^{\delta}_{\mu} of the discrete affine Grassmannian \mathrm{Gr}_{G,x}^{\delta} and discrete geometric Satake equivalence (verified for G = \mathrm{SL}(2)) ; Global Framework: On equipped single-loop graphs (b_1(\Gamma)=1), construct the discrete moduli stack \mathrm{Bun}_G^{\delta}(\Gamma) of principal G-bundles and the discrete {}^L G-local system stack \mathrm{LocSys}_{{}^L G}^{\delta}(\Gamma), proposing the categorical equivalence conjecture D_{\text{ét}}(\mathrm{Bun}_G^{\delta})_{\sigma\text{-eigen}} \simeq \mathrm{QCoh}(\mathrm{LocSys}_{{}^L G}^{\delta}) ; 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.