A Discrete Geometric Langlands Program: A Neutral, Axiomatic Framework from Six Functors

We propose a foundational framework for a discrete geometric Langlands program, built from the ground up on an axiomatic discrete geometric world. The guiding principle is that a serious discrete theory should not begin by imitating the final statements of geometric Satake or Hecke eigensheaves, but by specifying a discrete six-functor environment on finite geometric models and correspondences. Only within such an environment do we construct automorphic and spectral categories, formulate Hecke actions, and investigate Langlands-type functoriality. The framework is deliberately neutral: it relies on neither the arithmetic tools of the classical number-theoretic Langlands program nor the derived algebraic geometry of the characteristic-zero geometric Langlands proof. Instead, it begins from a purely combinatorial, category-theoretic foundation—strict discrete curve prototypes, refinement systems, and finite moduli prototypes—and builds the six operations, Verdier duality, and correspondence calculus at a finite, explicit level. The abelian and toric sectors are treated as the first rigorous testing ground, yielding complete finite-level Langlands transforms via explicit Fourier kernels and Pontryagin duality. The non-abelian rank-one problem (for SL₂) is isolated as the first decisive bottleneck, and the passage to higher rank is formulated as a structural extension problem. The program is organized into two branches: a characteristic-zero branch focused on refinement compatibility and comparison with the established geometric Langlands theorem, and a characteristic-p branch where Frobenius data and arithmetic exactness may provide a more rigid constructive environment. No non-abelian discrete Langlands theorem is claimed in this draft. The purpose of the present manuscript is to supply a precise, non-circular, and independently verifiable blueprint for such a theory, to separate verified input from programmatic conjecture, and to make the remaining constructive gaps mathematically explicit. --- Keywords Discrete geometric Langlands program; six-functor formalism; axiomatic discrete geometry; finite moduli prototypes; refinement systems; automorphic categories; spectral categories; Hecke correspondences; Langlands duality; neutral framework; combinatorial Langlands; characteristic p; characteristic zero