{"id":"doi:10.5281/zenodo.19761735","kind":"article","title":"A Discrete Geometric Langlands Program: A Neutral, Axiomatic Framework from Six Functors","authors":["Jianming Wang"],"author_count":1,"year":2026,"published":"2026-04-25","venue":"Zenodo (CERN European Organization for Nuclear Research)","cited_by":0,"doi":"10.5281/zenodo.19761735","arxiv_id":null,"nct_id":null,"source_url":"https://doi.org/10.5281/zenodo.19761735","archive_url":"https://space-kitty.com/papers/doi:10.5281/zenodo.19761735","updated":null,"pmid":null,"pmcid":null,"openalex_id":"W7155599733","pdf_url":null,"abstract":"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 $SL_2$) 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","abstract_license":"CC-BY (publisher license, via OpenAlex)","license":"cc-by","is_open_access":true,"retracted":false,"peer_reviewed":null,"categories":["Advanced Algebra and Geometry"],"details":null,"provenance":"openalex","attribution":"Abstract of \"A Discrete Geometric Langlands Program: A Neutral, Axiomatic Framework from Six Functors\" by Jianming Wang, https://doi.org/10.5281/zenodo.19761735, used under its Creative Commons Attribution license (CC-BY); markup removed, otherwise unmodified.","topics":[{"slug":"landmark-math-results","name":"Landmark Results in Pure Mathematics","domain_slug":"math-cs"}],"cited_in_dossiers":[]}