Higher-Categorical Lattice Gauge Theory

This article establishes an exact synthesis between higher category theory, discrete exterior calculus, and the topological soliton description of hadrons, resolving the continuity pathology of non-Abelian topological charge in lattice gauge theory. By embedding finite-dimensional multiplicative bundle gerbes and Kan simplicial manifolds onto the canonical flag complex of a discrete Johnson association scheme, the framework defines local Chern-Simons connections and integer instanton density operators without relying on cooling, gradient flow, or artificial fermion spin structures. The work resolves the open delooping obstruction of higher gauge theory—in which higher-dimensional hypercube boundaries fail to close due to ordering ambiguities—by showing that boundary anchor nodes dictated by the graph spectrum host interlocking Borromean bridge defects whose braiding monodromy identically satisfies the Yang-Baxter relation. In the gauge sector, the discrete Hodge Laplacian possesses an unconditional positive spectral gap, while complex multiplication over the maximal one-class imaginary quadratic field stabilizes the modular geometry, advancing constructive field theory by establishing a non-perturbative mass gap in the continuum limit. Within this substrate, the nucleon emerges as a stable, defect-free Borromean topological soliton carrying strictly conserved baryon number, accounting for the anomalous missing proton spin through boundary orbital precession and guaranteeing absolute mechanical stability against spontaneous decay through a rigid ten-dimensional internal moduli space.