Defect-Induced Möbius Holonomy and Quantised Wilson Loops

This work establishes a continuum bridge between discrete defect dynamics and gauge-theoretic holonomy by introducing a Möbius-twisted parallel transport framework. Starting from a multi-layer system with sign-alternating interactions, we define a discrete Möbius holonomy and prove its convergence to the path-ordered exponential, i.e., the Wilson loop, in the continuum limit. A central result of the paper is that the Möbius sign alternation is not merely an imposed boundary condition but emerges dynamically from defect formation. When curvature concentrates into localised defects, each defect acts as a holonomy wall that reverses the orientation of parallel transport, generating an alternating domain structure. In the continuum limit, this structure converges to a Möbius twist. Within this framework, trivial holonomy corresponds to a quantisation condition on the total phase, leading to a flat connection regime that we interpret as a “prime state.” Deviations from this quantised condition produce residual holonomy phases, identified with defect mass and interaction load. This provides a unified interpretation of defect-mediated blow-up suppression as a holonomy quantisation mechanism, linking discrete anchor dynamics to continuum Wilson loop structures and offering a geometric pathway connecting nonlinear PDE dynamics and gauge theory.