Laplacian Locking of Integer Winding

We develop a unified topological–dynamical framework for a three-layer Möbius phase-loop system with π/2 interlayer constraints. Continuous perturbations are re-encoded as integer holonomy winding via the decomposition Δ = 2πq + ε, separating a topologically conserved winding number from a bounded residual phase. The central principle is that complete U (1) independence occurs if and only if the relative winding is integer. A key new result is the formulation of independence as a Laplacian locking condition. The residual phase ε satisfies a defect-weighted sine-Gordon equation, ∂tε=DΔε−κ (x) sin⁡ε, ₜ = D - (x), ∂tε=DΔε−κ (x) sinε, whose unique coupling-free steady state is ε ≡ 0. This condition is shown to be equivalent to the vanishing of an inter-strand coupling energy functional, thereby providing an energetic and dynamical characterisation of independence. Defects act as spatially localised locking potentials: they absorb continuous mismatch into discrete winding quanta through a threshold mechanism while preserving the integer winding skeleton. In this picture, continuous divergence is not amplified but converted into topologically protected winding. The framework admits a non-Abelian extension in which the three-layer system lifts to SU (2) and SU (3) gauge-field descriptions. The transition from torus to sphere topology corresponds to the progressive truncation of off-diagonal gauge components, yielding the cascade su (3) → su (2) → u (1). Prime factorisation naturally appears as a discrete set of irreducible holonomy generators, providing an arithmetic encoding of the winding structure. These results establish a bridge between cohomological obstruction, defect-mediated quantisation, nonlinear PDE dynamics, and gauge-field reduction, offering a structural mechanism for divergence control via topological locking.