Instanton–Knot Correspondence in Semi-Classical Gauge Theories

A topological classification of nonperturbative gauge configurations via an integer–valued knot invariant \( I(K) \) is developed. The configuration space decomposes into disjoint knot sectors, yielding a block–diagonal operator algebra, discrete vacuum energy splitting \( \Delta E \sim 1/I(K) \), and sector–dependent renormalization group flow. Instanton tunneling amplitudes acquire knot–dependent corrections \( \mathcal{A}_K \sim e^{-S_0 - \delta S(I(K))} \), enforcing confinement without spontaneous symmetry breaking. Quantized shifts in lattice observables, mass gaps, and decay rates provide definitive, testable signatures of knotted gauge fields.