Discrete Graph-Cuts and Evanescent Quantum Walks: A Bipartite Derivation of Alpha Decay and Gamow Tunnelling

We reformulate alpha decay as a two-stage algorithmic process on a discrete Z3 ⊗ Q3 topological lattice. Standard continuous wave mechanics models the propagation of an alpha particle through a Coulomb barrier (Gamow tunnelling) but relies on phenomenological fits for the preformation prob- ability and assault frequency (the Geiger-Nuttall intercept). In this framework, the pre-tunnelling rate is derived as a calculable geometric edge-cut problem. Feshbach projection of the lattice walk operator yields a detachment rate of 1/36 per severed macroscopic gauge bridge. Assuming rapid many-body decoherence renders this a Markov process, the detachment rate scales as 36−k , where k is the integer number of boundary bridges. For a 2 × 2 cluster detaching via a minimal topological neck (k = 2), the framework predicts a pre-tunnelling emission rate of 1.14 × 1020 s−1, recovering empirical preformation phenomenology without fitted parameters. Post-detachment, the evanescent spatial propagation takes the form of a discrete tight-binding Riemann sum, which converges to the standard macroscopic WKB phase integral, explicitly recovering the 2πα√2mαc2 Gamow slope. Evaluating this unified equation end-to-end for 238U and 212Po yields effective barrier radii perfectly consistent with standard diffuse nuclear surfaces.