Far-from-equilibrium topological phase transition in one dimension

What the math gives to ML

The paper identifies a concrete nonequilibrium mechanism in which compact phase variables support spacetime vortices, while phase-shift symmetry creates a homogeneous fixed point that suppresses vortex nucleation. The competition between defect proliferation and relaxation toward homogeneity produces a continuous transition in the directed-percolation universality class. This can transfer to phase-valued recurrent or state-space networks by measuring hidden-state spacetime vortices and controlling recurrent gain or relaxation strength relative to the defect-proliferation threshold. The most testable prediction is that active-defect density should vanish continuously at a critical gain, rather than changing smoothly without a sharp transition.