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Description
Free-electron interactions with optical near fields are conventionally described as one-dimensional energy ladders, in which multicolor photon-exchange pathways can become spectrally degenerate. Here, we show that this degeneracy can be unfolded by coupling a single free electron to commensurate two-color optical vortices, thereby realizing a programmable synthetic lattice spanned by the electron energy and orbital angular momentum (OAM). The resulting single-particle platform maps the free-electron light interaction onto an effective tight-binding Hamiltonian whose primitive hopping vectors and coupling anisotropy are controlled by the optical OAM charges and near-field amplitudes. We use this construction to simulate a Lifshitz transition in the synthetic quasienergy band structure, identified by a topological reconnection of the zero-quasienergy contour and the associated Van Hove singularity in the density of states. We further show that the generated electron state exhibits tunable intra-particle energy--OAM mode nonseparability, which serves as a diagnostic of the synthetic-lattice geometry. This work unlocks the potential of free electrons as a versatile, reconfigurable synthetic-lattice simulator compatible with ultrafast electron microscopy and nanoscale near-field engineering.