Phase-Mediated Coupling in Oscillator-Based Ising Machines: A Preliminary Theoretical and Numerical Study

This work presents preliminary theoretical and numerical groundwork (Technology Readiness Level 2) for a coupled-oscillator Ising machine architecture in which phase-change material dynamics serve as both the nonlinear oscillator substrate and, as an exploratory hypothesis, the coupling medium between oscillators. Using a modified Kuramoto phase model, we demonstrate robust in-phase locking under positive coupling, consistent with established synchronization theory. We identify that the corresponding anti-phase state is structurally unstable for symmetric oscillator pairs, and show numerically that introducing a small frequency detuning (as little as 1%) resolves this limitation, enabling stable dual-state encoding required for Ising spin representation. We situate this work relative to recent experimental literature on phase-transition-material-based Ising solvers (notably Dutta et al. 2021, Maher et al. 2024), and outline next steps including circuit-level (SPICE) validation, lattice-scale simulation, and material characterization. All numerical methods, simulation code, and a transparent account of methodological limitations encountered during this work are included, in the interest of reproducibility and scientific transparency. Keywords: Ising machine, coupled oscillators, phase-change materials, Kuramoto model, Van der Pol oscillator, combinatorial optimization, CMOS-compatible computing