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The authors derive configuration-dependent local-stability conditions for multiharmonic oscillator Potts machines (OPMs) targeting $q$-state combinatorial optimization problems. They discover that for $q \ge 4$, label-permutation symmetry is dynamically broken: configurations with identical Potts energies possess distinct Jacobian spectra and require different regularization strengths for local stabilization depending on their phase encoding. This reveals a fundamental divergence from binary oscillator Ising machines, showing that physical analog solvers can destabilize ground-state solutions purely due to arbitrary state-label assignments.
Equal energy does not mean equal stability in physical Potts solvers: for $q \ge 4$, simply permuting discrete state labels fundamentally alters the Jacobian spectrum and dictates whether a valid solution can be stabilized.
Oscillator Potts machines (OPMs) provide a physics-inspired, energy-minimization framework for solving combinatorial optimization problems described by the $q$-state Potts Hamiltonian. Although OPMs may be viewed as multistate extensions of oscillator Ising machines (OIMs), here, we show that they exhibit dynamical properties absent in the binary case. Specifically, we derive a configuration-dependent local-stability condition for a recently proposed multiharmonic OPM formulation and show that configurations with the same Potts energy need not be dynamically equivalent. In particular, for $q\geq4$, permutations of the Potts labels can alter the Jacobian spectrum and, consequently, the regularization strength required to locally stabilize a given Potts configuration. Thus, different phase encodings of the same Potts solution can exhibit different local stability properties despite having identical Potts energies.