Statistical Landscape Exploration and Weak-Field Defects in Selective Oscillator Ising Machines

Oscillator Ising machines (OIMs) provide an analog dynamical approach to combinatorial optimization, but comparatively little is known about the statistical structure of the ensembles they generate. Using the Sherrington-Kirkpatrick (SK) spin-glass model, we introduce a selective oscillator scheme, OIM+, in which successive annealing epochs are combined with energy-based elimination and repopulation of replicas. We characterize the resulting states through overlap distributions, Hamming distances, hierarchical clustering, and local-field statistics. OIM+ explores structured low-energy regions while retaining a nontrivial ensemble of configurations despite repeated selection. We identify sparse weak-local-field residual defects as a characteristic limitation of the dynamics. A short Metropolis-Hastings refinement preferentially corrects these defects, indicating that the oscillator dynamics performs the dominant collective exploration while the remaining corrections are predominantly local. The weak fields arise mainly from cancellation among competing interaction terms rather than uniformly weak couplings. The same behavior persists for an antiferromagnetically biased SK ensemble, where the hybrid method reaches a pooled mean energy comparable to a substantially longer pure Metropolis anneal. The complete protocol also achieves competitive performance on standard G-set Max-Cut benchmarks. These results suggest a natural division of labor between collective analog search and targeted local refinement.