Van Hemmen interactions in a one-dimensional swarmalator model
We study a one-dimensional swarmalator model with van Hemmen pair disorder in the phase coupling. Pair disorder has two effects. First, it splits the static ring-model states into sync, split, splay, and phase-wave branches organized by the rainbow order parameters $r,s$ and four sign-weighted glass order parameters. Second, because the oscillators move, it creates active macrostates absent from the immobile Kuramoto-van Hemmen model: a bursty active async state and a glassy phase wave with rotating glass order. For balanced sign patterns we derive a six-field reduction, the exact finite-$N$ sync boundary, a closed first split branch with its first spatial destabilization, and an exact antiphase phase-wave branch. The iid sign audits preserve the tested state ordering but shift finite-$N$ thresholds through sample imbalance. The remaining challenge is a nonlinear theory of the two active branches.