Mean-field limits of phase oscillator networks and their symmetries

The paper proves (i) for W=1, every measure‑preserving transformation is a symmetry and the subspace M of x‑constant maps is dynamically invariant with the induced flow equal to the mean‑field system (Theorem 4.2), leveraging Proposition 3.6 on symmetries and an ergodicity argument to characterize the fixed point set ; and (ii) TI (Dirac fields) is invariant and the induced dynamics is exactly the graphon ODE (Theorem 5.2) . The candidate solution establishes the same two conclusions: (1) by a direct change‑of‑variables argument for symmetry when W≡1 and by observing the vector field is independent of x to prove invariance of M and exact reduction; (2) by noting push‑forwards preserve Dirac measures and differentiating along the flow to obtain the graphon system. The approaches differ slightly—paper uses symmetry/fixed‑point‑set theory (with ergodicity), while the model uses a direct vector‑field argument and flow properties—but they agree in substance. Minor omissions in the model are technical hypotheses ensuring the flow is well-defined (e.g., D 1‑Lipschitz and the standing continuity assumption on W and the flow map), which the paper states explicitly (Lemma 3.2, Def. 3.3) . Overall, both are correct.