(Non)displaceability in semitoric systems

Part (1): The paper proves nondisplaceability of focus–focus fibers with multiplicity ≥2 by a clean topological argument: an embedded Lagrangian S^2 has nonzero self-intersection (via the Lagrangian neighborhood identification with T* S^2), hence cannot be displaced by any map homotopic to the inclusion, in particular not by a Hamiltonian diffeomorphism (Proposition 3.3 and Corollary 3.4 in the paper ). The model instead asserts, without adequate hypotheses, that any embedded Lagrangian S^2 in dimension four is Floer-unobstructed with HF(L,L) ≅ H*(S^2), which is not generally valid absent monotonicity or bounding-cochain assumptions; thus its justification is flawed. Part (2): Both paper and model use a nodal trade to pass to a toric picture and then displace under explicit rectangle/half-length conditions; the paper displaces the square preimage directly (Lemma 3.5 ), while the model appeals to probes. So the paper is correct on both parts; the model is incorrect on Part (1) but essentially correct on Part (2).