ON THE AFFINE INVARIANT OF HYPERSEMITORIC SYSTEMS

The paper proves existence and invariance of an affine invariant for hypersemitoric systems by constructing J-preserving action-coordinate maps on the unfolded momentum domain, after introducing vertical cuts at rank-0 values, with uniqueness up to the subgroup T; it treats flaps (one cut per flap or per elliptic–elliptic value), pleats (two natural maps), and assembles the general case layer-wise (Theorems 5.1, 5.8, 6.1, 8.2). The candidate solution follows the same strategy, correctly invoking action-angle variables, vertical cuts, layer-wise construction on the unfolded domain, uniqueness modulo T, and invariance under isomorphisms. The only substantive mismatch is that the candidate asserts that, in the pleat case, the two maps differ by a vertical shear in T; the paper instead treats the two maps as distinct representatives (each unique modulo T) rather than explicitly T-equivalent (Definition 6.2 and Theorem 6.1). This discrepancy does not affect the core existence-and-invariance statement of Theorem 1.1, so overall both are essentially aligned, with the model requiring a minor correction for the pleat case.