Structure of the codimesion one gradient flows with at most six singular points on the Möbius strip

The paper enumerates all Morse (gradient-like) flows on the Möbius strip with up to six singular points via separatrix diagrams and uses a doubling/Poincaré–Hopf index constraint to screen possibilities; it then counts codimension-one bifurcations (SN, SC, BSN, BDS, HN, HS, HSC, BSC). Its stated results — one class for N=3; four for N=4 (with 2 HN, 2 BSN, 1 BDS, 2 HSC, 1 BSC); fifteen for N=5 (with totals 10 SN, 14 SC, 6 BSN, 2 BDS, 4 HN, 10 HS, 4 HSC, 2 BSC); and for N=6 (36 SN, 15 SC, 48 BSN, 21 BDS, 30 HN, 14 HSC, 2 BSC) — match the candidate solution’s counts exactly. The model’s argument is more structural: it justifies completeness via separatrix ribbon diagrams (with cyclic orders and a Möbius parity) and derives the index constraint by doubling to the Klein bottle. The paper does not present the invariant’s completeness formally and contains minor slips (e.g., a misstatement that node indices are 0, and a small inconsistency in the N=4 SN column of Table 1), but its main conclusions and the diagram-based classification agree with the model’s approach and totals.