Studying network of symmetric periodic orbit families of the Hill problem via symplectic invariants

The paper and the candidate solution use the same symplectic-invariant toolkit: planar/spatial splitting of the reduced monodromy and additive Conley–Zehnder indices with the iteration formula μp,s(ξn)=1+2⌊nθp,s/2π⌋, plus B-/C-sign guided crossing analysis and preservation of the local Floer Euler characteristic. The paper states these rules explicitly and applies them to build bifurcation graphs, including for n=5: at g(5,1) the index on the mother branch jumps 22→24 and two doubly symmetric spatial branches are born, one starting at 23 (OX∩OY symmetry) that bridges to f(7,1), and one starting at 22 (XOZ∩YOZ symmetry) that meets B(6,1)0 after two b–d events and one index jump; this is exactly the pattern asserted by the model. See the index-splitting/iteration rules and jump law (eqs. (23)–(24), “The index jump”) and the explicit n=5 graph and caption descriptions for g(5,1)↔f(7,1) and g(5,1)↔B(6,1)0, as well as the summary of low-energy anchors μCZ(g)=3+3 and μCZ(f)=1+1 used for bookkeeping. These match item-for-item the model’s claims and numerical labels, and the same mechanism is documented for n=3 (and n=4) in the paper’s earlier sections.