Superintegrability and choreographic obstructions in dihedral $n$-body Hamiltonian systems

We analyze planar $n$-body Hamiltonian systems with quadratic $D_n$-invariant interactions and identify the symmetry obstruction to choreographic motion. Choreographies are taken throughout to be collision-free solutions of the equations of motion in which all bodies traverse one closed curve with uniform time shifts. By diagonalizing the dynamics into discrete Fourier sectors, we show that superintegrability, periodicity, and choreography are governed by distinct conditions: commensurability of the active frequencies closes bounded motions, whereas a sectorwise $C_n$ phase-matching condition is required for full equivariance. At the configuration level this equivariance is already equivalent to a genuine simple choreography. Thus generic resonant multi-sector motions are periodic but multi-trace, while true choreographies occur only on phase-matched loci, in single irreducible sectors, or through effective one-sector reductions produced by exact degeneracy. The cases $n=4,5,6$ exhibit this mechanism explicitly, with $n=6$ marking the first distinction between nondegenerate commensurability and additional exact degeneracy.