Superintegrability and choreographic obstructions in dihedral $n$-body Hamiltonian systems
We study planar $n$-body Hamiltonians with quadratic pair interactions whose coupling pattern is invariant under the dihedral group $D_n$. After the center of mass is removed, the dynamics separates into label-Fourier modes. This makes it possible to distinguish three notions that are often conflated: commensurability of the normal frequencies, maximal superintegrability of the full relative Hamiltonian, and realization of choreographic space--time symmetry. We prove that a periodic configuration is $C_n$-equivariant if and only if every label-Fourier coefficient acquires under the time shift $T/n$ the character phase prescribed by cyclic relabeling. For harmonic motion this criterion determines the complete equivariant subspace of initial data and its codimension. Thus, a resonant orbit may be periodic without being a choreography. The cases $n=4,5,6$ illustrate the obstruction. In particular, $n=6$ is the first case with three inequivalent internal branches: the nondegenerate $1{:}2{:}3$ resonance admits a phase-matched three-branch choreography, whereas the exactly degenerate $1{:}2{:}2$ spectrum is incompatible with simultaneous $m=2$ and Nyquist $m=3$ excitation. A choreography on the latter surface survives only in a compatible invariant subspace. Resonant data outside the equivariant subspace can instead organize into synchronized multi-trace fragments.