 Translation invariants of Fourier-mode phases

If a translation symmetry acts on complex Fourier modes by $A_j\mapsto e^{i{q}_j\cdot s}{A}_j$, a linear combination of their phases is invariant precisely when its coefficient vector lies in the kernel of the matrix of wavevectors. For four amplitudes whose wavevectors span a plane, this leaves two independent phase variables after quotienting translations. At zero amplitude, polar phases are undefined, so smooth centre manifold equations should first be written in Cartesian amplitudes.