= Translation invariants of Fourier-mode phases
{title2=$n\cdot\theta,\quad\sum_j n_jq_j=0$}
If a <translation symmetry> acts on complex <Fourier modes> by $A_j\mapsto e^{iq_j\cdot s}A_j$, 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>.
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