= Translation invariants of Fourier-mode phases
{title2=$n\cdot\theta,\quad\sum_j n_jq_j=0$}
Let a <translation symmetry> with displacement $s\in\mathbb R^d$ act on nonzero complex <Fourier modes> by
$$
A_j\longmapsto e^{iq_j\cdot s}A_j,\qquad
A_j=|A_j|e^{i\theta_j}.
$$
Each <phase> changes by $q_j\cdot s$. Therefore a real linear combination of locally unwrapped <phases>, $\sum_j n_j\theta_j$, is unchanged exactly when $\sum_j n_jq_j=0$. This is a <kernel of a linear map> calculation for the matrix whose columns are the <wavevectors>.
A globally defined expression on angular <phases> requires more care: $e^{i\sum_j n_j\theta_j}$ is independent of the representatives chosen <modulo> $2\pi$ precisely when every $n_j$ is an <integer>. The globally defined translation-invariant characters therefore satisfy both $n\in\mathbb Z^m$ and $\sum_jn_jq_j=0$. An arbitrary real vector in the matrix's <kernel of a linear map> gives a local unwrapped invariant, but need not give a single-valued function of the angular <phases>.
For example, take four planar <wavevectors> $q_1=(1,0)$, $q_2=(0,1)$, $q_3=(1,1)$ and $q_4=(1,-1)$. The two independent angular combinations are $\theta_3-\theta_1-\theta_2$ and $\theta_4-\theta_1+\theta_2$, each interpreted <modulo> $2\pi$. Translations can set the first two <phases> to zero; the other two then equal these combinations.
More generally, when four planar <wavevectors> span a common <reciprocal lattice> of rank two, their translation action has a closed two-dimensional <torus> as its image and the quotient of the four-phase <torus> is two-dimensional. Real rank two alone guarantees only two local independent combinations; without a commensurability condition, the translation image can be dense in a larger <torus>. At zero <amplitude>, a <phase> is undefined, so smooth <centre manifold> equations should first use <real parts> and <imaginary parts> of the complex <amplitudes>.
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