= Small-vorticity phototactic orientation response
{title2=$\langle\mathbf p\rangle=K_0\mathbf e_x+\varepsilon K_1\mathbf e_z+O(\varepsilon^2)$}
For light along $\mathbf e_x$, <vorticity> along $\mathbf e_y$ and $\varepsilon=\omega/(2D_r)$, spherical coordinates $\mathbf p=(\cos\theta,\sin\theta\cos\phi,\sin\theta\sin\phi)$ give a first orientation-density correction proportional to $\sin\phi\,g(\theta)$. The linearized alignment-diffusion operator preserves azimuthal harmonics and the rotation of the axisymmetric equilibrium forces only this sine harmonic. Angular integration makes the first correction to the mean purely vertical. Uniqueness follows from the weighted identity $\mathcal A(f_0h)=\nabla_p\cdot(f_0\nabla_ph)$: <integration by parts> shows the only homogeneous smooth solution is a multiple of $f_0$, removed by normalization.
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