= Solution
<wave phase> labels the stage of a physical oscillation at a <spacetime event>. Two observers comparing the same event must identify the same crest, trough or intermediate stage: a change of coordinates cannot change the number of completed cycles between physically specified wavefronts. The continuous <wave phase> can therefore be treated as a <Lorentz scalar>, up to a common choice of <wave phase> origin. This does not assert invariance of the measured <angular frequency>, which is the rate of change of that <wave phase> along the observer's trajectory.
For <Lorentz invariance of plane-wave phase>, write
$$
\Phi=\omega t-\mathbf k\cdot\mathbf r+\Phi_0
=K_\mu x^\mu+\Phi_0,\qquad
x^\mu=(ct,\mathbf r),\quad K^\mu=(\omega/c,\mathbf k),
$$
using the metric with signature $(+,-,-,-)$. The <gradient> $K_\mu=\partial_\mu\Phi$ transforms as a <covector> because $\Phi$ is a scalar. Raising its index therefore gives the <four-wavevector>. Explicitly, substituting the inverse boost $t=\gamma(t'+vx'/c^2)$, $x=\gamma(x'+vt')$ gives
$$
\boxed{\omega'=\gamma(\omega-vk_x),\quad
k_x'=\gamma(k_x-v\omega/c^2),\quad k_y'=k_y,\quad k_z'=k_z}.
$$
These are exactly the <Lorentz transformation> laws for $(\omega/c,\mathbf k)$. In <vacuum>, $K_\mu K^\mu=0$; the <Doppler effect> and <aberration of light> are two consequences of transforming the same <four-vector>.
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