Polynomial-source photon transport solution ($\Delta (\tau )=[{\Delta }_R-B_R]e^{-ik\mu (\tau -{\tau }_R)}+B(\tau )$)

For $\dot{\Delta }+ik\mu \Delta =2{\mu }^2\dot{h}$ with $\stackrel{...}{h}=0$ and $k\ne 0$, a particular solution is $B=-2i\mu \dot{h}/k+2\ddot{h}/k^2$. Differentiating it cancels its $\ddot{h}$ terms and gives the prescribed source. The homogeneous solution transports the initial condition with a direction-dependent Fourier phase. The inverse powers of $k$ need not signal a physical singularity: use the original source integral to take a long-wavelength limit consistently.

A direction-independent term at a fixed observer contributes an angular monopole; a term linear in photon direction contributes a dipole. They do not directly generate higher angular multipoles. In a gravitational line-of-sight solution, separating such endpoint terms from a transported recombination term can clarify which contribution appears in a sky anisotropy map. The observable must retain all emitter/observer terms consistently with the metric gauge and the chosen brightness normalization.