= Endpoint terms in the synchronous Sachs-Wolfe formula
{title2=$W=e^{ik\mu(\tau-\tau_0)}$}
Use $h_{ij}n^in^j=h/3+(\mu^2-1/3)h_s$. Two integrations by parts of the anisotropic source give the exact <Cosmic microwave background anisotropy> amplitude $W_*[\delta_\gamma/4+3i\mu\delta_\gamma'/(4k)+i\mu(h'-h_s')/(2k)+h_s''/(2k^2)]_*+[i\mu h_s'/(2k)-h_s''/(2k^2)]_0-\int_*^0W[(h'-h_s')/6-h_s'''/(2k^2)]d\tau$. The emission velocity was eliminated using $\delta_\gamma'+4ik\cdot v/3+2h'/3=0$. The local observer bracket is a monopole plus a dipole and can be discarded only after explicitly restricting to the measured higher angular multipoles. This identity fixes endpoint signs and integral factors without importing a different scalar-metric convention.
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