Endpoint terms in the synchronous Sachs-Wolfe formula ($W=e^{ik\mu (\tau -{\tau }_0)}$)

Use $h_{ij}{n}^i{n}^j=h/3+({\mu }^2-1/3)h_s$. Two integrations by parts of the anisotropic source give the exact Cosmic microwave background anisotropy amplitude $W_{\ast }[{\delta }_{\gamma }/4+3i\mu {\delta }_{\gamma }^{\prime }/(4k)+i\mu (h^{\prime }-h_s^{\prime })/(2k)+h_s^{\prime \prime }/(2k^2){]}_{\ast }+[i\mu h_s^{\prime }/(2k)-h_s^{\prime \prime }/(2k^2){]}_0-{\int }_{\ast }^{0}W[(h^{\prime }-h_s^{\prime })/6-h_s^{\prime \prime \prime }/(2k^2)]d\tau$. The emission velocity was eliminated using ${\delta }_{\gamma }^{\prime }+4ik\cdot v/3+2h^{\prime }/3=0$. 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.