= Solution
Express the <Collisionless Boltzmann equation> in variables $(\mathbf x,q,\mathbf n,\tau)$ and divide the derivative along the photon ray by $d\tau/d\lambda$. Since $f_0(q)$ has no explicit position, direction or conformal-time dependence,
$$
0=\frac{\partial f_1}{\partial\tau}
+P^i\frac{\partial f_1}{\partial x^i}
+\frac{dq}{d\tau}\frac{df_0}{dq}
+\frac{dq}{d\tau}\frac{\partial f_1}{\partial q}
+\frac{dn^i}{d\tau}\frac{\partial f_1}{\partial n^i}.
$$
This is the chain rule in momentum magnitude and direction. At linear order, $P^i=n^i+O(h)$, while the last two terms are products of first-order perturbations and can be dropped. The <synchronous photon momentum redshift> from part i then gives, for spatial Fourier convention $e^{i\mathbf k\cdot\mathbf x}$,
$$
\boxed{f_1'+ik\mu f_1=\frac12q\frac{df_0}{dq}h'_{ij}n^in^j,\qquad
\mu=\widehat{\mathbf k}\cdot\mathbf n.}
$$
The factor $q$ is essential. It is missing in the PDF's displayed intermediate equation, but follows directly from part i and is needed to obtain its final brightness equation.
Let $I_0=\int_0^\infty q^3f_0(q)\,dq$. Absorbing the common phase-space normalization into $f$, the homogeneous photon energy density is $\rho_\gamma=4\pi I_0/a^4$, so the <photon brightness perturbation> is
$$
\Delta=\frac{\int_0^\infty q^3f_1\,dq}{I_0}.
$$
A small directional temperature change $\Theta=\Delta T/T$ in a <Planck distribution> gives $f_1=-qf_0'(q)\Theta$. Integration by parts yields
$$
\int_0^\infty q^4f_0'(q)\,dq
=\left[q^4f_0(q)\right]_0^\infty-4I_0=-4I_0,
$$
because the <Planck distribution> behaves as $q^{-1}$ at low momentum and decays exponentially at high momentum. Thus $\Delta=4\Theta$. Equivalently, frequency-integrated <blackbody radiation> energy scales as $T^4$. Identifying brightness with a single temperature perturbation assumes this <blackbody> spectral form; the brightness integral itself is defined more generally.
Finally, multiply the corrected transport equation by $q^3$, integrate and divide by $I_0$. The spatial streaming factor is independent of $q$, while its gravitational source integrates to $(1/2)(-4)h'_{ij}n^in^j$. Therefore
$$
\boxed{\Delta'+ik\mu\Delta=-2h'_{ij}n^in^j.}
$$
This derives the <synchronous photon brightness equation> with the normalization and sign fixed, rather than inferring it from the dimensionally inconsistent intermediate formula.
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