= Vacuum conservation of specific intensity
In a transparent, stationary <vacuum> with <Euclidean geometry> and observers at rest in the same frame, the <radiative transfer equation> has zero absorption and emission, so $dI_\nu/ds=0$ along a <light ray>. The same <specific intensity> reaches the observer without geometric dilution. More generally, $I_\nu/\nu^3$ is a <Lorentz invariant>; collisionless propagation preserves this ratio along a ray even with gravitational or cosmological <redshift>, rather than preserving $I_\nu$ at shifted <frequencies>. https://www.roe.ac.uk/japwww/teaching/ram5/ram5_2018.pdf#page=27[John Peacock, Radiation and Matter, 2018–2019, §4.1, Conservation of intensity and equation (140)]{ref}.
The elementary geometry is instructive. Two facing surface elements separated by distance $s$ exchange spectral <radiant flux>
$$
d^3\Phi=I_\nu\frac{\cos\theta_1\cos\theta_2}{s^2}\,dA_1\,dA_2\,d\nu.
$$
The receiver subtends $d\Omega_2=\cos\theta_2\,dA_2/s^2$ at the source. Describing that same traveling beam at the receiver instead uses $d\Omega_1=\cos\theta_1\,dA_1/s^2$. Both descriptions have identical geometric factors, so conservation of its <radiant flux> gives equal <specific intensities> at the two ends. This compares the same propagation direction at both ends, not emission in the opposite direction.
A uniform source remains equally bright per resolved direction under these assumptions, but occupies a smaller apparent <solid angle> farther away. For a small source of <projected area> $A_\perp$ at distance $d$ and a small receiver facing it, <irradiance> is approximately $E=L\Omega_{\rm source}\simeq LA_\perp/d^2$. The <radiant flux> intercepted by receiver <area> $A_d$ is approximately $EA_d$. With $L=10\,\mathrm{W\,m^{-2}\,sr^{-1}}$, $A_\perp=0.020\,\mathrm{m^2}$ and $A_d=10^{-4}\,\mathrm{m^2}$, distances $10$ and $20\,\mathrm m$ give $E\simeq0.002$ and $0.0005\,\mathrm{W\,m^{-2}}$, and received <radiant fluxes> of $0.20$ and $0.050\,\mu\mathrm W$. The <inverse-square law> concerns collected radiation from the whole source; $L$ remains $10$ at either distance.
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