Positivity limitation of a linear Eddington intensity ($I(0,-1)=-F/(4\pi )$)

For the affine angular ansatz $I=A+C\cos \theta$, the radiative flux is $F=4\pi C/3$. Setting the inward hemispheric flux to zero gives $A(0)=2C/3$, but then the formal intensity at $\cos \theta =-1$ is negative when $F>0$. Exact no-incoming radiation and nonnegative intensity cannot both be represented by a nonzero affine polynomial on all directions. The Eddington surface boundary condition therefore specifies approximate moments, rather than an exact emergent angular radiation field.