= Obstruction to a single smooth exponential mode in conservative grey transfer
Consider $I=\alpha[\beta+\tau+\mu+\gamma e^{-\delta\tau}/(1+\epsilon\mu)]$ with real $|\epsilon|<1$, $\alpha\gamma\ne0$, and $\delta>0$. Its angular mean includes $A_0=\operatorname{atanh}\epsilon/\epsilon$. Substitution into $\mu I_\tau=I-J$ requires $1-A_0+(\delta-A_0\epsilon)\mu=0$. Thus $A_0=1$ and $\delta=\epsilon$. For real $\epsilon$, $A_0=1$ only at zero, giving $\delta=0$, a contradiction. Independently, its flux contains a varying term proportional to $A_1=(1-A_0)/\epsilon$, which is nonzero for nonzero real $\epsilon$. Exact conservative transfer therefore needs a more general angular/depth structure than this one smooth exponential correction.
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