 Obstruction to a single smooth exponential mode in conservative grey transfer

Consider $I=\alpha [\beta +\tau +\mu +\gamma e^{-\delta \tau }/(1+ϵ\mu )]$ with real $|ϵ|<1$, $\alpha \gamma \ne 0$, and $\delta >0$. Its angular mean includes $A_0=\mathrm{atanh}ϵ/ϵ$. Substitution into $\mu I_{\tau }=I-J$ requires $1-A_0+(\delta -A_0ϵ)\mu =0$. Thus $A_0=1$ and $\delta =ϵ$. For real $ϵ$, $A_0=1$ only at zero, giving $\delta =0$, a contradiction. Independently, its flux contains a varying term proportional to $A_1=(1-A_0)/ϵ$, which is nonzero for nonzero real $ϵ$. Exact conservative transfer therefore needs a more general angular/depth structure than this one smooth exponential correction.