= Solution
At the <stellar surface boundary condition>, match the interior to a <stellar atmosphere>, rather than setting the <temperature> and <pressure> to zero at an arbitrary radius. At a <photosphere> of radius $R$, the <enclosed mass> is $M$, the <luminosity> is $L$, and the outward <radiative flux> defines the <effective temperature> through $F=L/(4\pi R^2)=\sigma_{\rm SB}T_{\rm eff}^4$. With no external illumination the incoming <specific intensity> satisfies $I_\nu(0,\mu)=0$ for $\mu<0$. A deeper boundary is matched to the nearly isotropic <radiative diffusion> field. A <grey atmosphere> commonly matches near $\tau=2/3$; under the <Eddington surface boundary condition> this gives $T=T_{\rm eff}$. For roughly constant gravity and <opacity>, the total <pressure> rises by $g\tau/\kappa$ relative to its outer boundary value; if radiation supplies appreciable support, the gas-pressure gradient uses the effective gravity $g-\kappa F/c$ instead.
Take $z$ increasing outward and $\mu$ the outward direction cosine. If $j_\nu$ is emission per unit path length and solid angle, the <radiative transfer equation> is
$$
\mu\frac{\partial I_\nu}{\partial z}=j_\nu-\rho\kappa_\nu I_\nu.
$$
Define inward-increasing <optical depth> by $d\tau_\nu=-\rho\kappa_\nu dz$ and the <source function> by $S_\nu=j_\nu/(\rho\kappa_\nu)$. Then $\mu\partial_{\tau_\nu}I_\nu=I_\nu-S_\nu$. This states the emission-coefficient convention; if $j_\nu$ is defined per unit mass instead, the emission term is $\rho j_\nu$.
For frequency-integrated intensity, introduce the <radiation-field moments>
$$
J=\frac12\int_{-1}^1I\,d\mu,\qquad H=\frac12\int_{-1}^1\mu I\,d\mu,\qquad K=\frac12\int_{-1}^1\mu^2I\,d\mu,\qquad F=4\pi H,\qquad cP_{\rm rad}=4\pi K.
$$
The first angular approximation writes $I\simeq J+3\mu H$. Its odd part does not contribute to $K$, so
$$
\boxed{K=\frac J3,\qquad cP_{\rm rad}=\frac{4\pi}3J.}
$$
This is the <Eddington closure approximation>, not an exact description of the escaping angular distribution. An approximate surface condition takes the outgoing hemisphere to have direction-independent intensity $I_0$ and the incoming hemisphere to be dark. Its moments give $F=\pi I_0$ and $J(0)=I_0/2=F/(2\pi)$. Integrating the moment equations with the closure then gives $J=3F(\tau+2/3)/(4\pi)$; in thermal equilibrium, $J=a_{\rm rad}cT^4/(4\pi)$ gives $T^4=(3/4)T_{\rm eff}^4(\tau+2/3)$. This derives the usual approximate photospheric matching condition rather than imposing a zero surface <temperature>. For conservative <coherent isotropic scattering>, $S=J$. Integrating the transfer equation gives $dH/d\tau=0$ and $dK/d\tau=H$, the moment conditions for <radiative equilibrium>.
The exponential ansatz in the PDF has a genuine consistency problem. Compute its moments before making any approximation. For a nonsingular real intensity on all $-1\le\mu\le1$, take $|\epsilon|<1$ and define
$$
A_0=\frac12\int_{-1}^1\frac{d\mu}{1+\epsilon\mu}=\frac{\operatorname{atanh}\epsilon}{\epsilon},\qquad A_1=\frac{1-A_0}{\epsilon},\qquad A_2=\frac{A_0-1}{\epsilon^2}.
$$
The continuous zero limits are $A_0=1$, $A_1=0$, $A_2=1/3$. Direct integration yields
$$
\boxed{J=\alpha(\beta+\tau+\gamma A_0e^{-\delta\tau}),\qquad F(\tau)=4\pi\alpha\left(\frac13+\gamma A_1e^{-\delta\tau}\right),}
$$
$$
\boxed{cP_{\rm rad}=4\pi\alpha\left(\frac{\beta+\tau}{3}+\gamma A_2e^{-\delta\tau}\right).}
$$
For a decaying nontrivial correction, $\delta>0$ and $\gamma\ne0$. Exact constant flux then requires $A_1=0$. But $\operatorname{atanh}\epsilon/\epsilon>1$ for every nonzero real $|\epsilon|<1$, so $A_1=0$ forces $\epsilon=0$. Thus exact <radiative equilibrium> cannot imply a small nonzero $\epsilon$ for this ansatz.
Substitution into the full transfer equation makes the obstruction sharper. The exponential coefficient must satisfy, for every $\mu$,
$$
-\delta\mu=1-A_0(1+\epsilon\mu).
$$
Comparing the constant and linear coefficients gives $A_0=1$ and $\delta=A_0\epsilon=\epsilon$. Hence regular real coefficients require $\delta=\epsilon=0$, eliminating any decaying correction. This is the <obstruction to a single smooth exponential mode in conservative grey transfer>. \b[There is no exact nontrivial real solution of the stipulated form with a small nonzero decay parameter.] Choosing, for example, $\epsilon=\delta=0.1$ and $\gamma\ne0$ supplies an explicit counterexample to the claimed exact flux constancy, because $A_1\ne0$.
Several requested conclusions remain useful as formal approximations, provided they are identified as such. If $\delta>0$, the large-depth exponential vanishes, giving $F\to4\pi\alpha/3$, hence $\alpha=3F/(4\pi)$ for the asymptotic physical flux. It also gives $cP_{\rm rad}\to(4\pi/3)J$. Since $A_0=1+\epsilon^2/3+O(\epsilon^4)$, neglecting second-order angular errors yields the formal relation $\delta\simeq\epsilon$. However $A_1=-\epsilon/3+O(\epsilon^3)$, so the finite-depth flux still varies at first order. The formal relation is not an exact transfer or constant-flux solution.
There is also a flux-normalization mismatch in the PDF's last displayed relation. With the same physical $F$ used above, the <Hopf function of a grey atmosphere> is defined by
$$
J=\frac{3F}{4\pi}[\tau+q(\tau)].
$$
The PDF's coefficient $3F/4$ would require redefining its $F$ as physical flux divided by $\pi$; it cannot simultaneously use the earlier $\alpha=3F/(4\pi)$ for the same physical flux. With the consistent physical-flux normalization, formal moment matching gives
$$
\beta=q(\infty)=0.71,\qquad\gamma A_0=q(0)-q(\infty)=-0.14.
$$
In the intended small-$\epsilon$ approximation this is
$$
\boxed{\beta\simeq0.71,\qquad\gamma\simeq-0.14.}
$$
The approximate outgoing surface ansatz then gives the <limb darkening> ratio
$$
\frac{I(1,0)}{I(0,0)}=\frac{\beta+1+\gamma/(1+\epsilon)}{\beta+\gamma}\ \longrightarrow\ \boxed{\frac{1.57}{0.57}\simeq2.75.}
$$
For finite $\epsilon$, $\gamma=-0.14/A_0$ gives a parameter-dependent ratio, not a unique exact value. The two Hopf endpoint values alone do not determine the exact emergent intensity: the <formal solution of the radiative transfer equation> uses the entire source profile, $I(0,\mu)=\int_0^\infty J(t)e^{-t/\mu}dt/\mu$ for $\mu>0$. Nor does the simple linear-plus-exponential ansatz satisfy the exact no-incoming-radiation condition for every negative $\mu$. Thus $2.75$ is a clearly labelled approximate disk-centre-to-limb result, not an exact consequence of inconsistent premises.
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