= Solution
For freezing, the liquid <concentration> decreases away from the interface, so the <liquidus> $T_L=-mC$ increases there. <Constitutional supercooling> means $T_l<T_L$ somewhere ahead of the interface. Since these <temperatures> agree at the interface, its local onset is
$$
T_{l,x}(a^+,t)<-m C_x(a^+,t).
$$
For the <similarity solution> and <salt rejection> condition, this is exactly
$$
\boxed{\frac{\epsilon(T_\infty-T_i)e^{-\epsilon^2\lambda^2}}{\sqrt\pi\operatorname{erfc}(\epsilon\lambda)}<m\lambda C_i.}
$$
It is also the global criterion for this freezing solution when $D<\kappa$ and $T_\infty>T_i$: the ratio $T_L'(x)/T_l'(x)$ is a positive constant times $\exp[-x^2(1/D-1/\kappa)/(4t)]$. For $x\ge a\ge0$ this ratio decreases with $x$. If the ratio is at most one at the interface, $T_l-T_L$ cannot become negative; if it is greater than one there, a supercooled interval appears immediately ahead. Equality marks onset, not a finite supercooled interval.
For fixed positive $\Delta\theta$ and $\lambda=O(1)>0$, the left side of the inequality is $O(\epsilon)$ and the right side is positive of order one, so freezing is constitutionally supercooled for sufficiently small $\epsilon$. To locate the boundary between the two regimes, resolve the smaller scale $\Delta\theta=O(\epsilon)$, $\lambda=O(\epsilon)$. The salt equation and <liquidus> then give
$$
C_i-C_0=\sqrt\pi C_0\lambda+O(\lambda^2),\qquad m\lambda C_i=\frac{\Delta\theta}{2\sqrt\pi}+O(\epsilon^2).
$$
Here $\epsilon\lambda=O(\epsilon^2)$, so the thermal equation improves to $T_i=(T_\infty+T_{-\infty})/2+O(\epsilon^2)$; this justifies the second expansion including <latent heat>. Also $T_\infty-T_i=\theta_++O(\epsilon)$ and the left side of the exact criterion is $\epsilon\theta_+/\sqrt\pi+O(\epsilon^2)$. Therefore the <small-diffusivity constitutional-supercooling threshold> is
$$
\Delta\theta>2\epsilon\theta_++O(\epsilon^2),\qquad \boxed{\theta_->\theta_+(1+2\epsilon)\ \text{to first order in }\epsilon.}
$$
This printed condition is an asymptotic onset condition, rather than an exact finite-$\epsilon$ inequality. In its $O(\epsilon^2)$ transition window the complete algebraic system and exact <gradient> criterion must be used.
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