$$V_\xi = \min_{ {\zeta\in\Omega:} \atop {{\mathcal{H}}(\zeta)<{\mathcal{H}}(\xi)} } \Phi(\xi,\zeta)-{\mathcal{H}}(\xi).$$ (c) The set of stable configurations is $$\Omega_{\mathrm{stab}}= \left\{\xi\in\Omega\colon\,{\mathcal{H}}(\xi)
= \min_{\zeta\in\Omega} {\mathcal{H}}(\zeta)\right\}.$$ (d) The set of metastable configurations is $$\Omega_{\mathrm{meta}}= \left\{\xi\in\Omega\backslash\Omega_{\mathrm{stab}}\colon\,
V_\xi = \max_{\zeta\in\Omega\backslash\Omega_{\mathrm{stab}}} V_\zeta\right\}.$$

It is easy to check that $\Omega_{\mathrm{stab}}= \{\boxplus\}$ for all $G$ because $J,h>0$. For general $G$, however, $\Omega_{\mathrm{meta}}$ is not a singleton, but we will be interested in those $G$ for which the following *hypothesis* is satisfied:

-   $\Omega_{\mathrm{meta}}= \{\boxminus\}$. \[itm:Hhyp\]

The energy barrier between $\boxminus$ and $\boxplus$ is $$\Gamma^\star = \Phi(\boxminus,\boxplus)-{\mathcal{H}}(\boxminus).$$

Let $({\mathcal{P}}^\star,{\mathcal{C}}^\star)$ be the unique maximal subset of $\Omega\times\Omega$ with the following properties (see Fig. [\[fig-protocrcr\]]{}):

1.  $\forall\,\xi\in{\mathcal{P}}^\star\,\exists\,\xi\prime\in{\mathcal{C}}^\star\colon\,\xi\sim\xi\prime$,\
    $\forall\,\xi\prime\in{\mathcal{C}}^\star\,\exists\,\xi\in{\mathcal{P}}^\star\colon\,\xi\prime\sim\xi$.

2.  $\forall\,\xi\in{\mathcal{P}}^\star\colon\,\Phi(\xi,\boxminus)<\Phi(\xi,\boxplus)$.

3.  $\forall\xi\,\in{\mathcal{C}}^\star\,\exists\,\gamma\colon\,\xi\to\boxplus\colon\,\\
    {\rm (i)} \max_{\zeta\in\gamma} {\mathcal{H}}(\zeta)-{\mathcal{H}}(\boxminus) \leq \Gamma^\star$.\
    [(ii)]{} $\gamma \cap\{\zeta\in\Omega\colon\,\Phi(\zeta,\boxminus)
    <\Phi(\zeta,\boxplus)\} = \emptyset$.

(15,15)(0,-1) [(0,0)(0,5)(0,10) (7,0)(7,5)(7,10) ]{} (-2,11)[${\mathcal{P}}^\star$]{} (5,11)[${\mathcal{C}}^\star$]{} (0.4,5.8)[$\xi$]{} (5.8,7.8)[$\xi\prime$]{} (-5,1.5)[$\boxminus$]{} (10.6,-2)[$\boxplus$]{} (-8.6,4.8)[$<\Gamma^\star+{\mathcal{H}}(\boxminus)$]{} (8.5,6)[$\leq \Gamma^\star+{\mathcal{H}}(\boxminus)$]{} (0,5) (7,7.2) (-4.5,3) (11,-.5) (0,5)[(3,1)[6.8]{}]{} (7,7.2)[(1,-2)[3.5]{}]{} (0,5)[(-2,-1)[3.5]{}]{}

Think of ${\mathcal{P}}^\star$ as the set of configurations where the dynamics, on its way from $\boxminus$ to $\boxplus$, is ‘almost at the top’, and of ${\mathcal{C}}^\star$ as the set of configurations where it is ‘at the top and capable over crossing over’. We refer to ${\mathcal{P}}^{\star}$ as the *protocritical set* and to ${\mathcal{C}}^\star$ as the *critical set*. Uniqueness follows from 