Proof of Lemma 2

$Z$ becomes in that case

$$Z = W_+ e^{- \frac{1}{2} \Delta} + W_- e^{\frac{1}{2} \Delta} \:\:,$$ (32)

where $\Delta=\vec{v}[1] - \vec{v}[0]$. There are five different values for $\Delta$, giving rise to nine different $\vec{v}$:

$$\begin{eqnarray*} \Delta = & +2 & \Rightarrow \vec{v} = (-1, +1) \:\:,\\ \Delta... ...0) \:\:,\\ \Delta = & -2 & \Rightarrow \vec{v} = (+1, -1) \:\:. \end{eqnarray*}$$

Fix $\Delta = k$ where $k \in \{-2, -1, 0, 1, 2\}$. $\forall k \in \{-1, 0, 1, 2\}$, the value $\Delta = k$ should be preferred to the value $\Delta = k-1$ iff the corresponding $Z$ is smaller, that is :

$$W_+\times e^{-\frac{k}{2}} + W_- \times e^{\frac{k}{2}} < W_+ \times e^{-\frac{k-1}{2}} + W_- \times e^{\frac{k-1}{2}} \:\:.$$ (33)

Rearranging terms gives $W_- < W_+ \times \frac{1}{e^{k-\frac{1}{2}}}$. This leads to the rule of the lemma.