A question about exchange pattern Exchange pattern, see Section 2 in "cluster algebras I: foundations" by Fomin and Zelevinsky or  How to understand exchange pattern?
Given an example  $\cdots \overset{2}{-} t_1 \overset{1}{-} t_2 \overset{2}{-} t_3 \overset{1}{-} t_4 \overset{2}{-} \cdots$, where 
\begin{align*}
\mathbf{x}(t_1) = (x_1(t_1), x_2(t_1)) = (x,y); \ (M_1(t_1), M_1(t_2)) = (y,1); \\
\mathbf{x}(t_2) = (x_1(t_2), x_2(t_2)) = (\frac{y+1}{x},y); \\
\mathbf{x}(t_3) = (x_1(t_3), x_2(t_3)) = (\frac{y+1}{x},\frac{x+y+1}{xy}); \\
\mathbf{x}(t_4) = (x_1(t_4), x_2(t_4)) = (\frac{x+1}{y},\frac{x+y+1}{xy}).
\end{align*}
By E3, we have 
\begin{align*}
(M_2(t_2), M_2(t_3)) = (\frac{y+1}{x},1),  \ (M_1(t_3), M_1(t_4)) = (\frac{x+y+1}{xy},1).
\end{align*}
But 
\begin{align*}
M_0 = (M_2(t_2) + M_2(t_3))|_{\frac{y+1}{x} = 0} = (\frac{y+1}{x} +1 )|_{\frac{y+1}{x} = 0} = 1; \\
\frac{M_1(t_3)}{M_1(t_4)} = \frac{M_1(t_2)}{M_1(t_1)}|_{y \leftarrow \frac{1}{y}} \Rightarrow \frac{x+y+1}{xy} \neq  \frac{1}{y}|_{y \leftarrow \frac{1}{y}} = y.
\end{align*}
This contradicts E4.  I don't know where is wrong. Any help is needed.
 A: Using Fomin-Zelevinsky's notation, we can start with $(M_1(t_1),M_1(t_2))=(x_2,1)$, $(M_2(t_2),M_2(t_3))=(x_1,1)$. This can be considered initial data, considering $t_2$ as the initial cluster: I don't think the axioms can be used to deduce the values of the second pair from the first. However, note that this choice is consistent with equation 2.6 in Fomin-Zelevinsky (which I am assuming to be E3 in your notation): we have $x_2|M_1(t_1)$ and $x_1|M_1(t_2)$, for example. 
We can then use the fourth axiom (equation 2.7 in FZ, i.e. E4 in your notation) to compute the pair $(M_1(t_3),M_1(t_4))$. We have
$$M_0=(M_2(t_2)+M_2(t_3))|_{x_1=0}=(x_1+1)|_{x_1=0}=1.$$
So
$$\frac{M_1(t_3)}{M_1(t_4)}=\frac{M_1(t_2)}{M_1(t_1)}|_{x_2\leftarrow M_0/x_2}=\frac{1}{x_2}|_{x_2\leftarrow 1/x_2}=x_2.$$
Hence $M_1(t_3)=x_2$ and $M_1(t_4)=1$. Note that this then gives the correct exchange from $t_3$ to $t_4$. We have
$$x_1(t_3)x_1(t_4)=M_1(t_3)(\mathbf{x}(t_3))+M_1(t_4)(\mathbf{x}(t_4)),$$
i.e. (using the values for $\mathbf{x}(t_3)$ and $\mathbf{x}(t_4)$ above):
$$\frac{y+1}{x}\cdot \frac{x+1}{y}=\frac{x+y+1}{xy}+1.$$
I think the problem arises from writing the $M_i(t)$ in terms of the initial cluster and not in terms of separate variables $x_1,x_2$ as in Fomin-Zelevinsky.
