Exchange pattern, see Section 2 in "cluster algebras I: foundations" by Fomin and Zelevinsky or [How to understand exchange pattern?](https://mathoverflow.net/questions/249230/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 question about exchange pattern
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.


