for every $n\in {\mathbb N}$. The projective sequence is fine if and only if for all $n \in {\mathbb N}$ and $a, b \in A_{n}$ with $d_{R^{A_n}}(a, b) = 2$, there is $m>n$ such that if $a' \in ({\varphi}^{m}_{n})^{-1}(a), b' \in ({\varphi}^{m}_{n})^{-1}(b)$ then $d_{R^{A_m}}(a', b') \ge 3$.

Let $a, b \in A_{n}$ with $d_{R^{A_n}}(a, b) = 2$, say $a {\mathbin{R}}^{A_n}c {\mathbin{R}}^{A_n}b$. If for each $m>n$ there are $a_{m} \in ({\varphi}^{m}_{n})^{-1}(a), b_{m} \in ({\varphi}^{m}_{n})^{-1}(b)$ with $d_{R^{A_m}}(a_{m}, b_{m}) = 2$, say $a_{m}{\mathbin{R}}^{A_m}c_{m}{\mathbin{R}}^{A_m}b_{m}$, let $$x_m\in {\varphi}_m^{-1}(a_m), \quad y_m\in {\varphi}_m^{-1}(b_m), \quad z_m, z'_m\in {\varphi}_m^{-1}(c_m),$$ with $x_m{\mathbin{R}}^{ \mathbb A }z_m, z'_m{\mathbin{R}}^{ \mathbb A }y_m$. Passing to a suitable subsequence, let $$x=\lim_{h\rightarrow\infty }x_{m_h}, \quad y=\lim_{h\rightarrow\infty }y_{m_h}, \quad z=\lim_{h\rightarrow\infty }z_{m_h}=\lim_{h\rightarrow\infty }z'_{m_h},$$ so that $x{\mathbin{R}}^{ \mathbb A }z{\mathbin{R}}^{ \mathbb A }y$. However, $x, y$ are not $R^{ \mathbb A }$-related (otherwise $a {\mathbin{R}}^{A_n}b$), so $(A_{n}, {\varphi}_{n}^m)$ is not fine.

On the other hand, if $(A_{n}, {\varphi}_{n}^m)$ is not fine there are $x, y\in \mathbb A $ such that $d_{R^{ \mathbb A }}(x, y) = 2$, say $x{\mathbin{R}}^{ \mathbb A }z{\mathbin{R}}^{ \mathbb A }y$, for $x, y, z$ distinct points. There is $n \in {\mathbb N}$ such that for all $m \ge n$ the points ${\varphi}_{m}(x), {\varphi}_m(y), {\varphi}_m(z)$ are 