Here is our second main result.

\[=Theorem \[T:torsion pair\]\] \[thm:A\] Let ${\mathscr{X}}$ be a set of diagonals of $\cZ$. Then $\add E({\mathscr{X}})$ is the first half of a torsion pair in ${\mathscr{C}}({\mathscr{Z}})$ if and only if ${\mathscr{X}}$ satisfies conditions PC1, PC2, and the Ptolemy condition.

Note that the first half of a torsion pair determines the second half, so our result does provide a complete classification.

\[rmk:PC\] The conjunction of PC1 and PC2 is equivalent to the following condition.

[PC:]{}

If there is a sequence $\{x^i_0, x^i_1\}_{i \in {\mathbb{Z}}_{\geqslant 0}}$ from ${\mathscr{X}}$ with $x^i_0 \to p$ from below and $x^i_1 \to q$ with $p \neq q$, then there is a sequence $\{x'^i_0, x'^i_1\}_{i \in {\mathbb{Z}}_{\geqslant 0}}$ from ${\mathscr{X}}$ with $x'^i_0 \to p$ from above and $x'^i_1 \to q$ from above.

It is clear that PC implies PC1 and PC2. To see the converse, note that the sequence $\{x^i_0, x^i_1\}_{i \in {\mathbb{Z}}_{\geqslant 0}}$ in PC will either have a subsequence with $x_1^i \rightarrow q$ from below, and then PC1 can be applied, or a subsequence with $x_1^i \rightarrow q$ from above, and then PC2 can be applied.

The paper is organised as follows: Section \[sec:admissible\_sets\] shows some properties of admissible subsets 