Theorem: E: cut monotonicity and exact conversion

Under Definition def:moves:

1. $\mu$ decreases by one under (m1)–(m3) and is preserved by (m4);

2. for every cut, ${\mu }_{X\mid Y}$ and ${\sigma }_{X\mid Y}$ are non-increasing;

3. activation preserves ${\sigma }_{X\mid Y}$ for every cut: one crossing token becomes one crossing identified edge;

4. along a run, the number of cross-cut activations is at most the initial number of crossing latent links, and therefore at most ${\sigma }_{X\mid Y}(g_0)$.

Proof. Use $1_A$ for the indicator of a condition $A$. For a deleted or activated link with endpoints $u,v$, let $c=1_{\text{endpoints cross}}$. The changes are:

Every entry in the cut columns is non-positive. The activation row counts an identified edge, so it remains valid for parallel endpoint pairs. Finite composition proves (i)–(iii).

No move adds a latent link. Each crossing latent identifier can be activated at most once. This proves (iv).