Lemma: Exact support update

At an edge event, a pair of empty sites remains empty. If at least one endpoint is occupied, the new two-site support has probabilities

These probabilities do not depend on the incoming nonzero pair.

Proof. Clifford conjugation acts transitively on nonidentity Pauli operators. One elementary justification is to use single-qubit Cliffords to turn nonidentity tensor factors into $X$ factors, then CNOTs and swaps to reduce their product to one $X$ factor. Inverting such reductions maps any nonidentity Pauli to any other. Uniform Clifford sampling therefore gives a uniform nonidentity two-site Pauli.

There are $q^4-1$ such Paulis. The support counts are $q^2-1$, $q^2-1$, and $(q^2-1{)}^2$, respectively. Divide by $q^4-1$.