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Realize the random graph process by giving every edge an independent uniform arrival time. For a fixed four-set $S$, only its six internal edges and its $4(n-4)$ outgoing edges affect whether it is an isolated clique. It is an isolated clique at some time exactly when all six internal edges arrive before every outgoing edge.

The relative ordering of these $4n-10$ edges is uniform. The chance that the first six are exactly the six internal edges is ${\left(\genfrac{}{}{0ex}{}{4n-10}{6}\right)}^{-1}$. A union bound over the possible vertex sets givesEdges entirely outside $S$ are irrelevant. Counting the whole arrival-order event avoids an invalid union over continuously many times.

This is an example of isolated clique appearance in a uniform random graph process.