I think the preamble should define expected count and initial expected count, the latter being what the current wording refers to as "what was specified by the expected argument to the constructor" (which should say "parameter" not "argument", or "argument that initializes the expected parameter", but should probably just be "the value of the constructor's expected parameter").

• The wording added to [thread.barrier.class]p1 says "the completion step is run on one of the threads that arrived at the barrier during the phase". What are we actually saying here?

That the thread that runs the phase completion step is not necessarily the last one to arrive at the barrier (except for the special case of the default completion function, which has a no-op completion function, and so I'm not sure whether it's observable which thread it runs on and so I don't know why we need to call out that special case).

Instead of " For the specialization with the default value of the CompletionFunction template parameter" could we just say "for the specialization barrier<>"?