In what way would Beeper's action cause Apple's customers to breach a contract with Apple? I would think most of the people who would purchase a service like this would be Android users, not iPhone users. Some of them might own Macs, but what would be the contract that the user would be breaching that would result in damage to Apple?

If they're "just Android users", they don't have iMessage accounts.

So your thinking is that these end-users have signed some sort of agreement with Apple, and that agreement says they won't use any unauthorized services to connect to Apple servers, or some such thing?

That's not "my thinking" so much as it is a fact.

If it’s a fact then it should be no trouble to share the relevant provision.

I was sharing that theory as a conjecture, since I have no reason to believe such a provision exists.