Sometimes the first action cannot literally continue until the later action. The endpoint interpretation then gives way to a purpose reading, expressed by kay, “so that.”
«عبدي حر إن لم آتك حتى تغديني»
My servant is free if I do not come to you so that you may give me lunch.
Coming to someone is an event, not something that necessarily continues until lunch is served. Thus the text reads hatta here as specifying the purpose of coming. Its conditional legal application is limited to the historical formula discussed.