There is no "real evidence" beyond the *A operating agreement which is a simple contract. It's largely as noted by cfisher above.

If you are entitled to lounge access as a benefit of the ticket, it is the operating carrier of the flight. Otherwise, the carrier which provides the benefit to you.

Because not all premium cabin tickets include lounge access, e.g. domestic F in the US, if you are *G via AC status and flying domestic F, it would be AC which pays (because the domestic F does not include lounge access with limited exceptions).