Hypothetical question: suppose I had a OsmocomBB phone modified to record all transmission done over GSM including encryption keys used during the communication. If I made a call and recorded all data transferred since the phone was turned on, would it be enough to prove that the phone call from one phone number to another took place? I'm interested in proving that without engaging the operator in this process.
If you mean "prove" in the sense of ensuring that the call is authentic and cannot be reputiadated by the other party then the answer is no.
Cellphone encryption is used only for the over-the-air portion of the call - between the mobile telephone and the base station. A GSM telephone has to authenticate to the base station but there is no guarantee the base station is genuine.