I found a collision on btckey.space, now how do I prove it?
What do you mean by this? Do you mean that you found two 256-bit ECDSA public keys that hash to the same 160-bit address?
So please, how can I prove that I found this address randomly?
How is it relevent that the process was random? To show a collision, it is necessary and sufficient to show two preimages that hash to the same value.