Some deep, technical bitcoin magic?
Would you care to elaborate on this more?
The problem is that to prove you are the owner of the originating bitcoin address, you need to sign something using the private key from that address. That's the only true bitcoin-like solution, as this key is the very thing that allows you to make transactions is your only technical proof of ownership.
Unfortunately, the way said key is stored is by being in your wallet (be it the wallet file of your bitcoin client or your brain wallet or whatever else you use). If you do not have this, then you lost that address, and with it access to any remaining bitcoins in that address, and the only purely technical way of authenticating yourself as the owner.
There may yet be a non-technical solution: Find someone whom the beneficiary of your transaction mishap trusts and who can vouch for the address being your address. Maybe someone has transferred bitcoin to that very address of yours and can assure the beneficiary that it is indeed yours?
Note that all this leaves another question: Does the beneficiary know what address he has been paid from? I may be wrong, but I suspect that each block in the blockchain describes only lumped inputs and lumped outputs, so the source(s) could(?) be any combination of the input addresses. So if I was the beneficiary, then without further educating myself about the technicalities of bitcoin transactions, I wouldn't be quite sure from which bitcoin address I'd need authentification from.