I disagree. Like a VIN number on a car, having knowledge of a serial number does not equate to proof of ownership, it simply means the person had physical possession of an item, whether stolen or owned. Since the serial number is usually printed on the physical item and easily accessible, i don't see how knowledge of a serial number is proof of ownership.
If the OP did NOT contact the police, it seems entirely fishy to me since that would be the first reaction of anyone in this situation with legitimate proof of ownership. If the OP did contact the police, the likelihood of him trying to track the IP address and recovering his equipment is not likely, as was pointed out. Best bet is to let the police work and claim it on insurance, yes?