Why it is not possible for a payee to verify double-spends of the coin?
As a participant in the peer to peer network, it is impossible for you to know that the state and timing you are seeing are consistent with what other people are seeing. You can't ask other participants in the network what they are seeing directly as they can lie, withhold information, or the request can be tampered with on its way across the network. The block chain and proof of work mining system are designed to ensure that in all probability, nodes in the network have consistent ordering and atomicity. Absent this, it would be very feasible for nodes to see a different state and make different decisions about validity.