Atomic Assignment

Atomic Assignment refers to a rule: when title to shares is assigned, all the shares are assigned at one time. This is not actually restrictive at all, because any assignment contract may have as many assigners and receivers as desired.
*up to a reasonable limit, perhaps about 1000, to prevent transaction spam from overtaxing the underwriters.
So, if (usually when) an assigner's source of shares has more shares than are needed for the transaction, the assigner provides a new public key to receive the excess shares in.

Among other things, this makes it simpler to verify no double spends: When each new ID (public key) is entered into the blockchain, it has a unique sequence number which may thereafter be used to refer to the public key value. Whether the shares associated any particular key are still available to be assigned is a simple binary state of the key's position in that list of all recorded public keys.