In all cash transfers, the notary gets involved at some level. In the simplest case, all the controlled pipelines already share signatures with their peers, work done at the start by notaries. After that we can consider the simplest case.
Consider:
send 5 Btc to P2 on the even money protocol
This is sent from P1, which is already stable so the Btc is valid bearer cash. In the simple case, the Btc is addressed to the pipeline wallet but under control of the user protocol. P2 could be a pit that trades Btc for Eth. The message causes P2 to accepts the Btc and start the even money trade bot.
The calling protocol is not allowed to run until P2 is done with the Btc. This is not a necessary condition, but I impose it.
P2, being a pit, compresses prices and the even money bot has bought Eth, owns them as bearer cash then says:
send 200 Eth to P1, return protocol
Now there are a number of other protocols, in oher processors that hold bearer Btc cash, refundable by P1. Not a problem, P2 or P1 can register the ownership, clear the transfer.
The original protocol, by arriving at the resolving protocol branch, has agreed to terms, accepts the result. attorney. The prior trace of protocol., when the Btc was controlled in P1, that trace has dropped out. The is no double spending, there i no stable double spending opportunity by proof, outcomes are finite, countable and testable.
But, notaries have to keep an eye out for keys.
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