Is the transaction complete even if we don't get the mint to give us a new one? Couldn't someone pay several entities with the same token offline. The first to redeem would get the new valid token from the mint, correct?

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This is true for token without locks. In the offline receive case the receiver would usually specify that the token needs to be locked to a key they control. If that is the case the token can not be double spend.

What if they build two coin simultaneously with locks to each? Both offline. Wouldn't that still be an issue?

If they build two token with two locks, they would have to spend two times the amount as well

I am saying if they made two with the initial token as the input, they'd still run into issues, correct?

That is not how it works. A token can not be used as an input twice. The mint will not allow double spending it

That doesn't matter. If someone two folks took the same token off line, there is a race to swap with the mint. So the person online first actually gets the value. Imagine selling something of value for a token you can't the value from. Isn't that a possiblity?