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I grabbed the "Digital Coin Technology" pdf file.
It looks to me like it fails in the second paragraph:
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If the DigitalCoin software can read the serial number, then so can other software. It is simply not true that "The DigitalCoin software alone can ..."
Either you have a central server running the DigitalCoin software and you have to trust that the people running that server won't double-spend DigitalCoins.
Or... you have Bitcoin, where the "serial numbers" are the public keys of coin generation transactions.