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You are wrong. They keep track. Until they found one on you...
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Cool story. Any evidence or even a plausible mechanism? What about coins? They don't have serial numbers.
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fiat coins are meaningless. Are just for fun to make sheeple believe they have money coins.
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So your answer is "no", I take it.
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People consider it private because no one is keeping track of who has which bill. They are effectively perfectly fungible.