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Fiat which is poofed out of thin air is not Proof of Work
@pony agreed there is no variable cost to make more fiat. However there is a fixed cost (aka Proof of War) to maintain the god like power of creating money from thin air.
The idea I have is that all money has a cost, including fiat, however it is important to note whether the cost is variable or fixed when creating more money. If the cost is only fixed the system has misaligned incentives.