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@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.
Fiat which is poofed out of thin air is not Proof of Work