Does the existence of any quantity changing money (specie or fiat) necessarily cause business cycles?
Rather, if one had any money, whose quantity was fixed, would that prevent the business cycle?
The Austrian business cycle occurs when unbacked money (fiduciary media) is injected into the credit market, artificially lowering the rate of interest.
So, if you are saying, “Suppose that the stock of money–including bank deposits–is held constant,” then I agree you couldn’t have an Austrian business cycle.
But just be careful: Even if the US government held the number of paper dollars fixed, the banking system could still expand and contract “the money stock” if we include bank checking accounts in our definition (and those *are* included in M1 for example).