Money supply (ie total $ in the system) times velocity of money equals GNP. Basic macroeconomics.
Look up Money supply on Wikipedia. It how the Fed works. It how the US prints money and controls inflation. It works I theory and in decades of practice.
Call it crude and imprecise, and I will agree. But call it nonexistent (which you did) and I'll call you wrong.
How about I call you wrong instead.
Here is what you said
The dollar literally represents a portion of the US GNP. Which is guaranteed by law. Is that abstract? Yes, but not so abstract that I can't say that, for example, my one dollar represents 1 / the total amount of dollars in circulation as a fraction of US GNP.
The dollar representing a portion of GNP is meaningless. The claim that it's guaranteed by law is what I was saying is nonexistent. The dollar is not guaranteed by law to be the value of a portion of GNP, it just is based on some silly estimate but that isn't a valuation of the dollar itself because GNP just represents some total volume of economic activity.
If you can find such a law, let me know. I suspect you might be referring to the law that requires the Fed to have as it's objectives in managing the banking system, full employment and low inflation. Those are simply objectives and they rarely meet the objectives (the mid-late 90s were the closest they got in my lifetime).
I think that law is based on the quantity theory of money which goes (in it's simplified form) something like
MV=PQ
where in simplified terms P is some price level and Q might be GDP (not GNP) and M is some measure of money in the economy while V is the velocity of money. However, this isn't what the Fed targets, they quit using money supply as their target a long time ago and started focusing on interest rates. Lately, they switched to the unemployment rate.
So I am afraid you aren't up to date on this stuff but suffice it to say, a dollar is not guaranteed to have some portion of the value of GDP by law.
What M amounts to is anyone's guess. It's not currency in circulation. In case you didn't know it, the Fed used to have three measures of M called M1, M2 and M3. Now it has M1, M2 and they track something else they call MZM which is closer to the M in the equation.
So while you have some notion of how you think things are, they are not that way. The shadow banking has totally screwed the Fed along with other financial "innovations". The idea that some single entity can control the economy by controlling the money supply is an academic fantasy. It's not reality and the quantity of money theory hasn't been proven to have any validity in the short term which is why the Fed hasn't been effective using it in it's management of the banking system.