"If garbage is not scarce, then neither is any arrangement of that garbage."
Your argument here seems to be that you cannot get a scarce good out of a non-scarce good. That's not true. Is air scarce? If I fix nitrogen from the air to make fertilizer, is the fertilizer scarce? Is saltwater scarce? If I extract salt from the saltwater, is the salt scarce?
"The only way any "scarce" arrangement of non-scarce goods (or patterns) STAYS scarce is through the creation of a government-protected monopoly."
That's also not true. If I write a novel (a scarce arrangement of non-scarce words), I can simply not release any copies of it. It can forever remain scarce. Or I can make copies for 10 trusted friends, who I know will not release any copies. Thus, it remains scarce. Or I can sell it with a restrictive covenant (copyright), which does not give any buyer the right to copy it.
Now, if you are opposed to the concept of restrictive covenants, that's a different argument. The question is, would a private law system recognize the validity of restrictive covenants? I say yes.
“With laws shall our land be built up, but with lawlessness laid waste.”