"It would be trivial to demonstrate, though it would likely take longer than I am willing to donate CPU cycles on my machines to generate a long enough noise stream (it depends on the size of the program that has to be generated to prove it).
The reason I am basically dropping this discussion is that we are arguing whether or not proven theorems of basic mathematics are true or not."
No, Illya Prigogine has already conclusively proven that random noise is incapable of producing levels of "order" more complex than a certain point.
And it is because of that scientific fact that you can never create a useful computer program of any serious complexity from mere random noise. So while you choose to pretend that the exercise is trivial (above) and unnecessary (i.e., already been proven mathematically and other such nonsense), the real reason is that you can't do it.
At all.
Ever.
Nobel laureate Prigogine has already shown us in Order Out of Chaos that increased levels of order (i.e. complexity) are increasingly likely to self-form naturally as you remove useful energy from a system, but this trend runs against the axiom that increased levels of order require more energy for their very composition. Clearly there is a point at which a system must have more energy removed, than would be required to comprise a certain high level of order, rendering that chaotic system wholly incapable of creating said level of order naturally.
The irony here is that Prigogine showed mathematically how complex systems could self-organize out of diffuse chaotic environments. You apparently never even read the abstracts of his papers or at the very least didn't have a clue as to what they actually said.
You are very confused. You are mixing and matching mathematics and engineering concepts willy nilly in ways that are never proscribed. For example, in algorithm complexity (what we are talking about, whether you knew it or not), there is no concept of "energy". None at all. It isn't a mathematical concept. Energy does apply to the computing hardware (which is used to feed the entropy pump), but that is utterly irrelevant to what you posited. The computing hardware doesn't change its energy characteristics whether you are pumping random bits or a Fourier transform through it. Bits are bits.
Clearly there is a point at which a system must have more energy removed, than would be required to comprise a certain high level of order, rendering that chaotic system wholly incapable of creating said level of order naturally.
This is a variant of the very tired (and fundamentally ignorant) argument trying to mix axioms of information theory and thermodynamics. First, nothing in information theory requires energy to generate order or the lack thereof. A word processing program 100kBytes in length require the exact same amount of energy to create as a stream of random noise of the same size. Period. As I noted above, information theory does not create any physical laws with respect to energy. Thermodynamics does have a number of rules regarding entropy (Note: "order" is a meaningless term; there is either more or less entropy in a system). As has been mentioned many, many times: local reductions in entropy are permissible in thermodynamics in the presence of an enthalpy gradient. This is absolutely true of every biological system we have evidence of. In fact, the definition of life in most biology texts alludes to this (all living things have a metabolism).
As I have stated in previous posts, don't address heavyweight mathematical topics with pedestrian definitions of the relevant terms. You are not understanding the mathematics at all and are engaged in a very common misapplication of these concepts. Worse, you whip out sources like Prigogine which actually refute your position. Fortunately, this discussion touches on an area of mathematics that just happens to be my field of expertise so I am pretty damn confident about the basic and elementary definitions that I am using. It would be easier to have a useful discussion if we weren't arguing ancient mathematical definitions that can be looked up in any decent and relevant mathematics text.