Thanks for your concern, but don't worry. I am always very careful.
***The Maxwell/Boltzmann analysis of velocity distribution in a chaotic gas uses the idea that, whatever that distribution is, it must be statistically stable. Essentially, they imagine an unknown statistical distribution, calculate the resulting new statistical distribution obtained via statistically random interactions. Then set the final distribution identical to the original distribution getting an equation; the solution of which established a unique distribution.
Now the question arises, would mathematicians regard what they did to be a theorem? ***
Good question. I am not competent to answer it, but I do have an opinion. I think that if a suitably rigorous definition of 'statistical stability' were made -- along the lines that statistical distributions must be the same -- then I think mathematicians would/could accept it as a theorem.
***(to date) you have failed to mention one of the most central aspects of what I have done. That is the fact that I use a very similar feed back analysis to yield my final result.***
Yes. I am stuck. And, that may very well be where I am stuck. I feel just like I did most of the time in Grad School. I get wonderful glimpses of the power and grandeur of the subject matter, but I had a heck of a time working the problems. I feel privileged to see the concepts to a certain degree, but I feel stupid for not being able to master the details. It's OK, though, because I never depended on those details to make a living, and glimpsing the concepts has enriched my life (or given me delusions -- either way it's pleasurable.)
The same goes for your work. I think I see the general concepts of what you have done. I think I see the general concepts of what it takes to be a theorem. And I think you have discovered a theorem. I know you know that, and unfortunately I can't offer any more evidence at this moment. But, let me try to sketch in a few more generalities in the conceptual picture anyway.
***For all practical purposes, what I prove is that there exists no collection of numbers which cannot be so "explained".***
I agree that this is what you have proved, and I think I understand what you mean by "so "explained"". It means that every collection of numbers must obey the constraints expressed by your fundamental equation.
***The issue then is, very simply, how do you propose to express the abstract idea "explained" as a mathematical entity.***
Yes, that seems to be the issue. Your description of the explanation being a transformation or an isomorphism would be right at home with the entities accepted by mathematicians all the time. I think what we need is for a competent mathematician to read your paper, not a physicist.
Now, let me be as bold as Alan and offer some far-out general ideas about this question.
***Looking at my picture from the perspective of "something A" (what is to be explained) and "something B" (the explanation), I use the idea that an isomorphic image of "something A" exists in "something B". In addition, "something B" (the explanation) also includes two other components: the mechanism by which "something A" (or actually, the isomorphic image of "something A") is transformed into the rest of "something B".***
It seems to me that there is another component missing from this picture, and that is the sentience to whom the explanation is to explain "Something A". Thinking in familiar terms, we can imagine an explanation in a book launched into space with no prospect of anyone ever reading it. Do the words in the book still constitute an "explanation"? Well, if anyone had ever read the explanation and learned or understood anything as a result, I guess we could say that, yes, the book really does contain an explanation. Even if no one had ever read the book, if the author wrote the explanation in such a way that it would be understandable if it were ever read, and if the author were competent, then again I think we could say that the book contains an explanation.
But what if, by some highly improbable fluke, a perfectly good explanation of something happened to get produced by a totally random machine but it was never known to any sentient being, would that then constitute an explanation? I don't see how it could.
In other words, I think sentience is a necessary component of any explanation, and thus must be part of your picture.
I don't know if you have read Chris Langan's CTMU paper yet, but I wish you would. I think the two of you should share ideas. Chris has concluded that the universe itself is sentient and that it comes to understand itself in a reflexive way. In his picture, this understanding adds information back into the universe enlarging it. The thus enlarged universe again comes to understand itself. This continues in a feedback loop that is reminiscent of the feedback you describe in your analysis. The universe cannot understand itself without an "explanation" and so, if at some point, the universe consists of "Something A", then "Something B" is required in order that the universe can understand itself. I think you have demonstrated and proved the way in which this "Something B" is produced, but you don't comment at all on the entity for whom it is produced.
I think that between you and Chris Langan, you have worked out the mechanism underlying all of reality. I think this is exciting and fun!
Note to Aurino: It is only exciting and fun. It is not important. The important thing is to live our lives in the way in which we all know we should. Knowing how the universe works might reinforce that attitude, but it certainly is not necessary.
So, in that spirit, I need to break away from this fun, and go attend to wife, kids, grandkids, Cubscouts, pets, the Waldstein (yes I am still struggling with it), my health and well being, and having fun in lots of other ways.