You wrote to Harv, "Paul, you think what I have done is some theorem because you can follow the math and understand the correctness of the procedure but I suspect that, when it comes down to actual fact, you also miss the nature of the original problem."
You are correct that what I think you have done is some theorem, but I don't think you stated the correct reason. The reason I think your result is a theorem is because what you have done is consistent with what I understand a mathematical theorem to be. In the second half of this post I will try to explain that reason.
Before I do that, though, I will try to allay your suspicion. You say that you suspect that I have missed the nature of the original problem you have solved.
I have read with interest and delight your recent posts in which you have tried to explain "the nature of the original problem you have solved". In all cases everything you said made sense to me. Nothing you said seemed puzzling to me. Of course, as you know, when I went through the details of your paper, there was much that puzzled me and it took the better part of two years for you to clear those things up for me. And, as you also know, when it got to the very end of Chapter 1, I got over my head and I cannot say that I understand the details beyond that point.
But as far as understanding the problem you have solved, I am convinced that at some level I understand what you have done. At least, I am convinced that you have discovered a profound result which could be eminently useful if it were to be adopted and used by theoretical physicists in their work.
I don't expect that expressing my conviction will convince you, and that was not my purpose. I simply wanted to tell you where I stand: I think I understand what you have done and I know that you doubt that I do.
You recently wrote, "Well, I know a little math. And Harv, I will define "mathematics" to be the construction and study of self consistent systems because I think that covers what mathematicians do! They define things and then see what they can deduce from those definitions!"
As I see it, Dick, you have defined things and then seen what you could deduce from those definitions. You discovered that by defining a completely general and unspecified set of numbers, you could deduce the constraints expressed in your differential equation. As you say, "I think that covers what mathematicians do!" And, I might add, when they do that, they call the results theorems.
You logically begin your work on the foundation of mathematical analysis and assume that foundation was properly laid by mathematicians. You proceed from that starting point by defining a few terms strictly in the context of mathematics with no appeal to anything real (the possibly unfortunate choice of one of your terms, viz. 'Reality', notwithstanding.) With those definitions, you develop equations strictly using logical deduction. There is no use of induction in any part of your development. I can find no part of your logical development that violates any of the rules of mathematics. In my mind, there is no question that your result qualifies as a theorem.
There are two things, however, that cloud this issue. One is the discussion of scientists, their methods, and their motivations, in the early part of your paper. I think this has distracted and confused many people who have tried to read it. Whether or not what you say in those paragraphs is true has no bearing on the validity of your mathematical results. I think it would be better to remove them.
I think these paragraphs have caused much of the problem for Yanniru, Harv and probably Alex.
The second thing that clouds the issue is the possible applications of your theorem. As you have demonstrated in your chapters 2 through 5, it can be demonstrated that certain solutions of your fundamental equation can be seen to be familiar equations of physics which have been arrived at by completely different methods. I think these chapters give us a glimpse of the awesome potential of your theorem to model extremely complex aspects of reality. But here, you have only given a primer.
This second cloud is the part that gives Harv particular trouble. He thinks that your work is somehow intended to address or explain some aspects of what he considers to be "reality" and he takes you to task for not conforming to what he sees as the correct approaches to that objective. As you continually try to point out to him, these are diversionary concerns and have nothing to do with the profound result you have obtained.
Before I post this, I want to say that your conversations with Harv, Aurino, Yanniru, Alan, and all the rest have exceeded my wildest hope when I first introduced your work to them. This level of intense conversation was what I was hoping for. Unfortunately, it has been a lot harder for people to understand what you did than I ever imagined it would be.
I still think there is hope though. Aurino, Alan, and I all claim that we have some level of understanding of what you have accomplished. We are not able to convince you that we understand, and maybe we don't. But, speaking for myself, you have given me the greatest single increment in my current idea of what is going on in the universe since I learned about differential equations and saw how much of our world they can model. I now really think I see the whole elephant, and you know that I am deeply grateful to you for that. I just hope I can help spread the insight that your result provides to some other people.
I am going to start on a post to Harv trying to explain to him how and why your result is a mathematical theorem. I would appreciate it if you would follow along in that discussion and correct me if, and when, I go off course.
P.S. Your color scheme to identify quotes works well on the screen, but when I print your posts to take up to the mountains to read, I lose the color. I'd like you to include quotation marks or something that shows up on black and white printing.