I have been slow to answer you because I really didn't know how to put it so you could understand. Though this is an answer to your e-mail, I post it in the forum because I think the issue should be understood by anyone interested in what I have done.
You have not been keeping your end of the deal and I am sure that has occurred because you do not understand the request I made of you. I said that you must understand mathematics in order to follow me and that, if you chose to proceed down that line, I would help you.
However, you keep wanting to put things in the form of verbal discourse which is, by its own nature, vague and inexact. Your explanations of equations 1.1, 1.2 and 1.3 are far more complex than what is being presented in those equations and yet entirely miss the fundamental issue (and that issue is actually quite simple). The problem is that the issue cannot be presented in English without consuming unbelievable quantities of space. I will try to give you a very very simple example so that you can perhaps understand the value of mathematics, but I have no intention of continuing the discourse on such a low level I am too old and to do so would take too much time.
Let us suppose, for the sake of argument, that a description of the entirety of the universe and everything the concept can include can be expressed in a message consisting of 15 numbers. "Reality is a set of numbers -- those 15 numbers!" Now that is quite ridiculous but the idea that it could be expressed with a very very large set of numbers is not. The power of mathematics is that the two very different cases can be handled in exactly the same way. I will pretend that the first case is correct because I haven't the time to explicitly write down an example of the second.
An observation is defined to be a subset of the numbers representing reality. In my example the size of the subset defined as an observation must be small. If the size of all observations were one number than the universe allows 15 observations. The past is defined to be the observations I have to work with and the future is defined to be the observations I do not know and wish to predict.
Now, I have proposed that there exists a rule which will tell me what observations are possible.
If I know the rule, then I know what those numbers are. (Actually, I do not know what the rule is and have no way of knowing what the numbers are except by an observation, but I will lay that aside for the moment.)
Instead of your vague representation "(data, data, data, ...)", (if I knew the rule) I could write down the 15 numbers: for example (2,6,35, 24,5, 2, 5, 12, 24, 5, 5, 6, 1238,1238,13). You must realize that with only 15 numbers I cannot even begin to exhaust the variety of patterns which might be of interest here but I have to lay that aside as expressing these ideas in English is long enough already.
Now I said that (if I knew the rule) I would know what observations are possible. What are they in this example? They are as follows:
(2,12),(2,24), ... , (2, 13), (6,35),(6,24), ... ,(6,13),(35,24), ... ,(35,13), ... ,(1238,13),(2,6,35), ... ,(2,6,13),
... , (2,6,35, 24,5, 2, 5, 12, 24, 5, 5, 6, 1238,1238,13).
So, the original problem was express the rule based on a set of observations. That being the case, let us look at a set of those observations (our past - the information we have to work with). Suppose we have 5 observations to work with and that they are as follows:
(5,5,5),(5,5,5),(5,6,5),(1238,1238,13,12),(2) --> observations .5, 100, 101, 2, and 27. (I have attached a tag to the observations which I will call time) so the order would be
Not very much to work with but exact examples are difficult to present without mathematics.
The first thing I notice is that my observations have different numbers of numbers, so I will add unknowable data to round the situation out. I will change the observations to
Now, I have just added numbers at random to fill in the holes. (Analogous to the assumption that just because it isn't in my observation doesn't mean it isn't there - something everyone does all the time.) In this case, I haven't actually been very random because if I hadn't taken care to create a certain problem it probably wouldn't have appeared: the two identical observations. (If the number of numbers in an observation and the total number of observations are reasonably high it becomes harder to prove the problem has not occurred.) The central issue is that, if the problem does occur, I can eliminate it by adding unknowable data. In this case we can just add another entry and the problem is eliminated:
Now all the observations are unique. Notice that the observations we are now working with are, on occasion, outside those observations available from the original set of 15 numbers: there is no way of obtaining an observation which is represented by five "5"s! This is not a real problem as all we do is add that additional five to the original set. It is totally fictitious, a creation of our mind, but, so long as it obeys the same rules which constrain the remainder of the set, its existence is logically defended by our observations.
We are now ready to take the final step: if the observations are unique even after one number is removed, then the rule will allow us to obtain the missing number. (If the observations are not unique even after one number is removed from each, just add some more unknowable data.) Notice that in the case above there are two ways to get four "5"s. So we add another number - more unknowable data!
At this point we can fulfill the content of equation 1.1: if we know the rule and are given five numbers from an observation, the sixth is known.
given The sixth is
etc. etc. etc ...
Such a list constitutes the definition of a function! In this case, the definition of the function "f" in equation 1.1. Equation 1.2 merely states that the value of "f " (which is the sixth number in this case) minus the sixth number is zero. That expression defines the function F. And the rule which specifies all our observations may be written F=0.
It follows that a rule which predicts the entirety of our observations can be written F=0 even when the number of numbers in the observation and the number of observations become astronomical. It is the issue of that fact which is important, not the actual expression of it.
This is simple algebra and I do not intend to go through it again.
You need to learn some mathematics! The power of mathematics is very difficult to express in the vague concepts available in English.
Have fun -- Dick