Hi Dr. Dick,
Still more to figure out re: your expose of equations 1.1, 1.2, 1.3.
In the above reply I pointed out what you may have overlooked: that adding unknown data may make "uniqueness" dissapear again if order doesn't count. This happens when you add a 3 and a 7 say to two observations where the only difference between them is a mirror image of the difference in the data you added to them.
That is, you added say a 7 to something that was lacking a 7 but had a 3; but that was otherwise the same numbers set as something that lacked a 3 but had a 7 and to which a 3 was added!
How does that affect your system!?
My breakthrough applies to your system as you describe your system in its latter stages, it appears. All that may be relevant is the final add-ons that allow a subtraction that retains uniqueness. That margin activity is what I addressed.
What you call "observations" might be called "perspectives". How can they represent reality when they are so limited? I prefer "perspectives".
When you allow repeat numbers as a problem: I note that any "repeat" must by its very nature be a "generalisation", something than can be divided.
As uniqueness is inherent to existence at some level.
Take a function (a rule) R that gives (a,b,c,d).
Add e and add f to the set (a,b,c,d).
Juggle the contents so you don't know which of a,b,c,d,e,f were obtained by the function R.
Subtract one item from (a,b,c,d,e,f).
Suppose you know R. And you know 5 of the variables in (a,b,c,d,e,f).
Then the missing variable is:
(1) either one of the items that the rule does not give; (e or f) or
(2) it is one of the items the rule does give; (a, b, c, or d).
In the first instance (1):
you have isolated the other item the rule does not give to be on its own with the 4 rule-items remaining in a group (after subtracting one of the 6 shuffled items to leave a group of 5). Since these 4 are given by the known rule; the non-rule item stands out.
In the second instance (2):
you have a pair of items in the set of 5 remaining that are not given by the rule. The missing item must be the 4th rule-given item.
Basically, it appears that if you know the rule to give 4 items; and add 2 more arbitrary items: by taking away ANY one item you can figure out from the rule sufficient to identify that the taken-away item is either a specific rule-item or one of a pair of arbitrary items.
Now, what about repeat this process?
Now, couldn't you take the 4 known rule-items in instance (1) and juggle them again with 2 new arbitrary unknowns?
So: (a,b,c,d,g,h) and you have reshuffled so you don't know which ones the rule gave anymore!
But you just use the technique above again.
You get a new instance (1) which leaves you back where you started with 4 rule-items in your 5 remainder group. I overlooked that before.
You get a g or an h isolated outside; with the other new unknown standng out in the remaining set.
And in instance (2) your new adding of 2 unknowns followed by new shuffling, knowing the rule, using subtraction; isolates your 3rd rule-given item as the missing item.
And so-on; keep repeating adding two unknowns; shuffle; use the rule; do subtraction; isolate the pair of unknowns from each other; or isolate one rule-item from the others.
Eventually you are doing this process with just one rule item in the set; in the instance (2) lineage. The instance (1)s I now realise keep sending you back to square one.
For the instance (2) lineage: since the rule now is effectively the item; this system has effectively swallowed itself and given you a logical construct that might map any rule, any function? OR leave you back where you started in the instance (1) case.
And on analysis; this logical construct looks like QED! And the rest of physics laws very likely seem to fall out!
It is your system, it seems. I disregarded identical items problems because the more things are identical the less relevant the rule is, surely? If the added unknowns say e, f were identical to the (a,b,c,d) then is one as good as another?
You might ask how determine they are not identical? Well, if you know the rule, that will tell you as it should only give you 4 items.
I may be missing problems in this construct. Can you see any?
It is mathematical.
It looks like the coconut puzzle taken to the last coconut. The instance (1) repeating process looks like Dirac's solution to the coconut puzzle (I can explain later).