ah it didnt come out right well this part

Now suppose we continue this process to infinity. What proportion of Columns taken in the order of the bottom sequence t1, are such that the relative frequency of A's in said column is 50 percent (or approximately 50 percent). It appears to me to be 100 percent or almost all in the combinatorial sense. My question is this claim true? and if so what technical mathematics do i need to prove this rigorously as it certainly appears true, so as to make it definitive.

I need to know to how prove this result if indeed it can be proven (its in bold at the end)?

I don't believe you cannot make it shorter. Just take some time and make sure people do not spend their own time on reading so many lines of text.

As for the question itself - have you tried the binomial test?

Here's the example of how it works in R:

## Conover (1971), p. 97f.

## Under (the assumption of) simple Mendelian inheritance, a cross

## between plants of two particular genotypes produces progeny 1/4 of

## which are "dwarf" and 3/4 of which are "giant", respectively.

## In an experiment to determine if this assumption is reasonable, a

## cross results in progeny having 243 dwarf and 682 giant plants.

## If "giant" is taken as success, the null hypothesis is that p =

## 3/4 and the alternative that p != 3/4.

binom.test(c(682, 243), p = 3/4)

binom.test(682, 682 + 243, p = 3/4) # The same.

## => Data are in agreement with the null hypothesis.

the basic idea is that you have an infinite sequence of trials for A's at the same time, interspersed cross space (or modal space but lets use space to make it simpler) . Given my model if the probability  for A is 0.5 then you will get the limiting relative frequency for A to be exactly 50 percent

t0: AAAAA~A~A~A ~A ~A A~AA~A etc the limiting relative frequency here converging to 50 percent because each of these elements resulted from some number on an equidstributed sequence modulo 1 being assigned to each triall, such that the digit assigned to each trail element in the sequence at t, determines whether A occurs via the following procedure: if the digit is smaller than the Probability value 0.5, then A occurs on that trial, otherwise it does not.

Ie we had a trial of coin set ups say at time t, s1,s2, s3, s4, s5,s6,s7 where s1 randomly got assigned digit 0.1 because 0.1 is let us say the first number in the equidistributed sequence {0.1,0.2,0.3,0.4,0.5,0.6.....} As such we get that the first trial gives result A because 0.1

There are then further questions if we allow the equidistributed sequences to be different the probabilities at different times to be different, and if we take the sequences of digits to reflect all the real numbers  of the continnum (if that makes sense) so that we have uncountably many countably infinitely long temporal columns but they can wait. If you want more details it in the original text (which details the swapping method which is required for certain reasons, due to cantorian ordering problems and not just to ensure convergence and ordering) The basic question is what piece of mathematics do i need to rigorously prove this result

Have you tried this: https://en.wikipedia.org/wiki/Probability_axioms ?

Not sure I really got the question. The model how you pick the sequences is not clear at all to me. So you pick events A or its complement with probability 1/2. Let me use 1 for A and 0 for the complement. From your picture with flipping the coin, I suppose that they are all independent, or not?

Then you build groups of 3 trials at each point in time.

Isnt that just the same as picking independent 3 digit binary random variables X_n at each point n in time? With the property that P(X_n = 111) = 0.125 ?

Is that your question in short?

Then the answer should just be the strong law of large number. This implies that the relative frequency of the event X_n = 111 converges almost surely to its probability in the limit n to \infty.

see my result is something within the confines of standard probability theory; I am building a model and analysis of what probability is, so i cant help myself to standard limit theorems. So pretend i didnt mention the word probability at all. It is a combinatorial question, rather than a measure theoretical one

What i am saying is suppose there is horizontal sequence of digits, equidistributed modulo 1. These number are assigned to a countably infinite series of trials with the same chance value x for the outcome A; and by that i mean the event occurs if the number assigned to that trials< x otherwise if that number> x ~A occurs. So we can see that in the horizontal sequence the relative frequency will necessarily (almost certain) be x (this follows from the equidistribution of the sequence). Then we build another infinite series of trials, each one occuring at the same spatial position for each of the proceedings trials (coin flips lets say). But now we split this new collective of trials in half, and have two equidistributed sequences modulo one, such that the first trial in the second collective which corresponding to the first A in the first collective gets the first digit in the equidistributed sequence, likewise the first trial in the second collective which corresponds to the first ~A also gets teh first digit (of the other equi-distributed sequence, but the two sequence can be identical), the trial in the second collective which corresponds to the second instance of A in the first collective gets the second digit in the equidistributed sequence, likewise the trial in the second collective which corresponds to the second instance of ~A gets the second digit in the (other) equi-distributed sequence. So this second collective will also have  a horiziontal frequency = x, but will be partitioned into two sub-collecitves with relative frequency x, the first of which corresponding to that sub-collective whose members/trials pertain to all of trials whose outcome was A in the first collective, and the second which corresponds to all of the trials whose outcome was ~A in the second collective, That way we get a kind of independence of vertical frequency results. But this is a combinatorial result, and so i am asking a question about 'almost all' not 'almost surely' that is, the horizontal limiting relative frequency(lim relative frequency of columns/vertical sequences =x)=1. No mention of measures or probabilities involved. It also not the same as the case of  a coin with two outcomes, each of which branch into further trials of another coin, such that you get after countably infinitely many branches, an uncountable number of countably infinite sequences whose limiting relative frequency = 1/2 , which follows (i believe) if get the name right (from borel normality theorem; that almost all binary sequences have lim relative frequency 1/2, almost all ternary sequences have limiting relative frequency 1/3)-- and that is because in my case, the number of trials or branches at each splitting is equal to the number of trials in the first collective (so it doesnt expand), each trial corresponds to only one trial in each subsequent collective at a later time; so its akin more to an urn model then an expanding sphere model

Can you please clearly (and briefly) define what you model is?

You take an equidistributed sequence x_n in (0,1)

Then you take the map that maps x_n to {A,~A} if it is either less or larger 1/2 and call the resulting sequence L.

Can you please state precisely how you define the next sequence?

My problem is, that at one point you say that you take another equidistributed sequence, then you say you take it at later times (so I assume ignoring the first few digits), and at other occasions that you order the original sequence L.

So the resulting sequence L as you can see was  formed from an equidistributed sequence  x_n that maps 1-1 to a simultaneous sequence of trials, U, at t0 such that for each Ui, if xi< 0.5 thenLi is A otherwise Li is ~A. This. What i mean is that given this first sequence, L you then consider another infinite ensemble of trials U2 all at the same time, t1, but where t1>t0 (ie at a later time then the original sequence U at time t0). This sequence U2, maps now 1-1 to the elements of L, so suppose that each element of L which is a coin toss with result A or ~A occurs a certain spatial position; thus the elements of U2 denote a second coin toss, the first toss in U2 occurring at the same spatial position as first element of L, the second coin toss in U2 occurs at the same spatial position as the coin toss which is the second element of L and so on. Then what happens is, we map the elements of U2 unto two new sequences, L1 and L2 lets call them which give the results of the trials in U2. How does this work, L1 considers the results of the trials in U2 which mapped 1-1 (occurrs at the same spatial position) as the elements in U (and thus L) whose results were A, L2 on the other hand considers the results of the trials in U2 which mapped 1-1 (occurs at the same spatial position) as the elements in U (and thus L) whose results are ~A (thus the first element of L1 is the result of a coin toss which occurrs at the same spatial position as the first A in L, that is the first element in U which happened to be mapped to the first A in L) the first element of L2 is the result of a coin toss which occurs at the same spatial position as the first ~A in L, that is the first element of U which happened to be mapped to ~A in L). To get L1 and L2 we consider two new equidistributed sequence on [0,1] which could be identical with each other, and to the original xn, call them xn-1 and xn-2, such that each element of L1i is A iff xn-1

Cant you give just a precise definition, than a long explanatory text?

Questions:

What is the time t_0 and what t_1, etc.. You have the sequence (x_i)_{i\in\N} and the corresponding trial U_i. What is now the time t_0 for the sequence (U_i)_{i\in\N} in this model?

You suppose there is a U2, then you map it 1-1 onto L1 and L2 which you then construct it from two other equididistributed sequences. But then you use the A's of U to define the L1 and L2 ... What now?

It seems you have some collection of equididistributed sequences (x^{(j})_i)_{i\in\N}, j=1,2,3.

The first sequence is defined by U_i = f(x^{(1)}_i), where f(x) =1 if x>1/2 and 0 otherwise.  OK.

Can you give the map in 3 lines that defines either U2, L1 or L2?

And please just the definition, not mixed with any explanations. These are not very helpful to understand the definition itself. You can add them later.

--- the time t_0 is just some first moment in time it is the same for U as it for X. U is just the collection of trials say coin flips so U= @to(coin toss at x=0, coin toss at x=1, coin toss at x=2....), where they all occur at t_0 hence the @, and where the x values denote spatial positions

, L, is just the outcomes of those same trials in U. so there is a 1-1 mapping between the two, ie L= HHHHHTTTTTHTHTHT, where H is heads and T=tails, the first element of L, is the result of coin toss that is the first element of U so that would be a coin toss (at t_0) at x-0

Its correct that I use the A's (which are now H's sorry about that, and ~A is now T for tails )of U, by which I mean the A's of L, which are the outcomes of the trials in U, to define L1 and L2 which are the outcomes of the trials in U2. U2= is just @t1(coin toss at space=0, coin toss at space =1, coin toss at space=2.........) where they  all occur at t1 where t1 is some time later than 'to'

So U= @to(coin toss at x=0, coin toss at x=1, coin toss at x=2....),

U2= is just @t1(coin toss at space=0, coin toss at space =1, coin toss at space=2.........)

U is mapped 1-1 to L which is = HHHHHTTTTTHTHTHT,  U2 is also mapped 1-1 to U, and to get L1 we just get all and only those elements of U2 that are mapped onto H in U (or rather L which are the results of U), assign their results according to an equidistributed sequence, if H if digit