Just to be clear I will reiterate the proof in slightly more detail

We will map the number 5.7
First we turn the 5 into binary i.e 101. Then we map the .7 which equals 111
this creates the number 101.111

Now we draw a line of length 1 with the halfway point marked. We can represent this by three points

A B C

B is the halfway point.
Now we map the number. If the digit is a 1 then we left shift from our current point half way to the next line. If it is a 0 we right shift

101.111

We start at B

First digit is a 1 so we left shift halfway to the next point and create point D as follows

A D B C

D is halfway between A and B

Next digit is a 0 so we right shift halfway to the next point which is B giving

A D E B C

Next digit is a 1 so we left shift halfway to the next point which is D giving

A D F E B C

For arguments sake we will say the line terminates here at point E

Now I measure the distance from A to E. This distance will be some multiple of a 1/2 x some a/b

We know this has to be the case because we are always dividing our distances by 1/2 so the final distance will be some multiple of a 1/2 x a/b. This will be true even if the number has an infinite number of digits

Now each number will produce a unique a/b coordinate.

This produces a set of fractions that I can then list in a 1-1 correspondence with the natural numbers

Err no. I only need one infinite quantity to map anything. If you can't map something when you have an infinite quantity then you are doing something wrong. Now you can have an infinity of infinities but I can map that to a single infinity

If you'd read my proof you would see that you can create a 1-1 correspondence with the natural numbers and the reals. If there is something you do not understand about the proof let me know

Ahh that is a good question and one I have not worked out the answer too. My assumption was that the homogeneous state has only 1 degree of freedom so to speak which is to change color.That is the assumption that I start with. As for the how and the why I'm not sure. All I'm saying is that if it can happen then my argument stands.

Its very easy you know. All the background would have to cycle its colours and hence time exists where there is nothing

Whats baseless about it? Is there something you don't understand about how clocks work and how easy it would be to create one using color.

I thought the whole enterprise of science was to have a curious mind. Furthermore the only leap I'm asking you to make is that a homogeneous state can change colour. That's a reality that is a lot simpler than this one..I'm not sure what you think I'm making up, I'm postulating a theory. Also you can create a clock and a form of language using this. I thought that was obvious but I can explain it more simply if you want.

Directions on a compass are not quite the analogy you need.Its not the same thing

So what were the physics before the big bang? I'm curious, I thought they popped into existence afterwards

Your defining laws that exist after the big bang to define colour. All I'm saying is that if the homogeneous state can change color by some means regardless of what that is then you can create a clock.The only thing I speculate is that the homogeneous state can spontaneously change color. That's all it needs.from there.