AuT-pp: dimension examined 1 of whatever & Hidden Figures

I'm going to have to make this into a thousand version, for whatever reason my ability to cut and paste directly into this blog has failed.   But the idea that we'll break this into a few pieces actually is appropriate to the process.  I would like very much to tell you how much I need you, how mad I am and how mad I am going, but I will instead just cover this mathematical undertaking.

The idea behind this series of posts will be to show why space time has to occur after ct1; why you can have space time based on a quantum size and still have measurements of smaller and smaller size.  These apparent inconsistencies can be solved easily and show why we have apparent linearity with infinite division even though it is based on quantum solutions where size doesn't even exist.
Sound difficult to prove?  It's not.

I'm going to show how I arrive at these solutions which involved an examination of dry mathematics.  In effect, looking for problems with my on mathematics forced me to determine where space time arose and why it could not exist prior to ct1 and how it came to exist afterwards.
This is, as a result several posts.  When we get to subsequent figures, all the proofs go to the same conclusion, but we are going to begin where I began with dimensional mathematics which fails in a quantum system.
Let's make traditional math fail.

But first, let give a nod to a movie I'd recommend to you.  Hidden Figures is an excellent movie on so many levels and is particularly timely in a world increasingly fractured, a country increasingly fractured.  If AuT teaches anything, it is that we are all equally irrelevant or, if you insist, relevant.
The fact that the movie dealt extensively with unappreciated mathematicians and was a little light on the actual calculations is a nice touch.  I am not very happy with the fact that they didn't spend more time on the actual calculations.
Euler's method is mentioned in the movie and that is relevant to the inquiry here because it is an estimating method much like the discussion which appears below (which predates me watching the movie) and also is not particularly relevant to the derivations set out herein which are according to a slightly different, but related method.

So let's start with some basic calculus understand the following:
1) The goal is to determine why ct1 cannot have dimension from a quantum state and
2) How can you have a quantum state based universe yield the infinite division we experience.

The first place to start is going to be a review of simple calculus and that is what this post will cover.  Next, and still in this post, the inconsistency of quantum mathematics in a dimensional universe will be raised.  It is really quite exciting and simple.
This is covered, by the way, even with many of the same drawings, in the published book, but this particular explanation is new material.

Theorem: Perceived curvature through the stacking of quantum states gives rise to distances less than quantum separation in an information based system.  This sounds a little on the impossible size, it’s like saying you have distances shorter than planck length, but the explanation is simple in concept although it will take several paragraphs to cover in detail.

The endpoint idea is that actual quantum length is not length at all, it is merely information, yes/no at ct1 states.  What we perceive as distance, i.e. the creation of space time, is the relative difference between two states which is a percentage of yes or no relative to the whole which can be infinitely divided (e.g. 22/7, the summary for pi, is an irregular number-note that pi is a true number capable of calculation for any total amount of information at a quantum time, any value of x; while 22/7 is not).

If there is a rule that every other solution is the opposite solution after the initial solution it looks like this:

Yes, yes, no, yes, no, etc.  This is a feature of the definition of pi as a converging series, albeit a small part.  One result of this equation is that there will always be more yes than no, but the amount of excess yes as a percentage of no will steadily decrease.  This is inherent in the geometric solution.  While a percentage change occurs at less than yes/no; no answer is anything other than yes and no.  This feature of logic and comparative logic is what allows dimension as we experience it to exist.

So having identified how you can start with a quantum block (yes/no) and end up with a comparative (% in this example) allowing for infinited division, let's look at some two dimensional models noting that the three or four dimensional models required by ct4 and ct5 (and higher models required by higher dimensional states) work the same way although with greater complexity.

Here are some curve solutions with my nomenclature:

Diff eq: sin(x+dx)/dx=sinxcosxdx+cosxsindx  sin(a+b)=with x=a and dx=b

So for diff eq:sin(x+dx)-sinx/dx as dx approaches zero is = [sinxcosdx+cosxsindx-sinx]/dx

=solving this normally assume that dx can go to zero which it cannot but if we could

=sinx([cosdx-1]/dx)+cosx(sindx/dx). 

As cosdx-1 goes to zero ad dx goes to zero LIMIT A

Sinddx/dx goes to 1 as dx goes to zero LIMIT B

So the derivative (as dx goes to zero) of d/dx(sinx)=cosx

COS(A+B)=COSACOSB-SINASINB

(Cos(x+dx)-cosx)/dx=cosxcosdx-sinxsindx-cosx/dx

Cosx(cosdx-1/dx)+(-sinx)(sindx/dx)

(cosdx-1/dx)=0

(Sindx/dx)=1

Same analysis leads to dcosx/dx as dx goes to zero= -sinx

d/dx(cosx) at x=0 then this is the limit as dx goest to zero

cos(dx)-1/dx is the limit that is zero according to A

d/dx(sinx) at x=zero then

Derivatives of sin and cosine at x=0 gives all values of d/di(sinx),d/dxcos(s)

Proofs of A AND B are well known, but let's take a look at this.

Proof of B

Only one way of defining sin and cos is by geometric proof: dx replaced with theta

Theta is the angle from the center to an arc of the circle.  Sin(theta) is a line from the top of the arc down to the bottom of the arc and if the curve is taken out it is the arclength as theta approaches zero

If you double this, 2sin(theta)(bowstring)/2theta(bow)=sin theta/theta which goes to 1 as theta goes to zero. You get this straight line at quantum separation, i.e. the curve ceases to exist at quantum lengths.  Short curves are nearly straight, but in AuT they actually are straight.

So let's prove that in a quantum universe, you cannot have linearity:

The Distance as shown between the straight line and the curve approaches zero.  The line from the center is the cos, so the gap between the straight line and the curve is 1-cos(theta) for 1-cos(theta) tends to 1 (the line is 1(cos) and it approaches 1-1 as theta (the gap) goes to the same number.
A special case of this is where you use information as a part of the calculation.  Here, we use the quantum length (yes or no) 1-cos(theta) as theta approaches the quantum length of yes/no. This "gap" is the gap between the straight line estimation (like euler's method) and the actual curved line, the arclength.  In later drawings we'll show this falling apart, but for purposes of this drawing, it's interesting to see the estimation leads to a ratio of information to compressed information in a discernable degree, calculable depending on the value of pi.  This method fails, by the way, but it shows the mechanism of how compression occurs within a comparative framework.

You can pick any point to show the failure of dimension.  If the baseline, for example is reduced to quantum length, then you cannot have a circle, try to draw it.  Let's focus, however on the point in question, which takes us to an ever decreasing distance between the line between a point on the curve and the baseline at a right angle and the arclength.  
The drawing shows, in AuT, that the answer goes to [1-(yes/no)]/[a finite number of yes/no answers].  It is important to note that if this solution holds true then you cannot force the circle beyond this point.  It is also true, however, that finite number changes very subtly with the amount of curvature not just locally but also in the entire universe.  
Note there are two thetas (above and below the base line) to get this solution.

To show this you can do costheat-1/theta= -(1-costheta/theta) as theta goes to zero, but stops at yes/no.  There is a ratio based on arclength to the straight line between the arc and the straight line between the two values of theta that is measurable and specific for any value of pi, although it becomes very small it remains a ratio.

Looking at theta in radians, the length along the arc length.

2pir*min lenth/max radians for a circle where pi is solved to a particular, albeit large number, of places as a converging series.  This makes the length of the baseline (r) (given as 1, a much longer number unless the circle is very small.
If the value of pi changes, however slightly, the distance (yes/no) decreases or increase which is, impossible.  But this is not as clear as it will be shortly, but that is for the next post.