Aut-The rough solution of curvature and compression

So there was only one missing element in AuT which was covered but without much detail. That is the reason for the evolution of compression and decompression.
In this post the evolution of curvature from a quantum value of pi, or more precisely from a precursor to what we call pi based on the evolution of positive and negative solutions (-1)^x.
Compression, at least based on the suggestion of observed models, evolves from an evolving value of e to to this same function and decompression results from the opposite value of this same function.

Pi is an evolving equation set forth with specificity as:

Pi= N+(from 2 to max x)N/F(pix)]

What this equation allows is that you can change N to any number and get a pi result and the rate of change for F(pix) being a constant in the calculation of pi never changes. While it doesn’t correspond or converge on 161% its rate changes in a way that suggests a relationship.

Figure 7 pi vs F-series

          This F-series is not F(n) for the compression equation, but the relationship remains.  Both have points of equivalence both in terms of the rate of change and based on the rate of change of the F-Series to F(pix) and pi for n=1.  

          The % change (for pi where n=1) and pi vary closely, and regardless of the value of n, the variation of F(pix) remains closely related.

Why change? Because curvature is change, in quantum states there is no change, no curvature, no dimension.  But when you move between quantum universes, you create history, distance, curvature and velocity.  F-series are infinite diverging series, but built into the divergence as a percentage is the amount of change between states.
The relationship of these values to exponential evolution and pi lies in the ratios of change and in the way that these features change as the solution to the single variable algorithm evolves and compression states increase.
Convergence:

The converging series for Pi as derived in this work when this series is used for N converges on 1, quantum change, x what have you.  It is miraculous in terms of proof and while not dispositive, it is strong evidence in support of the underlying mathematical structure. 

F-series diverge on the prior two numbers being added together.  Pi converges based on F(pix):

Pi= N+(from 2 to max x)N/F(pix)]

F(pix)=[(-1)^x]+[2x(-1)^x-1] or

Where f(x)=2x(-1)^x-1; F(pix)=n/[n+n/fx]

          The percent change for this converging series for Pi as derived in this work when this series is used for N converges on 1, quantum change, x what have you.

The equations we are left working with are these:

Base equations N=n-1+n-2;f(n)=n+n-1+n-2 I=2^n	Converts under compression to (F(n)^(2^n)
F(pix)=[(-1)^x]+[2x(-1)^x-1] F(pix)=n/[n+n/fx] F(pluspix) based on -1^n X^1/x and e=sum(x=0 to inf)1/x!=1/1+1/(1*1)+1/(1*2)+1/(1*2*3)+… vs the alternate definition of (1+1/n)^n	This combination results in compression y(t)=A*sin(wt+phi) where w=2pif(rad/s) for pi(n) where n=4 and with one observed model for a lower ct state being: C(t)=E(1-e^-t/RC) where 1/x corresponds to –(t/RC) corresponding to a an evolving value of e and the resulting evolution of pi and decompression is defined by D(t)=QRCe^-t/RC which is the opposite effect.
The global max of f(x)=x^1/x is at e compare (f(n)-^2^n).  as a location of the  phase shift which generates pi (figure 8 below). Sin and cos vary between 1 and negative 1 just as f(pluspix). At the quantum level these are the same, but they diverge and hence the consistence (pluses in a row) of f(pluspix) must increase as you get compression.	C(t) corresponds to C(x-1) in the compression approximation.

          Pi= N+(from 2 to max x)N/F(pix)]  This is different from the classical definition of pi which is the ratio of the circumference of a circle o its diameter.

          F(pix)=[(-1)^x]+[2x(-1)^x-1]

          The solution to F(pix) converges on 1 from either side.  What this equation allows is that you can change N to any number and get a resulting value for pi converging on a different value

          We focus on the value where N=4, ct4 geometry, but the true significance of pi is tied to a more complete analysis.  The change in Pi also converges on 1, but from another position, 2 (and change based on the rough analysis).  This is consistent with a universe tending towards, but never reaching complete compression.

          e=sum(x=0 to inf)1/x!=1/1+1/(1*1)+1/(1*2)+1/(1*2*3)+… vs (1+1/n)^n

          The rate of change for each of these approaches 1, but it happens a lot faster in the first method of calculating e

Looking at this from the exponential function F(x)=a^x.  In AuT a=2.  However, we also see e in o-space at ct4 being approx. 2.718 the natural log.

https://en.wikipedia.org/wiki/E_(mathematical_constant)

          F(n)=(n-1+n-2)

Sin and cos vary between 1 and negative 1 just as f(pluspix).

At the quantum level these are the same, but they diverge and hence the consistence (pluses in a row) of f(pluspix) must increase as you get compression.

Figure 8 quantum pi

Quantum plus and minus values for pi: (+/-), (-/-),(-/+),(+/+).

Numerator: plus, minus, minus, plus

Denominator: minus, minus, plus, plus

These go donot;do;donot,do (N/D)

The evolution, of course, is from the linear model to the circular model based on increasing in quantum states in between positives and negatives which appears largely to be through F-series or Informational expansion.

The offset is most likely a result of The global max of f(x)=x^1/x being at e compare (f(n)-^2^n) the negative version to f(n)^2^n for the positive version which the shift being controlled by an expanding difference in f(pluspix) which is the -3,5,-7,8 which in turn derives from-1^x which is how the phase shift can occur within the equation.

Where f(x)=x^(1/x) the phase shift occurs at e.

For the F-series compression equation, this equates loosely to f(x)=F(x)^[1/(2^x)].

e like pi is limited by the amount of information for how far it is solved.

See, e.g. https://en.wikipedia.org/wiki/E_(mathematical_constant