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Starting from well established results in pure mathematics, mainly transfinite set theory, E-infinity algebra over operads, fuzzy manifolds and fuzzy Lie symmetry groups, we construct an exact Weyl scaling for the highly structured E-infinity rings corresponding to E-infinity theory of high energy physics. The final result is an exact expression for the energy density of the cosmos which agrees with previous analysis as well as accurate cosmological measurements and observations, such as COBE, WMAP and Planck. The paper is partially intended as a vivid demonstration of the power of pure mathematics in physics and cosmology.

There are a few scientists, the present author included, who feel quite strongly that at its deepest level physics in general and high energy particle physics in particular is governed by the laws of pure mathematics [

The main purpose of the present paper is twofold. It is intended first to make the mainstream applied physicists more aware of the treasures of pure mathematics waiting to be discovered by them for application [

The present paper will use transfinite set theoretical results [

The main pillar upon which our highly structured E-infinity golden ring [

is de facto replaced by an almost elementary E-infinity arithmetic manipulating

of union and intersections in a way analogous to a Suslin operation. For instance the vital

to be nothing but

Similarly one finds that the same value is obtained for the average Hausdorff dimension [

The above constitutes the required minimum to understand the rest of the paper which is devoted to a major cosmological problem considered by many as the greatest mystery of our time [

We start from the obvious fact that Einstein’s space is 4 dimensional, i.e. 3 spatial dimensions fused to 1 time dimension and that the classical photon of the electromagnetic field moves in these 4 dimensions [

and the corresponding Einstein special relativity formula is again formally given by the trivial tautology

Now we move to the superstring picture [

i.e. it is approximately 1/22. This is a remarkable result because it means

which is complete agreement with all known cosmological measurements [

The preceding result and conclusion needs a great deal of further elaboration and refinement. To start with one could find it strange to set

In the above equation

exactly as should be. Having established that 137 is a dimension or particle-like state we should use then the exact E-infinity value which is

where

It is clear that the compactification of the 6 dimensions is best dealt with using a Calabi-Yau manifold [

As an aside we mention that at the beginning and due to inaccurate measurements

The difficulty in the above is that we are involving 'tHooft’s

We start this time from the bulk E8E8 and note that

However if we work without super symmetry with Einstein and Kaluza-Klein spacetime [

where

where we have taken the positive root only as a start. That way we see that

We recall that the number of states in Witten’s model are given by [

Furthermore this model represents a maximally symmetric manifold [

Similarly we know from the Heterotic superstring theory that the number corresponding to 66 in Witten’s model is 63 and that (2)(33) = 126 is exactly the total number of the particle-like states of the standard model. The corresponding number for Witten’s model would consequently be (2)(66) = 132. This is thus the dimension of the modular manifold M_{6,22} discussed ealier. From the above we see that the scaling factor

which is the exact integer approximation of cosmic dark energy density and as mentioned repeatedly, is very close to all cosmological measurements and observations while the ordinary cosmic energy density is clearly the ratio of 66 − 63 = 3 to the 66 total which means

as should be [

as reasoned earlier. Writing

and

are two results obtained a few years ago [

By taking internal and external isometries of spacetime carefully into consideration [