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The Egyptian engineering scientist and theoretical physicist Mohamed El Naschie has found a definite resolution to the missing dark energy of the cosmos based on a revision of the theory of Relativity. Einstein’s equation of special relativity E = m_{0}c^{2}, where m_{0} is the controversial rest mass and c is the velocity of light developed in smooth 4D space-time was transferred by El Naschie to a rugged Calabi-Yau and K3 fuzzy Kahler manifold. The result is an accurate, effective quantum gravity energy-mass relation which correctly predicts that 95.4915028% of the energy in the cosmos is the missing hypothetical dark energy. The agreement with WMAP and supernova measurements is astounding. Different theories are used by El Naschie to check the calculations and all lead to the same quantitative result. Thus the theories of varying speed of light, scale relativity, E-infinity theory, M-theory, Heterotic super strings, quantum field in curved space-time, Veneziano’s dual resonance model and Nash’s Euclidean embedding all reinforce, without any reservation, the above mentioned theoretical result of El Naschie which in turn is in total agreement with the most sophisticated cosmological measurement. Incidentally these experimental measurements and analysis were awarded the 2011 Nobel Prize in Physics to Adam Riess, Brian Schmidt, and Saul Perlmutter.

Special relativity presupposes a smooth space-time with Lorentzian symmetry group invariance [_{0}c^{2} may be elevated to a quantum relativity, i.e. a quantum gravity equation when scaled by _{}

This prior intuitive expectation noted first by El Naschie was confirmed later on by him on two counts, namely first experimentally using the cosmic measurement of Ries, Schmidt and Perlmutter [

In this paper we show following El Naschie that for a fuzzy Kähler [10,13], the scaling factor changes from

to. In addition to giving a derivation of where m_{0} is the controversial rest mass and c is the speed of light, we show that this result is in exquisite agreement with the cosmological measurement of COBE and WMAP as well as the analysis of certain supernovas which led to the award of last year’s 2011 Nobel Prize in Physics [

As aptly noted by El Naschie, an equation based entirely on a smooth space with Lorentzian space-time invariance developed years before quantum mechanics was formulated by Heisenberg, Schrödinger and Dirac could not possibly be expected not to break down at some quantum or intergalactic scale [1-3]. In the present short paper we show following El Naschie that the missing dark energy in the cosmos, discovered through various accurate cosmological measurements [_{0}c^{2} where E is the energy, m_{0} is the rest mass and c is the speed of light outside its range of validity [5,6]. We thought for a long time and understandably so that gravity cannot have that crucial effect on elementary particle physics. Similarly we thought that quantum mechanics also has very little effect on cosmology except maybe when it comes to incredibly shrinking objects such as black holes [

However when we started asking very deep questions regarding the unification of all fundamental interactions [7,8] we recognized suddenly that at the extreme small distance such as the Planck length (10^{−33 cm)} the feeble gravity becomes as strong as the other three fundamental forces, i.e. the weak force, the strong force and the electromagnetic force [1,3,7]. On the other hand we now just realize that quantum effect such as quantum entanglement has an impact on physics at extremely large intergalactic distances. It is so profound that the classical equation of Einstein E = m_{0}c^{2} is off the correct result by almost 95.5% [

In the present work we trace back the deficiency in E = m_{0}c^{2} and prove that this is the case because of the real non-classical geometry and topology of the actual fabric of space-time [2,7,8]. This non-classical topology is essentially the cause of amplifying what we perceive as

quantum effect which screens the energy by as much as 95.5% in full agreement with measurements. In particular we will show that the ratio of the two Betti numbers [9-12] fixes the homology of space-time’s de Rham topology of smooth classical space-time of relativity and the rugged K3 Kähler [_{QR} = b_{2} (smooth)/ b_{2} (Kähler) which accounts for the 95.5% missing dark energy [_{2} counts what we may call three dimensional holes (voids) [

Following super strings and related theories [

We consider a K3 Kähler manifold with four complex dimensions used extensively in theories with hidden dimensions particularly super and Heterotic string theory [12,13]. The manifold is fixed by the Betti numbers which determine the Euler characteristic and the signature. In case of non-fuzzy (crisp) K3 the Betti numbers are [10,13]

It follows then that the Euler characteristic is [10,13]

while [10,13]

and the signature is [10,13]

We stress once more that b_{2} counts the 3 dimensional holes in K3 and will play a crucial role in our derivation.

Now we look at an even more exotic version of K3 [_{3} as the previous crisp Kähler. Only and which measure a sort of average number of 3D fractal voids are given by [13,14]

and (5)

where. It follows then that [13,14]

It is important to note the following: The small numbers = 0.05572809014 as well as = 0.236067977 and all have various physiccal, topological and geometrical interpretations. For instance is the exact value of the vital Immirzi parameter of loop quantum gravity without which nothing would fit in this theory [

That means

which is a deep and useful relation utilized in various E-Infinity derivations.

We said that b_{2} is an important homological invariant of a manifold [9-11] and that it basically counts the 3 dimensional voids in the manifold [9,14]. For a two sphere S^{2} or any connected manifold b_{2} is equal to unity b_{2} = 1. On the other hand for our classical Kähler b_{2} = 3 + 19 = 22, and this number already indicates that this manifold is almost a Swiss cheese full of 3 dimensional holes [10, 13]. Compared to the smooth S^{2} manifold akin to the space-time of Einstein, K3 has 22 times less space-time and following general relativity, less energy. Now following, for instance, Nottale’s scale relativity principle, we could define a scaling λ_{QR} to be:

and use it to scale E = m_{0}c^{2} to

This implies that the missing hypothetical dark energy is

This is extremely close to the cosmological measurement [

In fact when using the fuzzy Kähler we notice immediately a quantum mechanical interpretation of the result because

means that

However is nothing else but Hardy’s generic quantum entanglement [16,17] so that our λ_{QR} may be viewed as the screening of a substantial part of the energy in the cosmos by quantum entanglement reducing the Newtonian action at distance by as much as 95.4915%. Finally there is an even more immediate interpretation when we invoke string theory and M-theory. The largest number of dimensions in Heterotic string theory is 26 in the classical case [

or more accurately [12,14]

Thus our scaling exponent is

or in the fuzzy case [13,14]

Within this mental picture we could say that the missing dark energy is concealed and hidden inside the dark extra dimension [7,12,14]. It is a deep philosophical and ontological question to consider something which we cannot measure nor detect to be real or not.

The homology of K3 Kähler and what El Naschie calls extra “dark” dimensions is the definite cause behind what we call the missing dark energy [_{0}c^{2} by a scale relativity factor λ_{QR} defined as the ratio of two second Betti numbers [10,11]. Since the Betti number of fuzzy Kähler b_{2} is 22 + k and since b_{2} = 1 for Einstein space of special relativity, our λ_{QR} becomes equal to and one finds [_{QR} may also be written as that means half Hardy’s quantum entanglement probability found using orthodox quantum mechanics and confirmed through sophisticated quantum information experiments, we feel that the ordinary sharp non-fuzzy K3 Kähler manifold approximates quantum gravity space-time geometry and topology to an astonishing extend and must be real. Seen that way, we must infer with El Naschie that the designer of the universe is a mathematician [