different form earlier on by the remarkable Italian Physicist L. Sigalotti [35] has conse- quence not only for us humans but for the entire universe and its formation and **energy** density. It is amazing how pure mathematics and mathematical logic can unite the un-unitable such as feeling and facts and all that via the magic of a Golden section based computation [236].

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probability of quantum entanglement [11]. This fact re- veals the quantum roots of our classical theory and we note on passing that **dark** **energy** is the absolute value of the negative **energy** of the quantum Schrödinger wave while the positive ordinary **energy** is that of the quantum particle, a subject which will not be discussed within the present work but is explained in some detail in Overview Charts No. 1-3 as well as Figures 3 and 4 [8,11].

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Now is this a trial and error solution? The answer is no. Then the next logical question is how could one recog- nize the meaning of all of these numbers? The answer is because we know the answer from so many other dif- ferent exact solutions. One could then retort sharply by saying that this is then more or less numerology, is it not? The answer to this crucial question is a definite no because if 90 percent of all the exact answers in high **energy** physics are found using numerology, then either this word numerology should be a misnomer and the correct name is number theory [21]-[26] or all theoretical physicists should forget the rest and engage themselves exclu- sively with numerology and elevate the word from a devaluation scorning word to a respectable and in this case highly effective method to come to exact results in agreement with experiments and **cosmic** observations. Num- bers are not just arbitrary things to do calculation with, they are probably the most basic things of nature and in- teract together following laws and theorems, some well known and some not yet discovered in the exact science of number theory applied to real world physics [21]-[26].

In the present short analysis we started from an infinite dimensional geometry and topology and ended with a finite expression for both the ordinary and the **dark** **energy** density of the cosmos. The result, as expected, is in full agreement with measurements but we gained additional evidence for the true physical fractality of spacetime at both the quantum as well as the cosmological level [1]-[10]. It is crucial to appreciate the deep meaning and implications of obtaining a quantum gravity result in agreement with **cosmic** measurements using essentially Newtonian classical mechanics and without involving quantum mechanics. We stress that this is only possible because of our use of nonclassical fractal-Cantorian spacetime geometry and infinite dimensional topology [4]-[10]. We note on passing that we have very recently discovered that all the above results could also be ob- tained using the nonlocal elasticity theory of A. C. Eringen [6] [11]-[14] which is a welcome confirmation of the basic philosophy of the present work, namely the unity of physics, mathematics, engineering and philosophy [14].

Our space unites the ununited and what was separated “by man” is no more “asunder” Our space is open and closed at the same time. That elevates the simple concept which we take as self evident to the higher reality of topology, namely a clopen space. Similarly our space is infinite and yet finite. Again the obvious contradiction is resolved by the hyper- bolic projection onto the plane “created” by “nature” via the work of Poincaré, Beltrami and Lobachevski in France, Italy and Russia (Kazan) respectively [11,12,47]. In our space what seems divergent may be readily resummed to give a finite answer via Borel techniques [41] as shown in ‘t Hooft’s analytical work and Wilson computer “simulation” [41]. The E-infinity spacetime is not fixed by a single dimension but by a host of various dimensions fitting to- gether in a mathematical symphony. Formally the space has infinite dimensions yet because of its fundamental Cantorian hierarchal structure, it has a topological Menger Urysohn dimension D = 4 in addition to an expectation topological dimension < n > = 4 + φ 3 = 4.23606799 which happens to be equal to its Hausdorff dimension [47]. This is not all. It still has a scaling dimension equal to 4 − ε where ε is equal to twice Hardy’s quantum entanglement as well as a spreading “spectral” dimension equal to 4.0199999 first discovered by R. Loll and J. Ambjørn and confirmed as distinct by L. Marek-Crnjac and the present author [11,12,40,41]. The reader must have noticed that we have descended from the global to the particulars. However this is only momentarily due to the complimentarity of the situ- ation. To wrap the eagle eye view in a few mathematical terms may be a little hard to do because n categories, E-infinity and Grothendieck K-theory [38-47] are too ma- thematical for most physicists including the present author. An exception in this regard may be the beautiful work of A. Connes on noncommutative geometry. In philosophical terms our space may be the geometrical topological incar- nation of Hegel’s dialectic and his coincidentia oppositorum [44]. At the end real experimental confirmation is what decides about what a potent scientific philosophy is. We show here beyond any reasonable doubt that our theoretical derivation of **dark** **energy** is in superb agreement with ac- curate **cosmic** measurements as well as the ultra symmetric logic of an all inclusive discrete nonlocal granular and physically real spacetime [12-18,39].

As is well known, there are many situations in which summing over all Feynman graphs is not only a daunting task but actually an impossibility [1]-[6]. In such cases, one of the best ways to solve the problem is to follow the non-perturbative strategy developed by the late Nobel Laureate J. Schwinger and E. Fredkin [1]-[6]. Another possibility similar in its non perturbative character is the E-infinity theory which resorts to summing over infinite hierarchal golden mean weighted dimensions [2] [3] [4] [7] [8] [9] [10]. It is the aim of the present work motivated by the Schwinger-Fredkin and E-infinity methodology [1]-[10] to see how the afore mentioned methods can be used to drive the ordinary **cosmic** **energy** as well as the **dark** **energy** density of our universe [10] [11]. Starting from QED and in particular by applying transfinite renormalization and P-Adic quantization [12] [13] [14] to the inverse electromagnetic E-infinity fine struc- ture “constant” ( )( ) 4

wave where φ is the inverse golden mean 1 φ = + ( 1 5 2 ) [6] [7] [8] [9]. Since at this resolution scale densities is a topological volume it was possible to calculate the volume corresponding to the pre-quantum particle density by in- tersection in five dimensional Kaluza-Klein spacetime and similarly but via un- ion operation for the pre-quantum wave [6] [7] [8] [9]. Proceeding in this way one finds that the **energy** density of the pre-quantum particle which is correlated and thus measurable [6] [7] [8] [9] is given by

The most popular candidate for **dark** matter is Weakly Interacting Massive Particle(WIMP), and WIMP has 2 → 2 annihilation channels into the Standard Model particles. WIMP relic density is determined when the 2 → 2 annihilation channels are frozen out. There have been a lot of experiments undertaken to find the WIMP, but we have not found any experimental evidence to date.

E = γ mc (1) where E is the **energy**, m is the mass and c is the speed of light while γ = 1 is evidently related to the absolute maximal **energy** density possible E = mc 2 [40]. The deep meaning of the above as well as the controversial as- pect connected to rest mass, real mass conversion to **energy**, difference and similarity to Newton kinetic **energy** E(k) = (1/2)mv 2 where v is the classical velocity of a particle as well as Einstein’s leap to a fully fledged gener- alization of similar earlier discoveries of E = mc 2 by Poincare and others will not be discussed here in the depth it requires [35]-[39]. Instead to cover these fundamental aspects, the reader is advised to consult first the out- standing work of Prof. W. Rindler and Prof. L.B. Okun [40] [41] and second the earlier work of the present au- thor and the references cited therein [35]-[39].

In the present work, we introduce E-infinity theory in which we can, in a manner of speech, “differentiate” and “integrate” the non-differentiable and non-integrible, namely geometrical structures made up entirely from transfinite point set of higher dimensional Cantor sets [12]-[16]. At a minimum, this is shockingly unconven- tional at least in the first few moments. However noting the use of non-standard analysis in the context of Not- tale’s theory of scale relativity as well as similar tools used in the work of the great French mathematical phy- sicist A. Connes, the inventor of noncommutative geometry [24] [28], the situation may start appearing slowly but surely in a different light. On this optimistic note, we will give in the present paper some more detail and explanation of the essential tools of the E-infinity Cantorian spacetime trade [20]-[34] before embarking upon applying our theory to the major problem of the discrepancy between the measured **energy** density of the cosmos as compared to the theoretical expectations. We are of course not divulging prematurely any secret when we say that one of our main tools in reaching our exact result is the fractal version of Witten’s eleven dimensional theory as it is clear from the title of the present work and it goes without saying that this fractal M-theory [18] [19] is as much the central piece of the work as it is the accurate determination of the ordinary and the **dark** **energy** density of this cosmos which turns out to be quite a surprise connected to what is probably the most famous equation in the history of theoretical physics, namely Einstein’s E = mc 2 where E is the **energy**, m is the mass and c is the speed of light [30]-[33]. The basic idea is that the fundamentally classical relativity theory E = mc 2 consists of two fundamentally non-classical quantum components that when added together, produce Eins tein’s beauty in the following manner: E ≅ mc 2 22 + mc 2 ( 21 22 ) = E O ( ) + E D ( ) = mc 2 where E(O) is the or- dinary **energy** and E(D) is the **dark** **energy** density of the cosmos [30]-[33].

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In a sense we are dealing here with a cosine of the butterfly effect. A small feeble local effect in the form of an attractive gravity induces at infinity an accumulated effect of anti-gravity adjacent to the horizon. In a sense ex- otic ideas that the universe may resemble a giant black hole or a knot compliment at infinity and therefore nei- ther open nor closed but topologically clopen may not be that far off after all [71] [74]. That way cosmological data and observation collected over a very long period culminating in several deserving Nobel Prizes in Physics has fused various theories together and confirmed the reality of Hawking’s radiation, Unruh’s temperature, anti- gravity and Rindler space-time all apart of completing the magnificent work of Einstein’s relativity, Planck- Bohr-Heisenberg’s quantum mechanics and Boltzmann thermodynamics [24] as indicated in the eminent work of T. Padmanabhan and his school [32] [33]. One could of course argue that the present derivation based on an analogy between metal forming, pinching of elastic types and the real behavior of a material space-time mi- cropolar elasticity [76] [78] is less fundamental than previous derivations starting from the zero set as a pre- quantum particle and the empty set as a pre-quantum wave [36]-[38] [77]. However, this is a largely subjective judgement and a matter of taste and personal philosophical stance [24]. In fact one could view the difference between the negative dimension of the empty set D T = 1 and the cosmological constant Λ = − 1 [34] as mere mathematical and physical tautology. In conclusion we have to admit that because of space limitation we have hardly touched upon many vital points which could have enhanced understanding the magnificent interconnec- tivity of mathematics, high **energy** physics and cosmology leading to the present synthesis. For instance we did not discuss the role of symplectic geometry which would have made the appearance of the golden mean and its derivatives and powers everywhere in our theory plausible, even unavoidable. However the reader may find all these points and more adequately covered in Refs. [55]-[75].

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We utilize homology and co-homology of a K3-Kähler manifold as a model for spacetime to derive the **cosmic** **energy** density of our universe and subdivide it into its three fundamental constituents, namely: 1) ordinary **energy**; 2) pure **dark** **energy** and 3) **dark** matter. In addition, the fundamental coupling of **dark** matter to pure **dark** **energy** is analyzed in detail for the first time. Finally, the so-obtained results are shown to be in astounding agreement with all previous theoretical analysis as well as with actual accurate **cosmic** measurements.

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The most complicated problems to solve in science are multidisciplinary problems, which include the problem of explaining of physical reality and physical essenceofimaginary numbers discovered more than five hundred years ago. However, the authors of the special relativity theory created more than a hundred years ago, were trying to unprovenly claim that they have solved this problem and that the imaginary numbers are not physically real with a principle of non-exceeding the speed of light. And physical experi- ments MINOS at the Tevatron collider and OPERA at the Large Hadron Col- lider were even seem to confirm such an assertion, since they could not refute the principle of non-exceeding the speed of light. However, since mathematics is the universal language of all exact sciences, the principle of the physical re- ality of imaginary numbers was still proven, but proven otherwise—with the theoretical and experimental studies of the vibrational processes in linear electric circuits that can be repeated and verified in any radioelectronic and electrical laboratory. And that is why they are definitely reliable and conclu- sive. Using the very general scientific principle of the physical reality of imaginary numbers allowed to correct the relativistic formula of special rela- tivity theory and on this basis to create a theory of hidden Multiverse con- taining more than two mutually invisible parallel universes. The nature of their invisibility is explained. It is proven that the existence of these unseen parallel universes explains the phenomenon of **dark** matter and **dark** **energy**. Using data obtained by the spacecrafts WMAP and Planck led to the discovery of quaternion structure of the hidden Multiverse and show that our hidden Multiverse is not the only one in nature.

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For energies less than Planck **energy**, classical description of space-time possible with scalar fields permitted by Particle Physics. As discussed above the concept of inflation in cosmology is therefore essential which is permitted if quantum fields are considered. In 1981 Guth [6] proposed inflationary Universe scenario using temperature dependent phase transition mechanism in order to get rid of the problems of the Big-Bang cosmology. In this scenario there was a phase of expansion of the Universe when a small causally coherent region grows to a huge size to encompass the present Universe. Subsequently it is found that a graceful exit from inflation is not permitted in the scenario [7]. A new inflation is then proposed by Linde [8], Albrecht and Steinhardt [9, 10] where the graceful exit is natural. The period of inflation was very short in which the Universe grows rapidly so that at a later epoch it allows the different parts we see to come in casual contact with each other, thereby explaining the isotropy of the observed Universe. Eventually, the quantum vacuum state decays, dumping its **energy** into the form of thermal radiation and the Big-Bang Friedmann model takes over. In 1983, Linde [8] proposed a suitable inflationary Universe model which does not require temperature dependent phase transition mechanism instead the Universe can be realized from a chaotic distribution of a homogeneous scalar field. In this scenario sufficient inflation required to solve the problems of the Big-Bang model can be realized if a causally coherent region grow out with an initial scalar field which picks up values ϕ > 3 M P ,

A main problem in astrophysics is the discovery, about two decades ago, that our Universe expansion is acceler- ating, instead of slowing down as predicted by the Big Bang theory [1] Scientists hypothesize the existence of an anti-gravity **energy** field dubbed **Dark** **Energy** (DE). It has been claimed that the identification and understand- ing of that DE would be the greatest accomplishment of the century [2].

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on the boundary of the Universe stand still, the Universe cannot either expand or contract as the velocity of light c means maximum deformation of space cells. In [2] we gave some evidence for the impossibility of proton decay, so the Universe cannot be destructed and barions will always exist in it. All the material in this article shows that the experiment on measurements of intensity i rather than anisotropy T of **cosmic** microwave background (CMB) in the mic ave frequency range far away from the Earth is of great importance for both cosmology and theory of elementary particles.

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The above analysis evidences that the space dynamic gravitational mechanism provides the base to account for the observed non-Keplerian galactic rotation as a function of the distance from the galactic center and can appro- priately settle the galactic gravitational dynamics without the need of **dark** matter. The velocity field of the QS (gravitational field) that rules the galactic gravitational dynamics of a system of orbiting bodies certainly is much different from the Keplerian velocity field of the QS created by a single, compact, homogeneous, spherically sym- metric and nearly static system of matter. The galactic rotation takes place within the self-consistent velocity field of the QS (gravitational field) generated by the whole ensemble of orbiting stars. Hence, from the viewpoint of the above analysis, it is not at all strange that the rotation of galaxies be a non-Keplerian one. Also in the central part of the galaxy the velocity field of the QS may be much reduced, analogously as shown by Equation (16) for the binary and becomes even more evident in the case of loops of four or more bodies [Figure 5]. Even the possibility of re- trograde rotation of the inner part of a galaxy, as observed in some galaxies, is predicted. Hence, from the view- point of space dynamics, the observed non-Keplerian rotation of the galaxies is in no way a surprise.

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It is worth noting that the experiments for the scanning the celestial sphere by the pulsed plasma generator [22,31] to detect some directions in space, where **energy** is more than the average value, are the final stage in the determi- nation of the direction of the new force. This direction was determined before by the using the high field mag- nets [20-22,32-35], by the investigations of the rate of β - decay for a number of radioactive elements [20-22,24, 25,36], by the investigations with high precision gra- vimeters [20-22,37], and by plasma generators of other types [20-22,38].

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Now the prevailing practice has been to consider ρ in Equation (6) as encompassing **dark** **energy**, matter and radiation. Since the Hubble parameter H has to decrease over time in Equation (6) for the age of the universe to increase, it is sufficient for matter and radiation density to decrease while **dark** **energy** density can remain con- stant. Indeed, so far the astronomical observational constraints are consistent with a time-independent cosmo- logical constant. However, it may be argued that since 8π

D ) moc 2 = 0.0450849 mc 2 and that this **energy** is the ordinary measurable **energy** density of the quantum particle. This means that almost 95.5% of the **energy** of the cosmos is **dark** **energy** which by quantum par- ticle-wave duality is the absolute value of the **energy** of the quantum wave and is proportional to the square of the cur- vature of the curled dimension of spacetime namely 26 k 4 2 where k 2 5 and 5 is Hardy’s probability of quantum entanglement. Because of the quantum wave collapse on measurement this **energy** cannot be measured using our current technologies. The same result is obtained by involving all the 17 Stein spaces corresponding to 17 types of the wallpaper groups as well as the 230 11 219 three dimensional crystallographic group which gives the number of the first level of massless particle-like states in Heterotic string theory. All these diverse subjects find here a unified view point leading to the same result regarding the missing **dark** **energy** of the universe, which turned out to by syn- onymous with the absolute value of the **energy** of the Hawking-Hartle quantum wave solution of Wheeler-DeWitt equa- tion while ordinary **energy** is the **energy** of the quantum particle into which the Hawking-Hartle wave collapse at cos- mic **energy** measurement. In other words it is in the very act of measurement which causes our inability to measure the “**Dark** **energy** of the quantum wave” in any direct way. The only hope if any to detect **dark** **energy** and utilize it in nu- clear reactors is future development of sophisticated quantum wave non-demolition measurement instruments.

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