In this paper Nottale’s acclaimed scale relativity theory is given a transfinite Occam’s razor leading to exact predictions of the missing dark energy [1,2] of the cosmos. It is found that 95.4915% of the energy in the cosmos according to Einstein’s prediction must be dark energy or not there at all. This percentage is in almost complete agreement with actual measurements.

Abstract:
Building upon the pioneering work of J. Bell [1] and an incredible result due to L. Hardy [2] it was shown that the probability of quantum entanglement of two particles is a maximum of 9.0169945 percent [2]. This happens to be exactly the golden mean to the power of five (?5) [3-7]. Although it has gone largely unnoticed for a long time, this result was essentially established independently in a much wider context by the present author almost two decades ago [3-6]. The present work gives two fundamentally different derivations of Hardy’s beautiful result leading to precisely the same general conclusion, namely that by virtue of the zero measure of the underlying Cantorian-fractal spacetime geometry the notion of spatial separability in quantum physics is devoid of any meaning [7]. The first derivation is purely logical and uses a probability theory which combines the discrete with the continuum. The second derivation is purely geometrical and topological using the fundamental equations of a theory developed by the author and his collaborators frequently referred to as E-infinity or Cantorian spacetime theory [3-7].

Abstract:
By religiously adhering to physics in spacetime and taking the final verdict of N.D. Mermin’s Ithaca interpretation of quantum mechanics seriously, Hardy’s paradox is completely resolved. It is then concluded that logical and mathematically consistent physical theories must be put in spacetime related formalism such as noncommutative geometry and E-infinity theory to avoid quantum paradoxes. At a minimum, we should employ the philosophy behind consistent quantum interpretation such as that of the famous Ithaca interpretation of D. Mermin.

Pure
gravity in eight dimensional super space possesses twenty degrees of freedom.
Gauging these generalized coordinates in terms of a single self entangled Hardy
type quantum particle we retrieve one hundred and thirty seven isometries
corresponding to the well known value of the inverse electromagnetic fine
structure constant. The deep implications of this thought provoking result are
discussed in a general physical context.

Abstract:
In a short, neat and credible analysis, it is established that time is a fractal effect of the Cantor set-like
topology of micro spacetime. This effect as well as the ordinary cosmic energy
density of the universe is shown to be a direct consequence of Hardy’s
probability of quantum entanglement. Finally and as a general conclusion, we point
out the importance of understanding the fractal origin of time as well as
spacetime for resolving certain types of paradoxes arising in quantum
information science.

Abstract:
An instructive analogy between the deformation of a pinched elastic cylindrical shell and the anti-gravity behind accelerated cosmic expansion is established. Subsequently the entire model is interpreted in terms of a hyperbolic fractal Rindler space-time leading to the same robust results regarding real energy and dark energy being 4.5% and 95.5% respectively in full agreement with all recent cosmological measurements.

Abstract:
Starting from Witten’s eleven dimensional M-theory, the present work develops in an analogous way a corresponding dimensional fractal version where ？. Subsequently, the new fractal formalism is utilized to determine the measured ordinary energy density of the cosmos which turns out to be intimately linked to the new theory’s fractal dimension via non-integer irrational Lorentzian-like factor: where is Hardy’s probability of quantum entanglement. Consequently, the energy density is found from a limiting classical kinetic energy to be Here, is ‘tHooft’s renormalon of dimensional regularization. The immediate logical, mathematical and physical implication of this result is that the dark energy density of the cosmos must be in astounding agreement with cosmic measurements and observations.

Abstract:
The work gives a natural explanation for the ordinary and dark energy density of the cosmos based on conventional quantum mechanical considerations which dates back as far as the early days of the quantum theory and specifically the work of Max Planck who seems to be the first to propose the possibility of a half quanta corresponding to the ground state, i.e. the energy zero point of the vacuum. Combining these old insights with the relatively new results of Hardy’s quantum entanglement and Witten’s topological quantum field theory as well as the fractal version of M-theory, we find a remarkably simple general theory for dark energy and the Casimir effect.

Abstract:
The quintessence of hyperbolic geometry is transferred to a transfinite Cantorian-fractal setting in the present work. Starting from the building block of E-infinity Cantorian spacetime theory, namely a quantum pre-particle zero set as a core and a quantum pre-wave empty set as cobordism or surface of the core, we connect the interaction of two such self similar units to a compact four dimensional manifold and a corresponding holographic boundary akin to the compactified Klein modular curve with SL(2,7) symmetry. Based on this model in conjunction with a 4D compact hy- perbolic manifold M(4) and the associated general theory, the so obtained ordinary and dark en- ergy density of the cosmos is found to be in complete agreement with previous analysis as well as cosmic measurements and observations such as WMAP and Type 1a supernova.

Abstract:
Einstein’s energy mass formula is shown to consist of two basically
quantum components E(O) = mc^{2}/22
and E(D) = mc^{2}(21/22).
We give various arguments and derivations to expose the quantum entanglement
physics residing inside a deceptively simple expression E = mc^{2}. The
true surprising aspect of the present work is however the realization that all
the involved “physics” in deriving the new quantum dissection of Einstein’s
famous formula of special relativity is actually a pure mathematical necessity
anchored in the phenomena of volume concentration of convex manifold in high
dimensional quasi Banach spaces. Only an endophysical experiment encompassing
the entire universe such as COBE, WMAP, Planck and supernova analysis could
have discovered dark energy and our present dissection of Einstein’s marvelous
formula.