A comparison is made between the curves displayed in Figure 1 and Figure 2 with the result of the IR theory in Figure 3, but in logarithmic representation.
Moreover, in a subsequent contribution a geometric analog to Hardy’s probability function will be presented with significance, besides geometry, to crystallography, electrostatic, botany, coding theory and other disciplines.
Infinite continued fraction representations of φ5 and its inverse yield
We notice that L5 = 11 is a Lucas number. It results from the definition
The Ln number series was named after the French mathematician François Édouard Anatole Lucas (1842-1891).
Figure 3. Fisher transform  of Hardy’s probability function (blue curve) , compared with the more convincing, but nevertheless slightly asymmetric redshift representation of the energy density according to the information relativity theory (red curve)  (see also Figure 4). The green curve is a further adaption in direction of the red curve.
Many researchers have found the golden ratio to be important by trying to uncover secrets of the universe and its mass respectively energy distribution  - . In the next Chapters this significance will be demonstrated.
If one deals with exponential functions, the author has learnt from Sherbon  that the Lambert W-function can serve as an analog to the golden ratio for exponential functions. Again, it is shown a sort of reciprocity, if one writes the relation as follows
The quoted publication of Sherbon  is highly interesting for all researchers, who want to learn more about of the fundamental nature of Sommerfeld’s fine-structure constant.
6. Golden Ratio, Archimedes’ Constant and Sommerfeld’s Fine-Structure Constant
Often you wonder why our world is what it is. Fundamental numbers such as the golden ration φ, the circle constant π as well as Sommerfeld’s fine-structure constant α and their obvious similarities play an important role. Some approximations should illustrate it. So one can connect the number π with the reciprocal of the Sommerfeld constant   or with Hardy’s quantum probability φ5 .
These approximations, believed to be accidental by others, may find now and then application in the following discussions.
7. Golden Mean and Madelung Constant for a Rocksalt-Type 2D-Lattice
In a previous publication the author drew attention to the numerical similarity between the golden mean and the Madelung constant  for a two-dimensional rocksalt-type lattice .
The Madelung constant α2D was iteratively determined with very high precision by Triebl  giving
The α2D value is very close to the quotient of two Fibonacci numbers, 21/13 = 1.615385, ... and can be adapted to φ−1 by only slight distortion of the square net along the two dimensions or by involving the third one to allow a quite flat curvature .
The difference to the inverse of the golden mean φ−1 is only marginal and gives
This almost numerical equality was applied to Villata’s lattice universe  consisting of matter and antimatter with gravitational charges of opposite sign at positions of a 2D rocksalt-type lattice. Then the ratio of repulsive contribution to the attractive one gives
This relation leads the author to a proposal for the golden mean based calculation of the mass constituents of the universe  independent from other approaches and with only marginal differences to such results  .
A certain reciprocity property may be suggested for a two dimensional rocksalt-type matter-antimatter lattice independent of whether it is a real possibility. The question is whether a conditionally flat lattice universe with a Madelung constant of φ−1 would guarantee sufficient stability to exist over long periods of time.
8. Golden Mean as Dimension of Empty Set and Zero Set
El Naschie’s E-infinity (ε∞) theory , not commonly known or accepted by physicists, originates from a fractal Cantorian set theory  as a number-theoretical route of physics for explaining the dualism between particles and waves that can help solving cosmological mysteries such as dark matter and dark energy . The quantum particle PQ is symbolized by the bi-dimension of the zero set, while the guiding wave WQ surrounding the quantum particle is given by the bi-dimension of the empty set according to
where n is the Urysohn-Menger topological dimension   and
represents the Hausdorff dimension , where φ is the golden mean as defined before.
It results for PQ (32)
respectively for WQ (33)
By using these dimensions a probabilistic quantum entanglement calculation   with velocity restriction delivers effective quantum gravity formulas for the cosmological mass (energy) constituents of baryonic matter eM, dark matter eDM, entire dark constituents eED, and pure dark energy ePD as follows
Recasting the matter amounts into a suitable form,
a reciprocity relation was confirmed between eM and eDM giving a persuasive equation for the pure dark energy 
Such quantum entanglement based coincidence means that the constituents of the cosmos should not be considered independent of each other, which was confirmed by the IR theory.
Importantly, if one compares the results given here with the following ones of the information relativity (IR) theory, then El Naschie’s set theoretical approach is restricted to , whereas the more general IR theory delivers results for the recession velocity in the hole range 0 ≤ β ≤ 1 (c is the speed of light).
9. Information Relativity Theory of Suleiman and Golden Mean
Many formal explanations or physical constructs that bothered long time the world of physics are overcome by the new exciting Information Relativity theory, developed by Suleiman  . It is not the intention of the author to keep the reader away from studying this theory in detail for himself. Therefore, only a sparse introduction was given. Suleiman found an overlooked flaw in Newton’s physics and corrected physical processes for time displacements between observer and moving bodies. Transformations for time duration, length, mass density as well as energy density were applied to a whole bunch of physical phenomena, which could be explained now in simple and beautiful clarity. For instance, Suleiman derived for the matter energy density eM of a moving body with velocity v and rest density ρo
where is the recession velocity respectively .
The matter energy density reached its maximum at a recession velocity of . Replacement of this special value in Equation (41) gives
Remembering, φ5 represents Hardy’s quantum probability at the maximum. This result was commented by the author in a publication before mentioned .
Suleiman aptly characterized the behavior at the critical point βcr = φ as phase criticality at cosmic scale . The dark matter density transforms as
The relations are depicted in Figure 4. If one calculates the energy density amounts (ratios) of matter and dark matter contributions at this point, one gets again a golden mean representation like Russian dolls nesting
The difference gives .
The case, where according to the Information Relativity theory of Suleiman   just at the recession velocity of β = 1/3 the matter and the dark matter density will be the same, delivers for the density amount the reciprocal of another Lucas number, namely L6 = 18 (see Chapter 5 and Figure 4)
Figure 4. Golden mean dominance in the evolvement of the energy density with the recession velocity according to the information relativity theory of Suleiman  . Φ5 represents Hardy’s maximum quantum probability. Red curve: matter energy density, black curve: dark matter energy density, green curve: energy density sum.
Furthermore, if the recession velocity at βeq = 1/3 is mirrored at βcr = φ, it resulted βmir = 0.9027. In its vicinity at β = 0.89297 the matter energy density would be exactly φ5/2 = 0.04508497∙∙∙ respectively the dual dark component 0.7523∙∙∙ ≈ 0.763932∙∙∙ = 2φ2 (Figure 1).
It approximately indicated a situation that is elaborated for by means of the fractal set theory summarized before in Chapter 8.
In Figure 5 the energy densities were illustrated via the redshift, which reads as z = β/(1 − β). It is suggested to fit the only slightly asymmetric red curve with the aid of a Cauchy function of exotic non-integer order on the basis of the golden mean introduced by the present author some time ago .
Suleiman’s IR theory validates once more the importance of the golden mean in solving physical phenomena. Reciprocity is given by the proposed duality between particle and wave.
As was demonstrated by Suleiman (Figure 6), an increase of the redshift z caused the matter density of the travelling corpuscular particle successively to diminish, while energy is transformed into the wave-like dark component and vice versa . This supports elegantly the concerns of the work here presented.
10. Mystery of the Electron and Golden Mean
The electron, considered as center of compacted information, still keeps its secrets, but not for long. Whereas the hydrogen atom problem was just solved by Suleiman  without any assumption of quantization of the electron’s orbits and using IR transformation of length as
Figure 5. Energy densities related to the red shift z = β/(1 − β) (logarithmic scale) according to Suleiman . Colored curves have the same meaning as in Figure 1. Now the coincidence point is at z = 1/2.
Figure 6. Suleiman’s famous reciprocal (complementary) duality between matter density and dark matter one. Logarithmic scale, red arrow at z = 1/2, blue arrow at z = φ−1 = 1.61803398∙∙∙ (see also Equation (39)). Applied matter density transformations in terms of the redshift  : , .
other constructs like the electron spin  or the measured anomalous gyromagnetic factor of the electron may be solved fractal-deterministic, supported by application of IR transformations. Also the fractal nature of electron pairing in superconductors should be reassessed this way.
The ge factor of the electron, conceived as a classical charged particle, is determined by the relation
where is the observable magnetic moment, is the Bohr magneton, and is the spin of the electron, e respectively m are charge respectively mass of the electron, and is the reduced Planck constant.
However, the spin as half-integer quantum number of the electron was introduced without any physical justification . Very recently, a first attempt has been undertaken by He et al.  to connect the golden mean with the ad hoc spin-1/2 construct. Such golden mean approach may be the result of dark halo movement around the stretched electron in the sense of the Information Relativity theory.
Remembering that the “anomalous part” of the gyromagnetic factor was recently given by a simple and solely golden mean representation with sufficient accuracy 
while a series expansion yields a value more accurate up to the tenth decimal place
This result may be compared to the high accuracy of the best known experimental value for ge determined as one-electron cyclotron transition for an electron trapped in an electrostatic quadrupol potential (Penning trap) 
In a subsequently presented seminal idea of He et al.  the spin quantum number s in the spin momentum term was replaced by a quantized golden mean giving
with the value as He-Chengtian average   to adapt the full accuracy of ge/2. One can calculate by a very simple formula, which resembles the representation for φ (Equation 7(a)) and delivers exactly the given value
and for the IR corrected value of ge = 2.0023190900 (see Chapter 11)
Using this formula, the gyromagnetic factor resulted simply as function of α/π 
where vK is the Klizing speed and c the speed of light.
The latest released values of Sommerfeld’s fine-structure constant α  respectively its reciprocal value is quoted according to NIST  being
An approximation using the α/π series expansion yielded
and further (60)
The deviation between this a bit underdetermined value and the newly relativistic corrected one is in the seventh decimal place as well as the corresponding g-factors. It is hoped that precisely re-determined experimental factors may lessen these deviations further.
One may ask, what the infinitely continued fraction representation of would result in. We can write similar to the golden mean 
The calculation with
. Indeed, the number 266.
With an assumed involvement of the fifth power of the golden mean in the continued fraction representation one may speak of a nested golden mean representation. This result supports once more the fractal-deterministic approach chosen for the physics of the electron beyond the ad hoc half-spin assumption, characterizing the electron as complexly nested resonating entity. An alternating approach for the gyromagnetic factor is given in Appendix II.
11. Alteration of Fundamental Constants
The calculation of the electron’s gyromagnetic factor is the prime example for application of the QED. However, a cascade of Feynman diagram calculation must be done to determine the pre-factors of systematic perturbative expansions in powers of α/π  . It is not so long ago that Gabrielse et al. asked “whether it is likely that other adjustments of the QED theory will shift the α that is determined from the electron g?” and answered “we hope not” . Nevertheless, the QED theory should be corrected for IR transformations to iron out some flaws, and the author suggests a considerable simplification of QED calculations as a renewed successful tool, altering the inferred α constant, and related to it, the charge of the electron. Also importantly, the experimental value of g must be corrected, too. The applied relativistic shift of the cyclotron frequency (ωc = cyclotron frequency, B = magnetic field strength in Tesla) was performed using the familiar relativistic factor γ. However, γ should be replaced by the mass transformation according to the IR theory 
For the classical case the corrected frequency ωc is
where the energy En of the nth quantum state of a harmonic cyclotron oscillator is given as
The classical relativistic shift δ in the cyclotron frequency per energy quantum was approximated by the level spacing of the harmonic oscillator giving 
For the IR theory one yields a much greater and positive shift because the cyclotron frequency yields now
The relativistic shift δ is approximated by
The gyromagnetic factor as ge/2 can be determined from the observed eigenfrequencies 
where the values are marginally modified with respect to the free-space values, is the anomalous frequency, and the spin frequency is , is the magnetron frequency, using the dip frequency ωz in Hz . For the experimentally chosen cyclotron frequency of νc = 149.2 GHz, the classical relativistic shift is calculated to be δ = −2π∙182.1 Hz compared to the IR corrected one giving δ = +2π∙14.78 Mhz. One can estimate that ge becomes noticeably smaller by a factor of approximately 1.0001, meaning a correction of ge from the seventh decimal point downwards to about ?
Now the scientific community is waiting for a most precise redetermination of the g-factor as well as the related Sommerfeld constant by experts  . The aforementioned Zitterbewegung approach of Niehaus  should be revised by that author himself. The comment of the present author may have fulfilled its true purpose, if research on this topic proceeds well with application of the IR theory .
12. Fractal Superconductivity
Nature presents much more relationships to keep in mind, where the golden mean is involved, and superconductivity is no exception. However, we must reassess the theory considering the dark matter surrounding the moving electrons, which dive into the dark after marriage, or in other words, become superconducting under special conditions. Before a golden ratio in the spin dynamics of the quasi-one-dimensional Ising ferromagnet CoNb2O8 was experimentally verified next a phase critical point by Coldea et al. , the present author suggested linking the optimum hole doping of high-Tc superconductors with the golden mean in the form of Hardy’s maximum quantum probability of two particles 
Obviously, this optimum is again near a quantum critical point in the phase diagram. In addition, the relation of the Fermi speed to the Klitzing speed comes out as
Both relations document the fractal nature of the electronic response in superconductors. It was suggested recently that the same is true for conventional superconductors . Also Prester had reported before about evidence of a fractal dissipative regime in high-Tc superconductors .
Interestingly, some time ago the present author connected the optimum transition temperature Tco of high-Tc superconductors with a Fibonacci number fi, proportional to a domain width, by the relation Tco = 12,000/fi . One yields the integer number 45, again as a product of solely Fibonacci numbers, when dividing this number by the number (see Chapter 10).
Quantum entanglement of two moving electrons may be influenced by local interaction of their interwoven dark matter surroundings, quoting the cogwheel picture of Suleiman . What happens, if two stretched electrons locally interact to become superconducting? May such particle stretching in the end lead to a double-helically wounded wavy entity, which escapes in the dark? Nature is known to copy itself again and again. So the double-helix approach is not only beautiful, if we quote Gauthier’s proposed entangled double-helix superluminal photon model . Therefore, a model calculation for superconducting electron strands is suggested based on this idea, addressing the problem of left and right (mirrored) strands as well as objections against an apparent superluminal velocity. The double helix strand in nature is a special fabric of duality.
13. Evolution of Life
Nature repeatedly applied its building plans, based on the hierarchical golden number system, from largest to smallest dimensions, from the cosmos to the smallest living cell. Inasmuch the golden ratio is involved, reciprocity is considered as a vital element of life. Recently, thoughts to the link between cosmology and biology are impressively formulated . Self-similarity as an element of chaos is intimately connected with self-organization of life producing compacted information and consciousness. However, visions about life such as England’s provocative approach of dissipation-driven adaption  or Pitkänen’s formulation  suffer from not considering duality of particle and wave or reciprocity of matter and dark matter  and should be adapted to the new physics.
The evolution of life may take place similar to the statistical bootstrap model of colliding heavy particles, so the Hagedorn temperature TH comes into play. I quote the formulation of Rafelski and Ericson  to explain this: “When a drop of particles and resonances is compressed to the ‘natural volume’, it becomes another more massive resonance. This process then repeats, creating heavier resonances, which in return consist of resonances, and so on”.
This nesting looks like a Menger sponge  . The process could explain the evolution of life with TH around ambient temperatures.
With respect to the entire energy density of φ2 at the phase critical point β = φ one may suggest formulating the Hagedorn temperature TH proportional to the squared golden mean φ, where is formally the tension of a string.
It remains to interpret the not liked string tension by a more appropriate thermodynamic quantity at ambient equilibrium conditions.
The duality between a compacted entity and its surrounding in general as well as the duality between a moving particle or body and the accompanying wave or reciprocity between matter and dark matter is the very spice of life. This was proven by the beautiful information relativity theory of Suleiman. Reciprocity is impressively formulated by the words of Wolfgang Pauli: “God made the bulk; surfaces were invented by the devil” (quoted from ). As a consequence of the IR theory some natural constants such as the gyromagnetic factor of the electron, Sommerfeld’s fine-structure constant as well as the charge of the electron are proposed to be marginally altered. The interpretation of superconductivity is influenced by the IR approach, too. Also the evolution of life may find a new basis. If we have fully understand the new IR physics with its particle-wave reciprocal dualism and intrinsic harmony, then we can shape our environment more effectively to achieve a balance between plants, animals and human beings, which enables a long-term life for all of us on earth. In this sense the golden mean should provide more beauty than chaos.
The author appreciated the critical reading of the manuscript by Prof. Ramzi Suleiman, University of Haifa, and the Triangle Research and Development Center (TCRD), who enriched the scientific community with his famous information relativity theory. The author is also grateful for the constructive criticism of a very creative reviewer.
About the Meaning of the Terms Reciprocity, Duality and Complementarity
These terms, omnipresent in many disciplines of science (physics, mathematics, philosophy, music, economy, etc.) can certainly have different meanings, even if they have something in common.
In mathematics, a reciprocal of a number is its multiplicative inverse, but an inverse is not necessarily a reciprocal. Reciprocity in the amounts of matter and dark matter is formulated according to Equation (39) of the main text. Important for mathematics and physics, a reciprocal vector system can be created by Fourier transformation. However, reciprocity in physics may have a more general meaning when describing a mutual dependence or influence.
Duality in mathematics can be demonstrated on platonic solids, also familiar for a crystallographer. The convex hull of the center points of each face of a starting polyhedron results in a dual polyhedron such that, for instance, the cube and the octahedron form a dual pair, but the tetrahedron is self-dual. In physics, the most prominent example for duality is that between matter and piloting wave in the sense of the De Brogly-Bohm approach . According to the IR theory of Suleiman  the relation between matter density and the dark matter surrounding may be quantified as “reciprocal duality”, where an amount of matter is transformed into an equal amount of dark matter depending of the recession velocity respectively redshift of a moving body (Figure 6).
Finally, the concept of complementarity in quantum physics has been formulated and coined by Bohr in his Como lecture of 1927, describing the familiar case of reciprocal uncertainty between position and momentum of an electron as conjugate variables . It means that it is hardly possible to know simultaneously with an arbitrary accuracy the outcomes of these variables. Another example of conjugate variables is the magnetic field strength in comparison to the electric one. An elaborated logical analysis of complementarity has been given by Bedau and Oppenheimer . In his late years Bohr was interested in philosophical aspects of complementarity as given in the Yin and Yang conjugate principle of the ancient Tao, and on his gravestone the Taoist symbol is engraved.
In mathematics, a number and the complement to a number add up to a whole of some amount. If one performs the reciprocal of these numbers and renormalize the resulting values, then complement and primal number change their values . In this way one may speak also of reciprocity when dealing with matter density and the dual dark matter density according to Figure 6.
Another approach for the gyromagnetic factor used the fifth power of with the value . An approximation is the inverse Lucas number L5 = 11 as combination of two inversely related irrational numbers (see Equations (13) to (15))
However, physically more convenient is the expression
where vK is the Klizing speed and c the speed of light. This term keeps no dimension, as required. If we are working with a speed, according to the IR theory the information offset has to be corrected. The speed transforms as
combining the length transformation with the time transformation , where β = v/c is the recession speed . Surprisingly, an additional term is needed to give more accuracy and the following simple formula