Reciprocity as an Ever-Present Dual Property of Everything

Abstract

Reciprocity may be understood as relation of action and reaction in the sense of Hegel's philosophical definition. In this contribution, a more mathematical than philosophical reflection about reciprocity as an ever-present dual property of everything was given. As a crystallographer, the author is familiar with the action of Fourier transforms and the relation between a crystal lattice and its reciprocal lattice. Importantly, integration of the function √1 − − 2 respectively its reciprocal leads to the area enclosed by a circle in contrast to its circumference or boundary. A generalization of the reciprocity term was stimulated by results of the famous Information Relativity theory of Suleiman with its proven physical manifestation of matter-wave duality, compared to the set theoretical E-Infinity theory developed by El Naschie, where the zero set represents the pre-quantum particle, and the pre-quantum wave is assigned to the empty set boundary surrounding the pre-particle. Expectedly, the most irrational number φ = (√5-1)/2 of the golden mean is involved in these thougths, because this number is intimately connected with its inverse. An important role plays further Hardy's quantum entanglement probability as the fifth power of φ and its connection to the dark matter. Remembering, the eleven dimensions in Wittens M-theory may be decomposed into 11 = φ-5-φ 5. If we have the wrong look at our world, we will completely fail.

Full text

1
Reciprocity as an Ever-Present Dual Property of Everything
Hans Hermann Otto
Materials Science and Crystallography, Clausthal University of Technology,
Clausthal-Zellerfeld, Lower Saxony, Germany
E-mail: hhermann.otto@web.de
Abstract
Reciprocity may be understood as relation of action and reaction in the sense of Hegel’s
philosophical definition. Quoting Kant, freedom and ethical necessities are reciprocally
limited. In this contribution, a more mathematical than philosophical reflection about
reciprocity as an ever-present dual property of everything was given. As a crystallographer,
the author is familiar with the action of Fourier transforms and the relation between a crystal
lattice and its reciprocal lattice, already pointing to the duality between particles and waves. A
generalization of the reciprocity term was stimulated by results of the famous Information
Relativity (IR) theory of Suleiman with its proven physical manifestation of matter‒wave
duality, compared to the set- theoretical E-Infinity theory developed by El Naschie, where the
zero set represents the pre-quantum particle, and the pre-quantum wave is assigned to the
empty set boundary surrounding the pre-particle. Expectedly, the most irrational number φ =
(√5 – 1)/2 of the golden mean is involved in these thoughts, because this number is intimately
connected with its inverse. An important role plays further Hardy’s maximum quantum
entanglement probability as the fifth power of φ and its connection to the dark matter.
Remembering, the eleven dimensions in Witten’s M-theory may be decomposed into the
Lucas number L
5
=
11 = φ
-5
– φ
5
. Reciprocity is indeed omnipresent in our world as piloting
waves that accompany all observable earthen and cosmic matter. As a side effect of the IR
theory some fundamental constants such as the gyromagnetic factor of the electron,
Sommerfeld’s fine-structure constant as well as the charge of the electron must be marginally
changed caused by altered relativistic corrections. Consequences also arise for our vision
about the evolution of life and consciousness.
Keywords
Reciprocity, Reciprocal Lattice, Fourier Transform, Archimedes’ Constant, Matter–Wave
Duality, Pilot Wave, Golden Mean, E-Infinity Theory, Information Relativity Theory,
Gyromagnetic Factor, Fine-Structure Constant, Quantum Entanglement, Dark Matter and
Cosmos, Superconductivity.

2
1. Introduction
Recently, the author reported on a reciprocity relation between the mass constituents of the
universe and Hardy’s maximum quantum entanglement probability of two quantum particles
[1][2][3][4]. Hardy’s maximum probability was before connected with the dark energy
question, using a fractal Cantorian set theory developed by El Naschie (E-infinity theory) [5]
and further elaborated by Marec-Crnjac [6]. However, in the last years a step was made in
direction of a unification of physical theories with the Information Relativity theory of
Suleiman [7][8]. Whereas great philosophers understood reciprocity as relation between
action and reaction (Hegel [9]) or as connection between freedom and ethical necessities
(Kant [10]), in this contribution a more mathematical as well as physical view on reciprocity
was given. Because the term reciprocity has something in common with the concepts of
duality and complementarity, their different meaning was shortly addressed in the Appendix I.
The following chapter deals with the Fourier transform and the reciprocal lattice of
crystallography that describes results of diffraction on crystals as a special kind of particle‒
wave duality. The reciprocity relation between boundary and enclosed area of a circle is
shortly treated, followed by the reciprocity property of the golden ratio in a separate chapter.
Then the fractal Cantorian set theory of El Naschie [5] and Marek-Crnjac was appreciated
[6]. The final chapter illuminates consequences of the famous scale-free Information
Relativity theory of Suleiman with its proven particle-wave duality, which explains for
instance the double-slit experiment [11] and suggests dark energy as piloting waves of
moving matter, touching up the old De-Broglie-Bohm theory [12][13][14]. The work was
supplemented by thoughts of the author about further consequences of the IR theory. It seems
that the secrets of the electron, spin, tremor and the anomalous gyromagnetic factor (g
e
) as
well as electron pairing in superconductors [15][16], can simply be explained or corrected
with this new theoretical approach to demystify the physics even more. The inferred
Sommerfeld fine-structure constant α may be altered as well as the experimental g
e
value,
concerning the relativistic shift correction. Related to it, the charge of the electron has to be
changed, too. In addition, a corrected version of the Niehaus EZBW (extended Zitterbewegung
may serve as a probabilistic model for the electron [17]. Consequences also arise for our
vision about the evolution of life and consciousness. Obviously the Hagedorn temperature
scales with the second power of the golden ratio. The conjecture is that all things are
interwoven with the reciprocal, where connection with the golden ratio indicates some system
stability.
The particle – wave duality is also essential in the quantum information theory, where the unit
of information is given by the quantum bit (qubit) coined by Schumacher [18], which exhibits
the aspects of particle localization (counting) and wave interference to represent a signal with
high fidelity [19]. Such two-state quantum system can be represented by the superposition
principle:  





, where z
1
and z
2
are complex numbers,
and 





 [19]. However, the quantum information aspect cannot be a main topic of
this limited contribution.

3
2. Fourier Transform and Reciprocal Space
The diffraction of X-rays of sufficient wavelength on crystals leads to a characteristic
diffraction pattern, where the electron density of a crystal structure is transformed to a
reciprocal lattice, weighted with intensities  

 where F(h) represents structure
amplitudes (Fourier coefficients) according to the transform



!"
#$% (1)
The vectors r = xa + yb + zc and h = ha
*
+ kb
*
+ lc* are position vectors of the crystal lattice
respectively the reciprocal lattice of a diffraction pattern. The lattice parameters a, b, c
respectively a
*
, b
*
, c
*
are the repeat units along the lattice axes. The reader may follow a
reciprocal lattice exercise in more detail, given as a lecture in [20].
The inverse Fourier transform, using the structure amplitudes as coefficients, delivers the
electron density of the crystal

&


'!"
#$
&
 (2)
where $
&
is the reciprocal volume.
In this way, a crystal structure can be completely solved by means of a diffraction experiment.
However, because only intensities I(h) are measured, phases of the structure amplitudes are
lost and must be recovered by elaborated crystallographic methods.
Applying a transform such as the Fourier transform one goes from the object space to the
image space or reciprocal space. If the original variable would be the time, then the
transformed reciprocal variable would be a frequency, exemplified by the Laplace transform
of electrical decay processes. Going from the object space to the reciprocal space one may
heretically ask, what could be the Fourier transform of the entire cosmos, delivering an
inverse universe or whatever else?
3. Relationship between Boundary and Enclosed Area of a Circle
Quoting the references [21] [22], the area enclosed by a circle of radius 1 yields
() *+,-


.
#- (3)
where π is Archimedes’ constant, the well-known circle constant. One obtains the
circumference C by using the reciprocal of the integrand
/  ) *

+'0
1

.
#-% (4)
This connection between the boundary and the enclosed area is of fundamental importance. It
may be thought of as a geometrical analog to the more general matter‒wave duality that is
being treated below. Besides, Archimedes’ constant π and the golden ratio φ as the fractal
numerical dominators of our existence show an intimate numerical connection, and we may
ask, in which manner nature makes use of this [3] [23]. An elegant continued fraction
representation of π was given by Lange [24].

4
4. Mushkolaj’s Reciprocal Transition Temperatures
In 2014 Mushkolaj [25] has presented a theory of critical temperatures for phase transitions
such as superconductivity, using an elastic atomic collisions model as well as an elastic spring
model. He found two inverse T
c
functions of the form
2
3
45566789

:9

:;-


'
2
3
<=67>89

:9

:;-

(5a,b)
where M
1
, M
2
are the colliding or by a spring connected masses, and ∆x is the distance
between atoms respectively the spring stretch length in Hooke’s region.
If we associate the atomic collision model with particles and the spring model with waves, we
are faced with a reciprocity relation between the two excitation variants and again with the
duality between particles and waves in a special form.
5. The Golden Mean Beauty and Its Intrinsic Reciprocity Property
The golden mean or golden ratio φ is an omnipresent number in nature, found in the
architecture of living creatures as well as human buildings, music, finance, medicine,
philosophy, and of course in physics and mathematics [26][27]. It is the most irrational
number known and a number-theoretical chameleon with a self-similarity property. On the
other hand, its infinite continued fraction representation is the simplest of all and is
represented by [28]
(6)
?+@,
 
 
 
A%

It impressively underlines the fractal character of this number. Most obviously, the golden
mean mediates stability of a system, because only ‘particles’ as the center of gravity of
vibrations with most irrational winding survive. Important relations involving φ are
summarized below. However, to prevent confusion, in textbooks of mathematics the
reciprocal value for φ is frequently used.

?
+B'

 C%DECFFGEEH?
'
? 
+BI

%DECFFGEEH(7a, 7b)

?

,?C%FEGDDCH
?
'
?%DECFFGEEH
(8a, 8b)
?


'
?
'
J@ (9a)
or equivalently ?
'

'
?

J@ (9b)
?
B

'K
IK
?

C%CGCDGG*FH(10)

5
K
L


B

?

 (11)
?
M
?


Hardy’s maximum quantum entanglement probability of two quantum particles [1][2] exactly
equals the fifth power of φ (Figure 1). This asymmetric probability distribution function with
p
τ
as entanglement variable, running from not entangled states to completely entangled ones,
is given by
N<
O


'P
Q
IP
Q
(13)
This function, displayed in Figure 1, turns out to be a central topic of the Information
Relativity theory of Suleiman [7] [8] by mapping the transformation of his relative energy
density (see Chapter 9 and compare the red curves in the Figures 3 and 4).
The probability function according to Equation 13 can be recast to an adapted distribution by
means of a varied Fisher transformation (Figure 2) [29]

+B

RST
IP
Q
'P
Q
U+@:V=WV7X<
O
 (14)
where the pre-factor was chosen as V
+B

?


. Then one gets for Y
YZ
[\]T
^
_
U'
[\]T
^
_
UI
`

abc
d
e

'
 (15)
Figure 1. Hardy’s quantum probability P for two particles [1], where p
τ
can be thought of
as entanglement variable, running from not entangled states to completely entangled ones.
Compare it with the equivalent red curve in Figure 3. [3][7]
A comparison is made between the curves displayed in the Figures 1 and 2 with the result of
the IR theory in Figure 3, but in logarithmic representation.

6
Figure 2. Fisher transform of Hardy’s probability function, representing an asymmetrical
distribution with its maximum now at z = φ
-1
(see Equation 15 and Figure 3).
Figure 3. Fisher transform [29] of Hardy’s probability function (blue curve) [1], compared
with the more convincing, but nevertheless slightly asymmetric redshift representation of the
energy density according to the information relativity theory (red curve) [8] (see also Figure
4). The green curve is a further adaption in direction of the red curve.
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 φ
5
and its inverse yield
(16)
?
@
 
 
 
A%


7
(17)
?
,@
?
@
 
 
 
A%
We notice that L
5
= 11 is a Lucas number. It results from the definition
f
g
?
'g
,?
g
(18)
The L
n
number series was named after the French mathematician François Édouard Anatole
Lucas (1842-1891).
Many researchers have found the golden ratio to be important by trying to uncover secrets of
the universe and its mass respectively energy distribution [3][4][5][6][7][8]. In the next
Chapters this significance will be demonstrated.
If one deals with exponential functions, the author has learnt from Sherbon [25] 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
-<h hJ (19)
Especially is ih -<j,hk C%@Dl*FGC*H , (20)
h


-<j,hk (21)
and mT


U:-<n,hT,


Uo%
(22)
The quoted publication of Sherbon [25] 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 p
'
qFl [23][27]
or with Hardy’s quantum probability φ
5
[23].
)q F
r
Ms't
F%*@GGH%, (23)
!'M
!
C%C*@ClHq
K
L

= 0.04508… (24)
M
!
C%G@*GGDHq
B

?

C%G@*G@H (25)
p
'
Fl

B
?
B
Fl%CFDCH (26)

8
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 [30] for a two-dimensional rocksalt-type lattice [4].
The Madelung constant α
2D
was iteratively determined with very high precision by Triebl [31]
giving
p
u'vewx
%D@@*DDlGGH. (27)
The α
2D
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 [4].
The difference to the inverse of the golden mean φ
-1
is only marginal and gives
?
'
,p
u'vewx
C%CC*GFDH (28)
This almost numerical equality was applied to Villata’s lattice universe [32] 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
p
u
q?
'
?
'
(29)
This relation leads the author to a proposal for the golden mean based calculation of the mass
constituents of the universe [4] independent from other approaches and with only marginal
differences to such results [5][6].
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 [33], not commonly known or accepted by physicists,
originates from a fractal Cantorian set theory [34] 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 [35]. The quantum particle P
Q
is symbolized
by the bi-dimension of the zero set, while the guiding wave W
Q
surrounding the quantum
particle is given by the bi-dimension of the empty set according to
#6yz7#
3
g
 (30)
where n is the Urysohn-Menger topological dimension [36][37] and
#
3
g
?
'

g'
(31)

9
represents the Hausdorff dimension [38], where
{
is the golden mean as defined before.
It results for P
Q
#6yN
|
C? (32)
respectively for W
Q
}~•h
|
,?

 (33)
By using these dimensions a probabilistic quantum entanglement calculation [6][33] with
velocity restriction €•4 delivers effective quantum gravity formulas for the cosmological
mass (energy) constituents of baryonic matter e
M
, dark matter e
DM
, entire dark constituents
e
ED
, and pure dark energy e
PD
as follows 

‚


'K
IK
?


K
L

C%C*@CE*Gl (34)

ƒu
,
‚

B

?

C%G@*G@C (35)

u‚

M

?
t
C%EE*l (36)

„u
?,


C%lFDCDE (37)

‚

u‚

„u
 (38)
Recasting the matter amounts into a suitable form,

‚


.
@?
B
 
u‚


.
@?
B

'
C%E (39)
a reciprocity relation was confirmed between e
M
and e
DM
giving a persuasive equation for the
pure dark energy [3]

„u
,

.
@?
B
@?
B

'
C%lFFlF%F… (40)
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€• 4,
whereas the more general IR theory delivers results for the recession velocity † 
‡
3
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
[7][8]. 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

10
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 e
M
of a moving body with velocity v and rest density ρ
o

‚



€





ˆ
4
'‰
I‰
†


ˆ'‰
I‰
†

 (41)
where †
‡
3
is the recession velocity respectively 
ˆ




ˆ
4

.
The matter energy density reached its maximum at a recession velocity of †?%
Replacement of this special value in Eq. 41 gives

‚

Še0

ˆ'K
IK
?


ˆ
?
B

ˆ
:C%CGCDGG*H (42)
Remembering, φ
5
represents Hardy’s quantum probability at the maximum. This result was
commented by the author in a publication before mentioned [3].
Suleiman aptly characterized the behavior at the critical point β
cr
= φ as phase criticality at
cosmic scale [8]. The dark matter density transforms as
‹
Υ
‹
Ž

‰
•
I‰
. (43)
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
?
M
?

C%FDCDlGlDHC%lDFGFCFH % (44)
The difference gives ?

,?
M
q+,
t
!
.
The case, where according to the Information Relativity theory of Suleiman [7][8] just at the
recession velocity of ß = 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 L
6
= 18 (see Chapter 5
and Figure 4)

‚

u‚
f
r
'


•
C%C@@@@@H ?
r
?
'r

'
qC%C@@lECEGH?
r
(45)
Furthermore, if the recession velocity at β
eq
= 1/3 is mirrored at β
cr
= φ, it resulted β
mir
=
0.9027. In its vicinity at β = 0.89297 the matter energy density would be exactly φ
5
/2 =
0.04508497… respectively the dual dark component 0.7523… ≈ 0.763932… = 2φ
2
(Figure 1).
It approximately indicated a situation that is elaborated for €• 4 by means of the fractal set
theory summarized before in Chapter 8.

11
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 [39].
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 [8]. This supports elegantly the
concerns of the work here presented.
Figure 4. Golden mean dominance in the evolvement of the energy density with the recession
velocity according to the information relativity theory of Suleiman [7][8].
Φ
5
represents Hardy’s maximum quantum probability. Red curve: matter energy density, black curve:
dark matter energy density, green curve: energy density sum.

12
Figure 5. Energy densities related to the red shift z = β/(1-β) (logarithmic scale) according to
Suleiman [8]. Colored curves have the same meaning as in Figure 1. Now the coincidence
point is at z = ½.
Figure 6. Suleiman’s famous reciprocal (complementary) duality between matter density and
dark matter one. Logarithmic scale, red arrow at z = ½, blue arrow at z = φ
-1
= 1.61803398…
(see also Eq. 39). Applied matter density transformations in terms of the redshift [8]:
‘
•
‘
’


dI
,
‘
Υ
‘
’

d
dI

13
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 [8] without any
assumption of quantization of the electron’s orbits and using IR transformation of length as
x
x
’

I‰
'‰

, (46)
other constructs like the electron spin [25] 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 g
e
factor of the electron, conceived as a classical charged particle, is determined by the
relation
“” >
‹
“
• –
”
—
˜
, “
•

‹
Š
(47)
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 X
™ is the reduced
Planck constant.
However, the spin as half-integer quantum number of the electron was introduced without any
physical justification [40]. Very recently, a first attempt has been undertaken by He et al. [41]
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 [42]
;>


57T

?
D
*
UC%CCFGFH% (48)
while a series expansion yields a value more accurate up to the tenth decimal place
>
‹

K
š
t
,



K
š
t


,

t

K
š
t

M
%CCFGFC*H
(49)
This result may be compared to the high accuracy of the best known experimental value for g
e
determined as one-electron cyclotron transition for an electron trapped in an electrostatic
quadrupol potential (Penning trap) [43]
>

%CCFGFC*FDE@ (50)
In a subsequently presented seminal idea of He et al. [41] the spin quantum number s in the
spin momentum term
–
”
—
˜
8 was replaced by a quantized golden mean ?› giving
œ
•

8?›?› (51)

14
with the value ?› = 0.6190713336307(34) as He‒Chengtian average [44][45] to adapt the full
accuracy of g
e
/2. One can calculate ?› by a very simple formula, which resembles the
representation for φ (Eq. 7a) and delivers exactly the given value
?› 


j8@>
‹
>
‹
,,k


8>
‹

, (52)
and for the IR corrected value of g = 0.0023190900 (see Chapter 11)
?› C%DGClFlH (53)
Using this formula, the gyromagnetic factor resulted simply as function of α/π [46]
>
‹
qž
‡
Ÿ
!3
ž
!
%CCF*l (54)
giving ?› C%DGClFC%% (55)
where v
K
is the Klizing speed and c the speed of light.
The latest released values of Sommerfeld’s fine-structure constant α [47] respectively its
reciprocal value is quoted according to NIST [48] being
pC%CClGlF@@DGF (56)
p
'
Fl%CF@GGGCE*% (57)
An approximation using the α/π series expansion yielded
>
‹
qž
!
,



!


%CCFEllEH (58)
or >
‹
qžRS
!
%CCFElE%%% (59)
and further ?› 


nž@*:RS
!
,o (60a)



nž@RS
!

t
,oC%DGClCGGH (60b)
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 [28]
(61)
?› +
@¡,
 
,¡


,¡


,¡

A%


15
The calculation with
¡

C%CCFl*ll*q

rr%r
˜
q
¢
K
q
K
L
t£
yielded ?› = 0.619071096. Indeed,
the number 266.D
™ is very interesting. Division of this number by integers frequently delivers
numbers with repeating decimals, exemplified by 266.D
™/ 24 = 11.
™. If one associates this
number with rounds, then one would need 27 ones to complete 20-times the full 360 degrees
extent.
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 the 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 pJ) [46][49]. 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” [50].
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 ¤
3

‹•
¥Š
(ω
c
= cyclotron frequency, B = magnetic field
strength in Tesla) was performed using the familiar relativistic factor ¦
.
However
,
¦ should
be replaced by
the mass transformation according to the IR theory [8]
Š
Š
’

'‰
I‰
. (62)
For the classical case the corrected frequency ω
c
is
¤
3
¤
.
,
ƒ
§
Š3
1
) (63)
where the energy E
n
of the nth quantum state of a harmonic cyclotron oscillator is given as
¨
g
T7


UX
™¤
3
(64)
The classical relativistic shift δ in the cyclotron frequency per energy quantum was
approximated by the level spacing of the harmonic oscillator giving [51]
¡,X
™¤
3

Jy4

. (65)
For the IR theory one yields a much greater and positive shift because the cyclotron frequency
yields now
¤
3
q¤
.
::ž
ƒ
§
Š3
1
) (66)
The relativistic shift δ is approximated by
¡q
©ª
«
©g
¤
.
ž

Š3
1
:
©8ƒ
§
©g
¤
.
ž

Š3
1
ƒ
§
 :X
™¤
3
:¤
.
ž
ƒ
§
Š3
1
(67)
The gyromagnetic factor as g/2 can be determined from the observed eigenfrequencies [51]

16
œ


ª
˜
_
'ª
˜
¬

ª
˜
«
'ª
˜
¬

(68)
where the ¤˜ values are marginally modified with respect to the free-space values, ¤
e
is the
anomalous frequency, and the spin frequency is ¤
-
¤
3
¤
e

ª
˜
^
1
ª
˜
«
¤˜
Š
is the magnetron
frequency, using the dip frequency ω
z
in Hz [51]. For the experimentally chosen cyclotron
frequency of ν
c
= 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 g
becomes noticeably smaller by a factor of approximately 1.0001, meaning a correction of g
e
from the seventh decimal point downwards to about >
‹
q%CCFGCG®
Now the scientific community is waiting for a most precise redetermination of the g
e
-factor as
well as the related Sommerfeld constant by experts [49] [51]. The aforementioned
Zitterbewegung approach of Niehaus [17] 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 [52].
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 CoNb
2
O
8
was experimentally
verified next a phase critical point by Coldea et al. [53], the present author suggested linking
the optimum hole doping ¯
.
of high-T
c
superconductors with the golden mean in the form of
Hardy’s maximum quantum probability of two particles [15]
¯
.
q
•
!
?
B
C%GF (69)
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
‡
°
‡
Ÿ
q

!
?
B
C%C@l (70)
Both relations document the fractal nature of the electronic response in superconductors. It
was suggested recently that the same is true for conventional superconductors [16]. Also
Prester had reported before about evidence of a fractal dissipative regime in high-T
c
superconductors [54].
Interestingly, some time ago the present author connected the optimum transition temperature
T
co
of high-T
c
superconductors with a Fibonacci number f
i
,
proportional to a domain width,
by the relation T
co
= 12000/f
i
[15]. One yields the integer number 45, again as a product of
solely Fibonacci numbers, when dividing this number by the number 266.D
™ (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 [8].
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

17
superluminal photon model [55]. 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 [56]. 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 [57] or Pitkänen’s formulation [58]
suffer from not considering duality of particle and wave or reciprocity of matter and dark
matter [8] 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 T
H
comes into play. I quote the formulation of
Rafelski and Ericson [59] 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 [56][60]. The process could explain the evolution of
life with T
H
around ambient temperatures.
With respect to the entire energy density of φ
2
at the phase critical point β = φ one may suggest
formulating the Hagedorn temperature T
H
proportional to the squared golden mean φ, where
p
±
is formally the tension of a string.
2
²eœ
q?

:

³
(71)
It remains to interpret the not liked string tension by a more appropriate thermodynamic
quantity at ambient equilibrium conditions.
14. Conclusion
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
[56]). 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.

18
Acknowledgement
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.
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Appendix I
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

21
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 [12].
According to the IR theory of Suleiman [8] 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 [61]. 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 [62]. 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 [4]. In this way one may speak also
of reciprocity when dealing with matter density and the dual dark matter density according to
Figure 6. Appendix II
Another approach for the gyromagnetic factor used the fifth power of ?› with the value ?›
5
=
0.09092922100312. An approximation is the inverse Lucas number L
5
= 11 as combination of
two inversely related irrational numbers (see Eq. 13 to 15)
?
'B
,?
B

'

'
C%CGCGCGCH% (72)
However, physically more convenient is the expression
!:‡
Ÿ
3
= 0.0917012… , where v
K
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
‡
‡
’
†, (73)
combining the length transformation
x
x
’

I‰
'‰
with the time transformation
´
´
’


'‰
, where
β = v/c is the recession speed [8]. Surprisingly, an additional 
´
´
’

'
,†
term is needed to
give more accuracy and the following simple formula
ž
!‡
Ÿ
3

!‡
Ÿ
3
,
!‡
Ÿ
3
L
) = ž
!‡
Ÿ
3
,
!‡
Ÿ
3



L
= 0.61907254…. (74)
leading to
œ
•

%CCDCC@GlCG=>
‹
%CCFC (75)
Remarkably, this value is almost identical to the result of [37], because

22

K
š
t
%CCFCCF (76)
Only now we are allowed to associate the term
!‡
Ÿ
3
with Sommerfeld’s fine-structure constant
α [41] applying
!‡
Ÿ
3
*)p (77)
where α is a measure of the strength of interaction of an electron and a photon in the quantum
electrodynamics theory (QED). The charge of the electron in QED (Lorentz-Heaviside) units
has the numerical value of  ,+*)p.
The accurate experimental value for the gyromagnetic constant could be attained from Eq. 74
using an adapted fine-structure constant of
p
±
C%CClGllGG@@DDG (78)
respectively p
±'
Fl%CFlFDDGEEGll* (79)
where p
±
,p D%*F:C
's
(80)
Tackling the problem of the not fully adapted accuracy in comparison to the experimental
value, one can multiply the term under the fifth root of Eq. 74 by a factor of 0.9999902180 or
alternatively reduce the Klitzing speed by a factor of 0.99999004931863 respectively the
charge of the electron by a factor of 0.99999502464694. This adjustment may result partly
from a correction of g as well as α with respect to the IR theory, besides needed radiative
corrections.