Quantum Walk on a Spin Network and the
Golden Ratio as the Fundamental Constant
of Nature
Klee Irwina∗, Marcelo M. Amaralab†, Raymond Aschheima‡, Fang Fanga§,
aQuantum Gravity Research
Los Angeles, CA
bInstitute for Gravitation and the Cosmos,
Pennsylvania State University, University Park, PA
Abstract
We apply a discrete quantum walk from a quantum particle
on a discrete quantum spacetime from loop quantum gravity and
show that the related entanglement entropy drives an entropic
force. We apply these concepts in a model where walker positions
are topologically encoded on a spin network.
Then, we discuss the role of the golden ratio in fundamen-
tal physics by addressing charge and length quantization and by
analyzing the ratios of fundamental constants−the limits of na-
ture. The limit of minimal length and volume arising in quantum
gravity theory indicates an underlying principle that we develop
herein.
1
Introduction
One of the principal results from Loop Quantum Gravity (LQG) is a discrete
spacetime−a network of loops implemented by spin networks [1] acting as the
digital/computational substrate of reality. In order to better understand this
substrate, it is natural to use tools from quantum information / quantum
computation. Gravity, from a general perspective, has been studied with
thermodynamic methods. In recent years, numerous questions on black hole
entropy and entanglement entropy have made this an active field of research.
In terms of quantum information and quantum computation, advances have
been achieved with the aid of many new mathematical tools. Herein, we
present the development of one such tool, which we call the discrete-time
quantum walk (DQW). We will see that the problem of a quantum particle
on a fixed spin network background from LQG can be worked out with the
DQW. This gives rise to a new understanding of entanglement entropy and
∗email: Klee@QuantumGravityResearch.org
†email: mramaciel@gmail.com
‡email: Raymond@QuantumGravityResearch.org
§email: Fang@QuantumGravityResearch.org
1
entropic force, permitting the proposal of a model for dynamics. In terms
of physical ontology, we suggest dynamics and mass emerge from this spin
network topology, as implemented by the DQW.
In summary, the first part of this paper is a reinterpretation of results from
LQG that emphasizes the quantum information perspective of a quantum-
geometric spacetime. That is, it adopts Wheeler’s it from bit and the newer
it from qubit ontologies−the general digital physics viewpoint.
One important view of reality is the digital physics paradigm−the idea
that reality is numerical at its core [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13].
In the second part of this paper, we defend the conjecture that reality is
information-theoretic at its core by presenting experimental and mathemat-
ical justification of our conjecture that the golden ratio is the fundamental
dimensionless constant of nature. Numbers include shape-symbols, such as
simplex-integers [14]. Generally, a symbol is an object that represents itself
or another object. Symbols can be self-referential and participate in self-
referential codes or languages, such as quasicrystals [14, 2, 15]. Generally,
symbols are highly subjective, where meaning is at the whim of language
users. However, simplexes1, as geometric number and set theoretic symbols
which represent themselves, have virtually no subjectivity. That is, their
numeric, set theoretic and geometric meaning is implied via geometric first
principles. If space, time and particles were patterns in a quantized geomet-
ric code, low subjectivity geometric symbols, such as simplex-integers, could
serve as both spatiotemporal and numerical quanta of space while also being
part of an organized code or language called a quasicrystalline phason code.
One element of digital physics we discuss here is the discreteness of space-
time and charge in general−something that a geometric code is well suited
to provide.
The voxel of space is an old idea that, today, has acquired a body of
experimental evidence as well as mathematical proofs [16].
Its early con-
struction starts with the birth of quantum field theory, where it was noticed
that models become more regular with a cutoff. When one considers quantum
gravity, like Bronstein did in 1936 [17], this question becomes fundamental
because, as opposed to smooth spacetime field theories, gravity does not al-
low an arbitrarily high concentration of charge in a small region of spacetime.
Bronstein concluded that this leads to an inevitable limit of the precision to
which one can measure the strength of the gravitational field. Advances on
different fronts in the following years lead to a general understanding that
this voxel of space, a minimal volume, must be the foundational building
block of any realistic proposal of quantization of the gravitational field. Such
deductions provide support for the digital physics view. Quantum mechanics
indicates that nature must have a minimum length, volume and time. We
can cite the rigorous developments of string theory, LQG, the Maldacena
conjecture, black hole physics, various thought experiments, the generalized
uncertainty principle and many other works as support of the discretized
1An n-simplex is an n-dimensional polytope which is the convex hull of its n+1 vertices.
For example the triangle in 2D, tetrahedron in 3D, 4-simplex in 4D...
2
spacetime view. For a review, references and historical details, we recom-
mend reference [16]. The discrete nature of quantum gravity theories implies
that time and motion are ordered numerical sets on point-like substructures,
which can be viewed as discrete pre-spacetime graphs acting as possibility
spaces for graph actions. Upon such graph drawings (a geometric represen-
tation of a graph), a quantum gravity code theoretic formalism can express
itself non-deterministically. This approach leads to a new ontology for the
substrate of reality that may replace older ontologies based on either random-
ness or determinism. The new ontology is called the code theoretic axiom
[2], where a finite set of irreducible geometric symbols and a finite set of non-
deterministic syntactical rules express as code legal and physically realistic
manifestations. Self-organizing expressions of a quantum code are inherently
non-deterministic because codes/languages are non-deterministic.
Herein, we address implications of minimal volume, the quantum of charge
of the gravitational field (along with minimal charge in general) in light
of the code theoretic view. The experimentally validated result of electric
charge quantization has a standard theoretical explanation in Dirac mag-
netic monopoles. This field, which applies topological arguments to address
charge quantization, follows Dirac’s work in general [18]. We go further here
to understand that there are underlying principles governing charge quanti-
zation. This is revealed when one considers the Planck volume limit of space.
One consequence we discuss is the special role of the golden ratio, where we
show that fundamental limits of nature can be understood in terms of the
golden ratio.
2 Methods
2.1 Particle interacting LQG
We start by considering a quantum particle on a quantum gravitational field
from LQG [19]. In LQG, spin networks define quantum states of the grav-
itational field. To consider a quantum particle on this gravitational field,
we consider the state space built from tensor product of the gravity state
HLQG and the particle state HP , H = HLQG ⊗ HP . The LQG state space
can be spanned by a spin network basis |s〉 that is a spin network graph
Γ = (V (Γ), L(Γ)), with V (Γ) vertices coloring which elements denoted by
v1, v2..., and L(Γ) → l1, l2..., ({ 12 , 1,
3
2 ...}) links coloring. For the particle
state space the relevant contribution comes from the position on the vertices
of graph Γ, spanned by |xn〉 where (n = 1, 2...). The quantum state of the
particle captures information from discrete geometry and cannot be consid-
ered independently from it. Therefore the Hilbert space is spanned by |s, xn〉
and the Hamiltonian operator can be derived by fixing a spin network to
the graph, and calculating the matrix element of this operator 〈ψ|H|ψ〉 with
ψ ∈ HP . Accordingly, for the interaction between the particle and the fixed
3
gravity state space [19] we have,
〈ψ|H|ψ〉 = κ
∑
l
jl(jl + 1) (ψ(lf )− ψ(li))2 ,
(1)
where κ is a constant that we can initially take as equal to one, lf are the
final points of the link l and li the initial points. The interaction term takes
this form because the relevant Hilbert space depends on the wave functions
at the vertices. So if the link l starts at vertex m and ends at vertex n we
can change the notation, relabeling the color of this link l between m and n
as jmn and the wave function on the end points as ψ(vm), ψ(vn). Now, for
H, we have
〈ψ|H|ψ〉 = κ
∑
l
jmn(jmn + 1) (ψ(vn)− ψ(vm))2 ,
(2)
The operator H is positive semi-definite. The ground state of H corre-
sponds to the case where the particle is maximally delocalized. This leads
to entropy [19]. In reference [20] it was considered that the classical random
walk is associated with (2), a Markov chain. The transition probabilities for
this random walk are
Pmn =
jmn(jmn + 1)
∑
k
jmk(jmk + 1)
.
(3)
and [20] shows that this reproduces an entropic force. This random walk is
implemented with the Laplacian solution (2) and differs from the Laplacians
coming from the discrete calculus work out in [21]. We will consider DQW
relations with more general Laplacians in future work.
2.2 DQW
As study the influence of discrete geometry on a quantum field. Here, it is
natural to consider the quantum version of random walks. To have DQWs
we need an auxiliary subspace, the coin toss space, to define the unitary
evolution
U = S(C ⊗ I),
(4)
where S is a swap operation that changes the position to a neighbor node
and C is the coin toss operator related to this auxiliary subspace. On some
graphs its clear how to define the coin toss Hilbert space [22] but for the spin
network considered above its not clear exactly what auxiliary space to use.
For a more general approach, we can make use of Szegedy’s DQW [23, 24],
which we can utilize in two ways: 1- consider a bipartite walk or 2 - a walk
with memory. For a bipartite walk, if we consider a graph Γ, we can simply
make an operation of duplication to obtain a second graphΓ̄. For our purpose
here, it is better to consider the second option − the walk with memory,
considering the Nv-dimensional Hilbert space Hn, {|n〉 , n = 1, 2, ..., Nv} and
Hm, {|m〉 ,m = 1, 2, ..., Nv}, where Nv is the number of the vertex V (Γ) .
The state of the walk is given as the product HNv
n ⊗HNvm spanned by these
4
bases. That is, by states at the previous |m〉 and current |n〉 steps, defined
by
|ψn(t)〉 =
Nv∑
m
√
Pmn |n〉 ⊗ |m〉 ,
(5)
where Pmn is the transition probabilities that define a classical random walk,
a Markov chain, which is a discrete time stochastic process without a memory,
with
∑
n
Pmn = 1.
(6)
Note that (6) is implied by definition of (3). For the evolution, we can
consider a simplified version of Szegedy’s DQW [25] by defining a reflection,
which we can interpret with the unitary coin toss operator2,
C = 2
∑
n
|ψn〉 〈ψn| − I,
(7)
and a reflection with inverse action of the P swap (previous and current step)
that can be implemented by a generalized swap operation
S =
∑
n,m
|m,n〉 〈n,m| ,
(8)
where we have the unitary evolution
U = CS,
(9)
that defines the DQW.
So using Szegedy’s approach is a straightforward way to obtain the dis-
crete quantum walk on the spin network Γ considered above. It is given by
equations (5 - 9) with P given by (3). Namely, for equation (5)
|ψn(t)〉 =
Nv∑
m
√√√√ jmn(jmn + 1)
∑
k
jmk(jmk + 1)
|n〉 ⊗ |m〉 .
(10)
We can interpret the coin toss space as the space of decisions encapsulating
a nondeterministic process possessing memory, such that we have a unitary
evolution. This approach makes the usual entropy convert into entangle-
ment entropy between steps in time. Szegedy’s DQW is a general algorithm
quantizing a Markov chain defined by transition probabilities Pn,m. These
transition probabilities are obtained directly from the Hamiltonian of the sys-
tem such that Szegedy’s DQW is used to simulate this system with quantum
computation [26].
2Szegedy’s DQW does not generally use a coin toss operator in the literature
5
2.3 Entanglement Entropy and Entropic Force
We turn now to calculate entanglement entropy. Consider the Schmidt de-
composition. Take a Hilbert space H and decompose it into two subspaces
H1 of dimension N1 and H2 of dimension N2 ≥ N1, so
H = H1 ⊗H2.
(11)
Let |ψ〉 ∈ H1⊗H2, and {
∣∣ψ1i 〉} ⊂ H1, {∣∣ψ2i 〉} ⊂ H2, and positive real numbers
{λi}, then the Schmidt decomposition would be
|ψ〉 =
N1∑
i
√
λi
∣∣ψ1i 〉⊗ ∣∣ψ2i 〉 ,
(12)
where
√
λi are the Schmidt coefficients and the number of the terms in the
sum is the Schmidt rank, which we label N . With this, we can calculate the
entanglement entropy between the two subspaces
SE(1) = SE(2) = −
∑
i∈N
λilogλi.
(13)
We can now calculate the local entanglement entropy between the previousm
step and the current n step (similar for current and next steps). Identifying
the Schmidt coefficients
√
λi with
√
Pmn, with Pmn given by (3) and, from
(10), we see that the Schmidt rank N is the valence rank of the node. Then
insert in (12, 13), and the local entanglement entropy on current step is
SEn = −
N∑
m
PmnlogPmn.
(14)
By maximizing Entanglement Entropy
SEn = logNmax,
(15)
where Nmax is the largest valence. Which gives the entropic force for gravity
worked out in [20, 27] with dS
dx = |SEn−SEm | proportional to a small number
identified with the particle mass M
dS
dx
= |SEn − SEm | = αM,
(16)
where, if we take the logarithm of (15) to be base 2, α is a constant of
dimension [bit/mass].
Each of these interpretations has related applications. For example, with
the DQW, we have unitary evolution by encoding a non-deterministic part of
the classical Markov chain. This gives an internal structure for the particle as
well as the entanglement entropy value. In such a digital physics substrate,
the particle walks in such a way as to maximize its entanglement entropy
based upon the inherent memory of its walking path (a measurement of the
6
system on n implies the state at previous m), which generates an entropic
force.
Interestingly, we can consider von Neumann Entropy (14) in the context of
quantum information. The probabilities are the Markov chain connecting the
steps. Accordingly, the particles construct “letters” of a spatiotemporal code
as expression of the allowed restricted but non-determined walking paths,
where the entropy measures the information needed. From this, we see how
the entropic force emerges.
3 Results
3.1 Entropy of Black Hole
The framework above is a discretized quantum field expressing phases as
algebraically allowed patterns upon fixed discrete geometry. Specifically a
moduli space algebraic stack. Gravitational entropic force suggests a unified
picture of gravity and matter via a quantum gravity approach. Consider now
a regime from pure quantum gravity, like the black hole quantum horizon, so
that there are no quantum fields but only quantum geometry. We can use
our DQW to simulate this regime.
Again, DWQs encode local entanglement entropy (14) in the sense that
the chains or network of step paths are always dynamically under construc-
tion as a "voxelated" animation. This resembles the isolated quantum hori-
zons formulation of LQG [28, 29, 30, 31, 32], which gives explains the origin
of black hole entropy in LQG. In this scenario, the Horizon area emerges
from the stepwise animated actions at the Planck scale that are simulated
by DQWs. We argue that DQWs are the Planck scale substrate forming the
emergent black hole quantum horizon, where particle masses composite to
black hole mass in the full aggregate of quantum walks on the spin network
of the event horizon.
In the isolated quantum horizons formulation, entropy is generally calcu-
lated by considering the eigenvalues of the area operator A(j) and introducing
an area interval δa = [A(j) − δ, A(j) + δ] of the order of the Planck length
with a relation to the classical area a of the horizon. A(j) is given by
A(j) = 8πγl2p
∑
l
√
jl(jl + 1),
(17)
where γ is the Barbero-Immirzi parameter and lp the Planck length. The
entropy, in a dimensional form, is
SBH = lnN(A),
(18)
with N(A) the number of micro-states of quantum geometry on the horizon
(area interval a) implemented, combinatorially, by considering states with
link sequences that implement the two conditions
8πγl2p
Na∑
l=1
√
jl(jl + 1) ≤ a,
(19)
7
related to the area, where Na is the number of admissible j that puncture
the horizon area and
Na∑
l=1
ml = 0
(20)
related to the flux with ml, the magnetic quantum number satisfying the
condition −jl ≤ ml ≤ jl.
The detailed calculation [29, 32] shows that the dominant contribution to
entropy comes from states in which there is a very large number of punctures.
Thus, it is productive to interpret this entropy as quantum informational
entropy (14). Let us investigate how the horizon area and related entropy
emerges from maximal entanglement entropy of DQWs. Condition (19) is
associated with these DQWs. From (14 and 15), considering edge coloring,
maximal entanglement entropy occurs for states on nodes of large valence
Nmax and sequence with jl =
l
2 (l = 1, 2, ..., Nmax). Accordingly, for the
entropy calculation, it is admissible that j, which punctures the horizon area,
respects these sequences and that associated nodes have large valence rank.
Therefore, we can rewrite condition (19) as
Nai∑
i=1
ai = ac,
(21)
where Nai is the number of admissible nodes with
ac =
a
4πγl2p
.
(22)
and each ai is calculated from
ai =
Nmax∑
l=1
√
l(l + 2),
(23)
and considering the dominant contributions given by the over-estimate each
ai and counting N(ac) that will give N(A), each ai is an integer strictly
greater than 1
√
l(l + 2) =
√
(l + 1)2 − 1 ≈ l + 1,
(24)
which means that the combinatorial problem we need to solve is to find N(ac)
such that (21) holds. This was discussed in a similar problem in [28]. It is
straightforward3 to see that
logN(A) =
log(φ)
πγ
a
4l2p
,
(25)
where φ = 1+
√
5
2
is the golden ratio4.
3Because the cardinal N(ac) of the set of ordered tuples of integers strictly greater than
1 summing to ac is the ath
c Fibonacci number F (ac).
4The golden ratio [33] is an irrational number (1.61803398875...) as the solution to the
equation φ2 = φ+ 1.
8
With this established, we can now conjecture that this entropy, relating to
states that give maximal entanglement entropy, is simply the entanglement
entropy of the DQWs. Accordingly, holographic quantum horizons can be
simulated by the DQWs with maximal internal entanglement entropy. The
walker moves are from nodem to node n, each each with large valence number
so that equation (16) works. If we consider a walker mass spanning I with
Planck mass mp, we can choose the proportionality constant so that
4SE = |SEn − SEm | = I,
(26)
so M = Imp gives the amount of information on the horizon, and 4SE is
given in bits. We can make explicit the log2 on (25) so that it is given in bits
too and propose that log2N(A) = 4SE and that this measure of information
gives the emergent Bekenstein-Hawking entropy. DQWs encode or express
black hole information, so to infer its entropy, one needs to make contact
with some frame of DQWs. The particles that simulate black holes will be
more probable on a node of maximal entanglement entropy encapsulating
logN(A). Note that logN(A) is not the usual statistical entropy because
we have not taken into account all of the micro-states or the flux condition.
From (18) the black hole entropy, changing the logarithm, is
SBH = log2N(A) =
log2(φ)
πγ
a
4l2p
.
(27)
Therefore, Bekenstein-Hawking entropy is recovered by setting the Barbero-
Immirzi parameter
γ =
log2(φ)
π
,
(28)
showing that the states of maximal entanglement entropy are dominant in
black hole entropy. So we can think of Bekenstein-Hawking entropy as emer-
gent from the local entanglement entropy above. A horizon area a spanning
4I Planck areas has I bits like (26).
3.2 A model of walker position topologically encoded
on a spin network
The Clebsh-Gordan condition at each node is realized by covering the graph
with loops. From (3) and (14), we can compute the local entropy from a
vertex as
SEn = logσ −
1
σ
N∑
m
jmn(jmn + 1)log (jmn(jmn + 1)) ,
(29)
where σ =
N∑
m
jmn(jmn+1) of neighbor links. For example, at a node {2, 3, 3},
j = {1, 32 ,
3
2} gives σ =
19
2 and SEn = 1.06187. At a node {2, 2, 2}, j =
{1, 1, 1} gives σ = 6, SEn = 1.09861, which is the maximum possible local
9
Figure 1: Loops.
entropy. Note that (29) is the local entropy formula given in [20] as 14 and is
in accordance with well known LQG formulas for quantized length and area.
See figure (1).
The local entropy at each node is color coded. From equation (16), a
massless particle moves on the same color and a massive particle moves along
constant absolute color differences.
In the figure (2) :
(blue){l1, l2, l3} = {0, 1, 1} or {0, 2, 2} (side effect)
(white) ... {1, 2, 3}
(yellow) ... {1, 1, 2}
(orange) ... {2, 3, 3}
(red) ... {2, 2, 2}.
Dynamics:
Photons can orbit on orange hexagons. No possible particles with constant
mass. Possible travel of massive particle, with mass = |S(red) − S(orange)|,
(= 0.037) interacting with photons at some orange vertex.
Figure 2: Entropy is color coded.
The walker position or the presence of a particle at one node is encoded
by a triangle. Its move is a couple of 3-1 and 1-3 Pachner moves on neighbor
positions, piloted by the walk probability. See figure (3).
10
Figure 3: Particle and Pachner moves.
4 Ontological discussion
4.1 Minimal volume and charge
The generic implication of quantum gravity, the existence of a minimal vol-
ume (vo) near the Planck scale indicates a new fundamental limit in nature.
This fundamental limit that arises during the quantization of gravitational
fields is similar to the fundamental charge qe that arises in the quantization
of electromagnetic fields5. For all these fundamental limits, there is one fun-
damental dimensionless constant, which relates them. It is often said that
the dimensional constants are more fundamental than the dimensionless con-
stants [34]. This is especially true when they are fundamental building blocks
of a theory, such as occurs when there are general limits found in nature such
the speed of light c in special relativity and the minimal action ~ in quantum
mechanics. The dimensionless constants depend on the model, such as the
standard model of particle physics, and they are generally determined ex-
perimentally, where there is a meaningful experimental margin of error. We
argue that there are dimensionless constants of nature that have a special
and fundamental role as ratios of the fundamental limits of nature.
In quantum electrodynamics (QED), the fundamental ratio is the fine-
structure constant α, wherein the relation is obtained from Dirac’s quantiza-
tion of magnetic charge [35]
α =
e
g
n,
(30)
and where n is some integer, g is the magnetic charge rewritten in Coulombs
and e the elemental electric charge. The magnetic charge is related to the
5The same discussion can be made for fundamental charges associated with electroweak
and the strong force that is in agreement with the understanding of a fundamental running
coupling constant using renormalization group equations.
11
non-trivial topology of charge space. Or, considering a discrete substrate,
for each lattice we use to describe the electric field, there is a dual lattice
corresponding to magnetic flux. So, for each site, we have the two funda-
mental charges living in respective dual spaces to one another. The charges
of QED are quantized such that the ratio between any two is α, and the cou-
pling constant is related to the strength of the force. We will not elaborate
further here. However, we note that the monopole magnet provides a new
understanding of the relation between topology, geometry and physics that
is captured in the simplest form by equation (30).
For more insight, we can use a similar formula without flux by fixing to
the Planck scale,
α = e2
ke
~c
=
(
e
qp
)2
(31)
where ke is Coulomb’s constant or the electric force constant, a constant from
the QED interaction, ~ is the reduced Planck’s constant, c is the speed of
light in vacuum and qp is the Planck charge. In terms of α−1 ,
α−1 =
1
e2
~c
ke
=
(qp
e
)2
.
(32)
The approximate value for this constant is based on experiments and QED
perturbation theory calculations. The CODATA [36] recommended value is
α = 0.0072973525664
(33)
and
α−1 = 137.035999139
(34)
and is associated with the QED scale. It it important to take note of the
fact that the CODATA value of α−1 in equation (34) is based on an average
of disagreeing results given by five different experimental techniques for the
measurement of Planck’s constant. To explain, α is a ratio relative to the
elementary charge. CODATA does not independently define the quantity for
α. Instead, a value is derived from the relation (31). Accordingly, the accu-
racy of measurement of Planck’s constant is proportional to our knowledge
of the fine structure constant. The CODATA value for Planck’s constant is
a weighted average of the following five experimental techniques that agree
only at the 4th place after the decimal (6.6260×10−34Js). See figure 4 [36].
Experimentally, we can therefore only be confident in the value of h to
the 4th decimal place, which limits our knowledge of what α is to the 4th
decimal place. If one wishes to assume a probable value greater than the
4th decimal place, it would be logical to use the most advanced and precise
experimental technique to date, as opposed to the CODATA approach that
uses older lower resolution results to down-grade the precision of the more
precise newer results. In the case of the gravitational constant, the precision
is only to the 2nd decimal place. The most advanced high energy technique is
based on atomic scale experiments not available when the earlier techniques
12
Figure 4: Planck’s constant. CODATA 2010.
averaged into the current CODATA value were developed. Specifically, the
most technologically advanced calculation for measuring the gravitational
constant was reported in Nature in 2014 [37]. It is a result derived by probing
the atomic scale. Herein, we will refer to this as “highest resolution value of
G”, denoted as GHR, GHR = 6.671× 10−11m3kg−1s−2.
In an attempt to discover what the actual minimum volume vo is that
nature uses, there have been many attempts to quantify it from string the-
ory [38], loop quantum gravity, (LQG) [1] and others [16]. However, none of
these unification theories have made successful predictions. So until a predic-
tive quantum gravity theory is discovered, one cannot make solid theoretical
assumptions using the CODATA value beyond a few decimal places for any
fundamental constant such as α. When precision is desired, one would want
to use GHR combined with c to generate ~ and, from there, to generate the
rest of the constants [39].
There is consensus that the quantum gravity scale is at or near the Planck
volume. However, the limit of our knowledge of the value of ~ directly limits
our knowledge to the current limit, which is GHR. Fortunately, actual results
permit us to understand that the gravitational constant, G, ties the three
following fundamental limits of nature together to build a dimensionless fun-
damental constant that we call β, which relates to α. The three dimensional
constants associated with these limits are the maximal local physical velocity,
the speed of light c, the minimal action (or a minimal amount of informa-
tion) ~ and this newly discovered minimal volume vo. It is the fundamental
dimensionless ratio defining the relationship between these fundamental di-
mensional limits. We write this as
β =
1
v2o
(
~G
c3
)3
=
(
vp
vo
)2
(35)
13
or in the inverse form
β−1 = v2o
(
c3
~G
)
=
(
vo
vp
)2
,
(36)
where we use the Planck volume, vp = l3p, with lp the Planck length that
defines the Planck scale and Planck units as
lp =
√
~G
c3
.
(37)
We will discuss in the next sections how these constants β and α are deeply
related to the golden ratio.
For more insight, let us focus on the quantization of the gravitational
field using results from LQG [1]. Quantization is performed using a gener-
alization of the lattice of QED − a triangulation of spacetime and its dual
two-complex. Here we a split of the gravitational “charge” into two com-
ponents; one related to flux6 (gravitational analog of magnetic field) with
area eigenvalues l2p
√
j(j + 1) where j are non-negative half integers. And
the other is the gravitational analog of the electric field in the dual space
that is the geometry of quantized spacetime, which we will call ll
l2l ∝ γl2p
√
j(j + 1),
(38)
where γ is the Barbero-Immirzi parameter. The proportionality constant in
this reference is 8π. From this quantization condition, we derive the eigen-
value of the volume to be
v20 ∝ γ3
(
~G
c3
)3
,
(39)
which gives
β ∝ 1
γ3
.
(40)
The proportionality constant in this reference is 6
√
3/(8π)3. The theory
has some ambiguities and does not permit us to calculate this constant or
γ. There are no experimental solutions possible yet. However, black hole
calculations based on the limit of general relativity and quantum mechanics
provide an approximate value for γ between 0 and 1. See equation (28) and
reference [32].
6A less discussed quantity in the literature which is used for dynamics in quantum
gravity formalisms are the volume flow rate, well known in hydrodynamics and magneto
hydrodynamics, related to constant QΦ = 4πhG
c2
[40] given in the SI units by m3s−1.
Following the discussion above about the most accurate value of Planck’s constant based
on the most precise experimental technology, we reiterate that the CODATA values agree
only at the 4th place after the decimal and the value for G agrees only to the 2nd place after
the decimal. Using the result of the most recent and high energy atomic scale experimental
techniques, GHR, together with the value of h only to the 4th place after the decimal gives
QΦ = 0.6180382×10−59m3s−1. This results in a value going out to the 5th decimal place
with 1/φ.
14
This indicates a relationship between β and the quantum gravity coupling
constant, γ, associated with the gravitational constant. That is, β leads to
the precise quantum gravity coupling constant−also a number between 0 and
1.
Correspondence between the two scales, QED and quantum gravity, is ex-
pected in a correct unification theory. Our interpretation is that it is related
to the quantization of charge and is manifest in the ratios of equations (30)
and (38). The result suggests an underlying principle governing this funda-
mental ratio. We suggest this principle is related to symmetry and graph
network theory efficiency, where nature exists as a maximally efficient graph
theoretic topological quantum code due to its requirement of classic efficiency
as exemplified in the principle of least action and other conserved symmetries
ubiquitous in nature, such as the gauge symmetry unification of particles and
forces in the standard model of particle physics. Put differently, presume the
digital physics view is true, where nature is information theoretic. That is,
nature is based on symbolic language and not merely described by symbolic
language. From the probability plot in 3-space of the quantum wave form, to
gravity theory to classic physics, this supposed language is simulating or ex-
pressing itself geometrically in the form of physics. Accordingly, a geometric
code would be a good conjecture for what form of symbols would be used in
the mathematical language of our geometric universe. And because nature
appears to be concerned with efficiency, a maximally efficient geometric code
would be logical. Graph drawing codes, such as phason algebraic codes in
quasicrystals, may be optimally efficient in the universe of codes capable of
expressing a geometric physical universe.
The efficiency of a code, such as a geometric spatiotemporal code, can be
ranked by the ratio of binary actions needed to express a given physical sys-
tem. A geometric based quantum gravity plus particle physics code would ex-
press spatiotemporal information using the minimum quantity of irreducible
geometric symbols. Trivially, the minimum number of symbols in any code
is two. Fundamental quasicrystals such as the Elser-Sloane quasicrystal [41]
with H4 symmetry, derived by projective transformation of a minimal slice
of the E8 lattice, or the quasicrystalline spin network (QSN) [42, 43] with H3
symmetry are constructed as geometric languages composited from geomet-
ric spatiotemporal symbols as points separated by two distances related as 1
and 1/φ. Such codes exist physically in nature as quasicrystalline phases of
matter, where generally a network of Fibonacci chains [15] in 3-space mani-
fests as energy wells separated by 1 and 1/φ instead of abstract points in the
mathematically ideal abstract analogue. At any given moment, some energy
wells are occupied by an atom and some are vacant. Over the time domain,
the atoms are known to tunnel at very low energy according to a phason code
that is inherently non-local.
The binary efficiency of a network can be ranked by the connectivity rank
of its nodes−specifically the average valence value of its vertices. Maximally
connected graph drawings can only be achieved via the use of the golden
ratio, due to it being the only ratio possessing the fractally self-similar quality
15
where
A
B
=
A+B
A
, where A = 1 and B =
1
φ
.
Figure 5: To optimally network 1D two-letter codes in 2, 3 and 4 dimensions,
we must use the Fibonacci chain with the two letters as 1/φ or the connec-
tivity breaks down. An infinitesimal deviation from the golden ratio spacing
renders the object (1) not a quasicrystal because arbitrary closeness of nodes
will exist and (2) a non-code because there will be an infinite number of
1D symbols as lengths and (3) not a quantum topological network because
non-local influence where an action on one point influences other non-local
point on/off states breaks down. We know that the reason for this is be-
cause, in the universe of all ratios closed under multiplication and division,
only φ possesses fractal self-similarity. The above diagram on the left shows
a network of five short Fibonacci chains, where nodes overlap perfectly due
to the fractal self-similarity of φ. To left, we see how a small deviation from
the golden ratio spacing destroys the connectivity rank by lowering the av-
erage near neighbor valance magnitude and introducing arbitrary closeness
between nodes.
Our conjecture is that this fundamental dimensionless constant β dis-
cussed above is
√
β = 1/φ3, which makes the ratio between the Planck
length and the minimum length lo, related to vo, φ and allows for networks
where nearest neighbor lengths are restricted only to 1 and φ, starting at the
Planck scale. Due to its power of network code efficiency, this fundamental
ratio would be the building block of the standard model of particle physics
and general relativity in a geometric graph-drawing code theoretic unifica-
tion formalism. Note that in a gauge unification picture, Lie groups, Lie
algebras and their associated lattices, such as the E8 lattice, play a central
role [44, 45]. Grand unified theories are elegant and successful. They embed
the Standard Model Lie group inside larger groups, where in 6D the largest
group possible is E6 [38, 46]. Going to higher dimensions, E6 is a subspace of
E87. E8 is a powerful tool unifying gravity and the standard model via the
most popular and foundational form of string theory, heterotic string theory
[38, 47, 48] or E8 Grand Unification by itself [49, 50, 51, 52, 53]. Remarkably,
E8 symmetry has been found in condensed matter physical experiments [54].
Furthermore, one can recover E8 gauge symmetry physics from an icosian
construction of the E8 lattice [55]8.
7SU(3)× E6 the maximal sub algebra of E8.
8Note that geometrically, E6 is a sub-lattice embedded in E8.
16
Summarizing this section, if nature is a geometric code, such as a qua-
sicrystal phason code derived from the higher dimensional E8 lattice, this
might explain charge quantization in nature and shed light on a universally
generalized optimization principle. Let us discuss this in detail next.
4.2 The principle behind fundamental ratios
To discover the correct quantum gravity and particle physics unification the-
ory, new insights and fundamental principles must be realized [56, 57]. The
code theoretic axiom may lead us in the right direction [2]. The existence
of a minimal volume implies violation or generalization of principles such
as Lorentz transformations, the uncertainty principle and the equivalence
principle [58, 59, 60, 61]. We argue for a new principle associated with this
minimal volume, a dimensionless constant.
It is trivially true that ratios are more fundamental than numerical values
based on arbitrary metrics. If we consider a model of quantum geometry, as
developed in various quantum gravity theories, these ratios will be geometric
as will any building blocks of a realistic model.
If this is the case, what
is the principle governing the organization of these building blocks at the
Planck scale? Or more generally, what is the principle behind charge and,
equivalently, length and volume quantization that can be deduced by looking
at the ratios of fundamental limits:
vp =
√
βvo
(41)
and,
e =
√
αqp
(42)
In Planck units:
vp = 1
vo = 1/
√
β.
(43)
For the elementary electric charge
qp = 1
e =
√
α.
(44)
So this building block ratio for charge and volume quantization is a specific
fundamental ratio based on the efficiency and symmetry principle we are
looking for.
It is generally agreed that fundamental physics relies on a fundamental
optimization principle, like we referenced in [62]: “The whole of theoretical
physics (classical and quantum) relies on a fundamental optimization prin-
ciple from which the basic equations of physics can be constructed under
the form of Euler–Lagrange equations, relating fundamental quantities, such
as energy, momentum and angular momentum. This principle of least ac-
tion becomes a geodesic principle in the framework of relativity theories (i.e.,
17
the action is identified with the proper time). The geodesic principle states
that the free trajectories are the geodesics of spacetime. It plays a very
important role in a geometric relativity theory, since it means that the fun-
damental equation of dynamics is completely determined by the geometry
of spacetime and therefore does not need to be set down by an independent
equation. Moreover, in such a framework, the action can be identified (mod-
ulo a constant) with the fundamental metric invariant, which is nothing but
the proper time itself. The action principle becomes nothing else than the
geodesic principle. As a consequence, its meaning becomes very clear and
simple: the physical trajectories are those, which minimize the proper time
itself”.
So the principle of least action is a principle related to efficiency and
optimization of resources and has a defined meaning at a classical level; the
minimal path. That is, the action is taken as a stationary value. Here we have
a clear first-principles understanding of “efficiency”. A similar optimization
principle can be found in computational systems in the principle of least
computational action [63]. Putting together the least action and relativity
principles, we are lead to a principle of maximally efficient ratio. However, we
are interested in discrete spacetime implied by quantum gravity. Accordingly,
at the quantum level, the classical notion of a sharp trajectory followed by
a physical system is rejected, leaving an opportunity. Specifically, the ratios
defining the discrete geometry at the level of quantum gravity. To make
this principle of maximally efficient ratio precise, we will define the discrete
substrate of quantum gravity. Before doing so, let us articulate the principle
in the context of discrete quantum geometric evolution:
↪→ The evolution of a system is such that it uses the most efficient ratio
relating its indivisible building blocks.
In section 2.1, we considered a model for a particle interacting with a
discrete substrate of the gravitational field, a fixed spin network from LQG.
The interaction is given by equation (1). The interaction term takes this
form because the relevant Hilbert space depends on the wave functions at
the vertices. From this, we derive the dynamics by implementing a unitary
evolution with a quantum random walk. Here we will go further to propose
that the transition from a node vm to a node vn is guided by one action
S(vn, vm). Equation (1), in fact, implies a Laplacian operator for the graph
because we have a discrete structure that derives energy values. Classically,
this action is a function that integrates the two steps leading from vm to vn,
i.e., between the two discrete times. In order to account for such a discrete
substrate, the action must be considered probabilistically. From random
walks on a graph theoretic spin network, we derive the probability P (vn)
for a particle ending up on an arbitrary vertex vn along with the transition
probabilities P (vn, vm) for transitions from vm to vn. That is, the probabil-
ities are used to construct the operators for unitary evolution. Considering
the action probabilistically, the average action is calculated, which can be
achieved locally by
∑
vm
P (vn, vm)S(vn, vm) and globally in the spin network
18
Γ by considering that random walks on arbitrary vertices generate a global
average action SΓ
SΓ =
∑
vn
∑
vm
P (vn)P (vn, vm)S(vn, vm).
(45)
This is the total average action. To find the most efficient ratio between
spacetime and particle building blocks, we must look for the probabilities
that minimize the action. This is intuitive because probabilities are ratios.
And the sum above implies the existence of a set of probabilistic matrices.
It is interesting to note that both the Pauli and Dirac matrix formalisms can
be reduced to binary matrices. The highest probability non-trivial eigenval-
ues for any n × n binary matrix are φ and −1/φ. See Appendix (A). The
precise definition of “non-trivial” depends on the dimension. At small dimen-
sions (less than 10), the domination of φ is very clear. After dimension 50,
additional “trivial” values can be the most probable with φ still holding a
relatively high rank. Naturally, when the dimension goes to infinite, only 0,
the singular trivial value, has a significant probability, all other eigenvalues
going to a zero measure.
The efficiency of discrete quantum geometry can be exploited by looking
at which of those matrices describe realistic evolution and minimize the av-
erage total action. In other words, find the matrix generating the minimal
action using approaches which can be implemented by a partition function
in the language of statistical mechanics or a quantum mechanics path inte-
gral formalism. At this level, it is the same object. So this is an efficiency
and optimization action principle that can work for quantum gravity and
for discrete versions of other force quantization models. Further investiga-
tion is planned to make clear how to derive, from the action principle, (45)
the building blocks for the charge and volume quantization equations (43)
and (44). However, for now, the above evidence implies that these founda-
tional matrices are deeply related to the golden ratio. See Appendix (A).
The code theoretic physics view, which implies an information theoretic uni-
verse, indicates that the choice of fundamental ratios and building blocks
must be guided by the notion of efficiency or computational least action [63].
This leads naturally to the golden ratio with its powerful properties of being
deeply associated with icosians [64], gauge symmetry physics, closure un-
der multiplication and division and the crucial fractal self-similarity quality
where
A
B
=
A+B
A
, allowing for maximally connected quantum-topological
non-deterministic graph codes. Interestingly, because symbolism traffics in
the quantity called information or meaning and, considering the principle of
efficient language [2], we see that the most principle may be one that takes
into account the rank of geometric meaning, which can be considered as com-
plexity of order. For example, in the entropic phase of some binary alloy, we
can have a high entropy phase with low complexity of order when order is
low. And, at the low entropic phase, we have high oder at the crystal phase
but low complexity rank because the network of objects is a homogeneous
arrangement with all near neighbor distances being equal. Code theoreti-
19
cally, this is an ordering of just one spatiotemporal symbol, which defines it
as a non-code. The maximum rank of complexity order is at the non-zero
limit of spatiotemporal symbols, which is 2. The only way to achieve this
in a network is by compositing a network of 1D two-letter codes known as
Fibonacci chains make of spacings that are 1 and the golden ratio.
↪→ If reality is code theoretic, its purpose is to efficiently express meaning,
such as physical information, with its ultimate conserved quantity−quantized
actions of the evolving substrate. Specifically, syntactically free binary choices
in the self-emergent code theoretic network. Efficiency is achieved by (1) op-
erating as a neural network code that generates maximal meaning from bi-
nary actions and (2) strategically placing those choices for maximal meaning
(generally, physical meaning) according to syntactically free code choices.
4.3 Golden ratio
From the information theoretic and code theoretic views, particles, mo-
tions, interactions and spacetime geometry are language based. This new
paradigm implies a non-arbitrary code operating at the Planck scale. We
suggest the problem of quantum gravity is the search for the discovery of this
non-arbitrary and efficient code of nature. With respect to physical codes,
much progress has been made in understanding quasicrystalline codes. The
most famous is the DNA code [65, 66], a golden ratio based quasicrystal
biomolecule. Quasicrystal codes in materials science have also been studied
[15, 42, 43, 67, 68, 69, 64, 70].
In 1944, Schrödinger predicted that the molecule encoding life would be
a “quasiperiodic crystal” before the term quasicrystal was coined in the 1970s
[71]. Amazingly, this was well before the precise molecular identification by
Crick and Watson [66]. Each of the two DNA strands is a 5-periodic helix
exactly expressed with the golden ratio. The two strands offset along the
shared helical axis by two sequential Fibonacci numbers closely approximat-
ing the golden ratio. DNA has rotational symmetry but not translational
symmetry−one of the defining characteristics of quasicrystals. DNA is a 3D
network of deep and narrow double well potentials allowed by golden ratio
based atomic organization. This aperiodic discrete energy landscape is part
of what defines DNA code, where efficiency is achieved via the maximum spa-
tiotemporal restriction of atoms without being fully restricted, as opposed to
a crystal wherein all energy wells of the same size are occupied in each atom
is locked into an EM trap. That is, a quasicrystal is a network of double well
potentials with some wells occupied and others vacant, wherein spatiotempo-
ral freedom of the atoms approaches the non-zero limit. For example, unlike
a crystal, the assembly rules for a quasicrystal allow self-organization or con-
struction choices within the rules that are not forced, i.e., non-deterministic.
And dynamically, the orderly pattern of vacant energy wells allows particle
dynamics not possible in crystalline or amorphous phases of matter. Due
to the fractal self-similar quality where
A
B
=
A+B
A
, the non-zero limit of
spatiotemporal freedom can only be achieved via golden ratio spacing ratios
20
between energy wells.
A quasicrystal is a structure that is ordered but not periodic. It has long-
range quasiperiodic translational order and long-range orientational order. It
has a finite number prototiles or “letters”. And it has a discrete diffraction
pattern indicating order but not periodicity. Mathematically, there are three
common ways of generating a quasicrystal: the cut-and-project method (pro-
jection of an irrational slice of a higher dimensional crystal) [15], the dual
grid method [15], and the Fibonacci grid method [42, 43]. Finite quasicrys-
tals can be constructed by matching rules. Quasicrystals were discovered in
nature via synthesis only in 1984 [72].
A hallmark and general characteristic of the 300 or so quasicrystals phys-
ically discovered is the golden ratio. Most of these quasicrystals are pro-
jections or subsets of the Elser-Sloane quasicrystal [65], which is a cut-and-
project of the E8 lattice using the angle arctan
(
1/φ3
)
≈ 13.28 and or the
angle π/4− arctan
(
1/φ3
)
≈ 22.24. The simplest quasicrystal possible is the
two length Fibonacci chain, as 1 and 1/φ.
The Penrose tiling, a 2D quasicrystal, is a network of 1D quasicrys-
tals. 3D quasicrystals in nature, such as a 3D Penrose tiling (Ammann
tiling) are networks of 2D quasicrystals, which are each networks of 1D
quasicrystals−generally Fibonacci chains. So the irreducible building block
of all quasicrystals are 1D quasicrystals. The “letters” of these 1D spatiotem-
poral codes are lengths between vacant or occupied energy wells. And a 1D
quasicrystal can have any finite number of letters. However, the minimum
is two. The Fibonacci chain is the quintessential 1D quasicrystal. It pos-
sesses two lengths related as the golden ratio. There are an infinite number
of 2-letter 1D quasicrystals. However, maximally code efficient quasicrystals
in higher dimensions that utilize only two letters must be constructed from
Fibonacci chain 1D quasicrystals with letters being 1 and the inverse of the
golden ratio as uniquely generated by a cut-and-project of the Z2 lattice, in
the direction of arctan
(
1/φ3
)
≈ 13.28 degrees from the diagonal direction of
the unit cubic cells.
Returning to the principle of efficient ratios and quantization conditions
(30) and (38), we note the complementary role of the dual space in the case
of discrete structures like DNA and quasicrystals in general.
In (30) and
(38), this relationship is obvious9. This dual space can be derived from the
first object by generating its diffraction pattern. The details of the duality
and quantization condition depend on the specific physics considered. The
diffraction pattern is quantized in relation to the distance between physical
objects in physical space. The interesting case is when distances are irrational
numbers such that the ratios allow aperiodic order evidenced by the discrete
Bragg peaks in the dual space as the diffraction pattern. Again, when the
irrational number is the golden ratio, values are closed under multiplication
and division and the ratio itself is fractally self-similar, which allows the max-
imally connected or densest network of 2-letter quasicrystals to be organized
9Let L be a lattice. All lattices L have an associated dual lattice L∗, the set of vectors
−→y whose scalar products with the vectors L, −→x are integers, −→x .−→y = n.
21
in dimensions 2D to 4D−the convergence of networks of 1D Fibonacci chain
codes into higher dimensional quasicrystal codes.
↪→ Let us bring together some of the above ideas to support the conjunc-
ture that the coupling constant at the Planck scale is 1/φ3 −the quantization
of charge and the proposed discretization of spacetime. Simplex integers, as
shape-numbers, are the fundamental building blocks of the QSN quasicrystal
(a 3D network of Fibonacci chains) [14]. The principle of efficiency of ratios
or the more general principle of efficient language states that fundamental
ratios allow maximally efficient geometric codes, specifically quasicrystals.
We presented the prominence of the golden ratio in two important natural
codes−the DNA quasicrystal and metallic quasicrystals.
Next, we shall present some results in fundamental physics that may be
clues that this direction is correct.
4.4 Golden ratio in physics
In this section, we review results that come from different fields of physics.
All indicate the preeminence of the golden ratio. These results suggest that,
with theoretical and experimental advances, the role of the golden ratio will
become more clear.
4.4.1 Atomic physics−the hydrogen atom
The hydrogen atom is important because it is the simplest and most abundant
element, having a single electron and nucleon. It serves as the foundation
for all atomic theory. With the hydrogen atom, we have a case where the
electron and atomic charge are identical, allowing us to isolate fundamental
atomic structure ratios to test for the principle of efficient ratios. First, let
us consider the hydrogen spectral series. When one electron goes from a
higher to lower energy state, spectral emission occurs. The wavelengths of
emitted/absorbed photons is given by the Rydberg formula in vacuum
1
λvac
= R
(
1
n21
− 1
n22
)
,
(46)
where R is the Rydberg constant
R =
α2mec
4π~
=
α2
2λe
,
(47)
and n1 and n2 are integers greater than or equal to 1 such that n1 < n2
corresponding to the principal quantum numbers of the orbitals occupied
before and after the quantum leap. For example, with n1 = 1, we have the
Lyman series [73], which we can rewrite as
λvac =
1
R
(
n22
n22 − 1
)
=
2λe
α2
(
n22
n22 − 1
)
,
(48)
22
where λe is the Compton wavelength of the electron. It gives for n2 = 2. For
example, 1.2156× 10−7m. If we use an approximation as α =
1
20φ4
λvac = 800φ
8λe
(
n22
n22 − 1
)
= 800(13 + 21φ)λe
(
n22
n22 − 1
)
,
(49)
we get 1.2158× 10−7m. As expected, the first excited state of the hydrogen
atom is φ-based [74]. When we add corrections of the energy levels of the
hydrogen atom due to relativity theory and spin-orbit coupling, the hydrogen
fine structure shows up. The fine structure is α-based and φ-based.
4.4.2 Standard Model: Neutrino Mixing and the Cabibbo angle
The first evidence for physics beyond the standard model (SM) of particle
physics comes from experimental neutrino oscillations. See [75] and references
therein for a review. One case of experimental evidence is the detection of
solar neutrinos. Only electron neutrinos are emitted by the sun. However,
only about 30 percent of the number predicted by theories that explain how
the sun works are actually measured. This disagreement was resolved by an
improved understanding of neutrino physics. Specifically, electron neutrinos
are converted into muons and tau neutrinos. This is in agreement with
various experiments. Neutrino oscillation is a verification that neutrinos are
massive, which is in disagreement with the SM, where neutrinos are expected
to be massless. So the understanding from experiment is that there is a lepton
mixing matrix U−the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix
UPMNS , which relates the basic SM neutrino states, νe, νµ, ντ , related with
the electron, muon and tau to the neutrino mass states ν1, ν2, and ν3 with
masses m1, m2, and m3.
 νe
νµ
ντ
 =
 Ue1 Ue2 Ue3
Uµ1 Uµ2 Uµ3
Uτ1 Uτ2 Uτ3
 ν1
ν2
ν3

(50)
This is similar to the usual CKM (Cabibbo–Kobayashi–Maskawa) quark
mixing matrix. The lepton mixing matrix UPMNS has large uncertainties.
Despite the lack of a deep knowledge of UPMNS , the aforementioned data
leads to an approximation of the first order, which allows for a theoretical
description. There are different patterns of the values and parameterizations
for this matrix that are in agreement with experimental data10. We suspect
that the most powerful mixing matrix models are golden ratio based, as shown
in [76] and was further developed in [77, 78, 79, 80], the so called golden ratio
prediction. But more important than the exact values of the numbers that
constitute the matrix is the idea that they can be derived via first principles
from the symmetry breaking relationships of fundamental symmetries at the
unification scale relating to hyper-dimensional lattices and their associated
Lie algebras. For example, one can use the rotational icosahedral group [78],
10Different constructions of this matrix can be found in [75].
23
the alternating group of five elements A5 [79] or A5 ×Z5 ×Z3 [80]. And one
can recover gauge symmetry physics from an icosian construction of the E8
lattice [55]. Of course, by definition, all these groups are golden ratio based.
In this context, we have the important concept of quark-lepton comple-
mentarity which proposes that one parameter of the UPMNS matrix, the
solar angle (θ12), is related with one parameter of the quark matrix mixing,
the Cabibbo angle (θc), by
θ12 + θc =
π
4
.
(51)
We conjecture that the Cabibbo angle is the universal parameter which con-
trols the entire structure of fermion masses and therefore appears in many
places, such as mass ratios and mixing parameters. Recent data and theo-
retical proposals indicate the possibility of θc ≈ 13.28 [75, 77, 81, 82, 83].
This fits well with the golden ratio prediction [77]. The findings of these
authors is not surprising because, as mentioned earlier, this angle appears
in the cut-and-project of E8 to 4D, breaking the crystal symmetry of E8 to
icosian associated H4 symmetry. Similarly, this angle is required to bring the
projection of E8 to 3D into H3 symmetry.
Both the 4D and 3D projections are golden ratio based quasicrystals. Sim-
ilarly, the Weinberg angle (θW ) of the electroweak interaction is an unknown
value experimentally suggested to be slightly less than ∼ 30◦. In practice,
what is measured is the quantity sin2(θW ). In a study of parity violation in
Møller scattering a value of sin2(θW ) = 0.2397 +− 0.0013 was obtained at mo-
mentum transfer, Q = 0.16GeV/c, establishing experimentally the “running”
of the weak mixing angle. LHCb measured, in 7 and 8 TeV proton-proton
collisions, an effective angle of sin2(θW ) = 0.23142, though the value of Q for
this measurement is determined by the partonic collision energy. The mean
of these measured values deviates from 1/φ3 at five 10,000ths. The measure-
ment uncertainty of 0.2397 is 0.0013, the 1,000ths part, so our conjecture is
well within the experimental margin of error.
To understand the origin of fermion masses and mixtures in the SM,
one must turn to the family of symmetries that restrict the form of the
mass generator matrices and models the hierarchies and mixtures through
small symmetry breaking perturbations. Recent investigations, such as in the
above references, indicate that icosahedral symmetry acts as an important
scale unifier of the electromagnetic, weak and strong forces (∼ 1025eV ). If
flavor has an underlying simplicity associated with unification symmetry, it
would be productive to attempt recognition of it from neutrinos instead of
from charged leptons and quarks, which have more complicated structures.
This suite of experimental and theoretic evidences for massive neutrinos and
our theoretical proposal of quasicrystal symmetry in unification scale physics
supports the conjecture of a quasicrystalline spacetime and particle structure
at the Planck scale.
We have found fundamental golden ratio values in the structure of E8
itself. The root vector polytope of E8 is the Gosset polytope11. One of the
11The E8 root system contains 240 root vectors spanning R8, each with the same length
24
building blocks of E8 is the 3-simplex, a regular tetrahedron. For example,
the E8 lattice and the Gosset polytope can be constructed, if one chooses,
solely from tetrahedra. Two face-kissing tetrahedra in E8 live in the same
4D subspace of R8. And certain pairs share one vertex that is the centroid of
the Gosset polytope while their other vertices are the vertices of the Gosset
polytope. The following table 4.4.2 shows the list of the cosine values of the
angles, most of which are golden ratio expressions. Notice that the angle
ArcCos 14 can be written as π/3 − ArcCos −
3φ−1
4
. It is worth noting that
every eight of these tetrahedra live in the same 3-space (grouped into two
dual orientations) and can be extended to an A3 lattice. Since, the Gos-
set polytope can be generated by composition of all the transformations of
one tetrahedron through these angles, the E8 lattice can be generated via
composition of the rotational transformation of the A3 lattices through these
angles.
1
2
7
(4φ−2)2 −
3
8φ−4
1
4
1
2(2φ−1)
2φ−1
4
The Cartan matrix of the E9 Lie affine algebra and the E8 Lattice is:
CE9 =

2 −1
0
0
0
0
0
0
0
−1
2 −1
0
0
0
0
0
0
0 −1
2 −1
0
0
0
0
0
0
0 −1
2 −1
0
0
0
0
0
0
0 −1
2 −1
0
0
0
0
0
0
0 −1
2 −1 −1
0
0
0
0
0
0 −1
2
0
0
0
0
0
0
0 −1
0
2 −1
0
0
0
0
0
0
0 −1
2

Its 9 eigenvalues are 4, φ+ 2, 3, φ2, 2, 2−φ−1, 1, φ−2, 0 and form the diagonal
matrix ΛE9 . The last eigenvalue is 0 because the Lie algebra is affine. Five
of the eigenvectors (columns of VE9) have integer coordinates and the four
other have coordinates which are 0 or a power of φ. We note ϕ for the inverse
of φ.
VE9 =

−1 φ −1 ϕ −1 −ϕ 1 −φ 1
2 −φ2 1 −ϕ2 0 −ϕ2 1 −φ2 2
−3 φ2
0 −ϕ2 1 ϕ2
0 −φ2 3
4 −φ −1 ϕ
0 ϕ −1 −φ 4
−5 0
1
0 −1 0 −1 0 5
6 φ
0 −ϕ 0 −ϕ 0 φ 6
−3 −1 0
1
1 −1 0
1 3
−4 −φ −1 −ϕ 0 ϕ
1 φ 4
2
1
1
1
0
1
1
1 2
 ΛE9 =

4 0 0 0 0 0 0 0 0
0 φ+2 0 0 0 0 0 0 0
0 0 3 0 0 0 0 0 0
0 0 0 φ2 0 0 0 0 0
0 0 0 0 2 0 0 0 0
0 0 0 0 0 2−ϕ 0 0 0
0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 ϕ2 0
0 0 0 0 0 0 0 0 0

(52)
DE9 =
1
120

1
0
0
0
0
0
0
0
0
0 12ϕ2 0
0
0
0
0
0
0
0
0
20
0
0
0
0
0
0
0
0
0 12φ2 0
0
0
0
0
0
0
0
0
30
0
0
0
0
0
0
0
0
0 12φ2 0
0
0
0
0
0
0
0
0
20
0
0
0
0
0
0
0
0
0 12ϕ2 0
0
0
0
0
0
0
0
0
1

(53)
and forming the vertices of the Gosset polytope.
25
V −1
E9
= DE9VE9
(54)
CE9 = VE9ΛE9DE9VE9 = VE9ΛE9V
−1
E9
(55)

2 −1 0 0 0 0 0 0 0
−1 2 −1 0 0 0 0 0 0
0 −1 2 −1 0 0 0 0 0
0 0 −1 2 −1 0 0 0 0
0 0 0 −1 2 −1 0 0 0
0 0 0 0 −1 2 −1 −1 0
0 0 0 0 0 −1 2 0 0
0 0 0 0 0 −1 0 2 −1
0 0 0 0 0 0 0 −1 2
 =

−1 φ −1 ϕ −1 −ϕ 1 −φ 1
2 −φ2 1 −ϕ2 0 −ϕ2 1 −φ2 2
−3 φ2
0 −ϕ2 1 ϕ2
0 −φ2 3
4 −φ −1 ϕ
0 ϕ −1 −φ 4
−5 0
1
0 −1 0 −1 0 5
6 φ
0 −ϕ 0 −ϕ 0 φ 6
−3 −1 0
1
1 −1 0
1 3
−4 −φ −1 −ϕ 0 ϕ
1 φ 4
2
1
1
1
0
1
1
1 2


4
φ+2
3
φ2
2
2−ϕ
1
ϕ2
0


−1 φ −1 ϕ −1 −ϕ 1 −φ 1
2 −φ2 1 −ϕ2 0 −ϕ2 1 −φ2 2
−3 φ2
0 −ϕ2 1 ϕ2
0 −φ2 3
4 −φ −1 ϕ
0 ϕ −1 −φ 4
−5 0
1
0 −1 0 −1 0 5
6 φ
0 −ϕ 0 −ϕ 0 φ 6
−3 −1 0
1
1 −1 0
1 3
−4 −φ −1 −ϕ 0 ϕ
1 φ 4
2
1
1
1
0
1
1
1 2

−1
(56)
Equation 54 gives a simple expression of the inverse of the eigenvector
matrix. Equation 55 states the diagonalization of the Cartan matrix. Equa-
tion 56, where the formal notation bve expresses the diagonal matrix form of
the vector v, (here bve = ΛE9), is unexpectedly a new formula not previously
reported in the literature.
Our intent was to look for a result in E8 similar to that found by Sirag
in the relationship between E7 and CE+7
[84]. We analyzed the spectrum
of the CE9 Cartan matrix to obtain our result− a systematical study of the
spectrum of all Cartan matrices though the Dynkin diagram of all simple Lie
algebras and their affine extensions. We found that the golden ratio appears
in the eigenvalues, outside of the exceptional E9 = E+8 , only for SU(5k),
where k is an integer. This crucial emergence of φ from the E8 lattice may
be the key explanation of three other facts:
• McKay correspondence [84] between E8 Lattice and double icosahedral
group
• Correspondence between icosians and E8 Lattice
• Discrete Hopf fibration S3 → S7 → S4 [85]
We can now formulate our ontological theorem:
Theorem 4.1. E8 generates the golden ratio.
Proof. The proof is given in equations 55 and 56.
Proposition 4.2. We propose the following ontological chain: B ⇒ T ⇒
E8 ⇒ φ⇒ D⇒ QC ⇒ h92 ⊕ a7 ⇒ QSN
26
• The bits 0 and 1, or equivalently, the sphere S0 emerges ab initio
• The trits, the integer, the matrix structure and the structured division
algebra emerge from the bits, as described in Annex A (which also sug-
gests a direct path to φ)
• The last division algebra is the octonions. Its integers, the Cayley in-
tegers, emerge as the E8 lattice
• From theorem 4.1, the golden ratio φ emerges from the E8 lattice
• Extending the integer by φ, our ring D of Dirichlet integers and its
structural tensor products quantize all the needed algebras
• From the E8 lattice, and in a simple way by using explicitly φ or in a
more subtle way implicitly, emerges the Elser-Sloane quasicrystal
• Spacetime geometry emerges from this structure on which a gauge al-
gebra is naturally inherited from the E8 Lie group
• A 3D representation made of only regular tetrahedra forming a qua-
sicrystal code made of Fibonacci chains, the QSN, emerges to support
the geometrical reality
Proof. We do not yet have a proof of this statement. However, each of
the eight steps in the proposed ontological chain is mathematically sound
and supported by evidence. The logical flow is strong. Accordingly, this is
more than a conjecture. It is a deduced hypothesis. Readers are invited to
think critically. This mathematical first-principles based emerging theory is
incomplete but yet evidenced as the right direction via the physical evidence
and equations presented herein.
4.4.3 Quantum theory
An important result in foundational quantum mechanics is that of Hardy
[86], which can be considered the best version of Bell’s theorem [87] because
of its simplicity and logical/mathematical rigor. While Bell’s proof of the
impossibility of Einstein, Podolsky and Rosen’s view is based on statistical
predictions and inequalities, Hardy’s test is a test of non-locality without
inequalities.
This test considers two qubits prepared in an entangled state, wherein
each is sent in the opposite direction by a source in the middle of two detec-
tors. One electron goes to the detector at left and the other to the right. The
entanglement can be on polarization, spin components or momentum. The
detectors have a switch between two positions that can be settled randomly
and independently after the particles are emitted by the source and before it
arrives at the detectors. We call the detector on the left L and on the right
R. So they can have two positions L1, L2 and R1, R2. The choice for the
switches are experiments with two possible outcomes, say +− 1, that depend
on entangled particles. Just four of the possible experiments are needed to
27
get the result of maximal non-locality, which we will present herein. These
four experiments are fixed positions of the detectors to L1, R1 (let us call
this experiment A); L1, R2 (experiment B), L2, R1 (experiment C) and L2,
R2 (experiment D). Running these experiments many times, shows that, in
experiment A, L1 and R1 being both −1 never occurs. In experiment B, with
L1, R2, and experiment C, with L2 and R1, both being +1 never occurs.
And in experiment D, L2 and R2 being both +1 occurs sometimes. Accord-
ing to local determinism, suppose that the atom going left has instructions
to come out +1 if it encounters position L2. According to experiment C, if
L2 gives +1, R1 can’t give +1. So it must give −1. According to experiment
A, if R1 gives 11, then L1 must give +1. And according to experiment B,
if L1 gives +1, then R2 must give −1. This instruction set is the only one
consistent with predictions A, B, and C and with L2 giving the result +1.
It shows that, in situation D (for which the detectors are set to L2 and R2),
whenever the left atom comes out +1, the right qubit must be −1. However,
according to quantum mechanics, there is a probability for the right qubit to
come out as +1. The exact probability P for L2 and R2 being both +1 is
P = φ−5.
(57)
This result is in strict accordance with both quantum mechanics experiment
[88]. That is, the probability for the simplest quantity relationship, two,
of the simplest fermions, electrons, in the simplest possible dynamic and
geometric relationship (moving apart at equal velocities on the same line)
is a probability of the golden ratio to the power of -5. This result is deep
and important because, in some sense, this is the probability for maximal
non-locality in nature.
4.4.4 Chaos theory: KAM theorem
The Kolmogorov, Arnold and Moser (KAM) theorem12, which is a result in
classical mechanics of the study of dynamical systems. It is about the per-
sistence of quasiperiodic motions under small perturbations. A dynamical
system can be expressed in configuration space in terms of action-angle coor-
dinates consisting of action and angle variables in terms of a torus defined by
its angle variables. Quasiperiodic orbits represent integral motion on a torus
and, if it is integrable, there is a constant or invariant of the motion associ-
ated with the torus that leads to the term invariant tori. What happens to
the invariant tori as the nonlinearity of the system increases? Say we have a
system with Hamiltonian H = Ho+εH1 + ..., where Ho is the non-perturbed
Hamiltonian and H1 the non-linear perturbation allowing ε to mediate the
force of the perturbation. For sufficiently small perturbations, virtually all
tori are preserved.
Consider the frequency of motion around each angular variable of a torus.
As a point moves, it rotates around the tube while revolving around the torus
axis. If we take the ratio of these frequencies, we get a quantity called the
12There are many references in the context of chaos theory. For example, see [89, 90].
28
winding number, σ = ω1ω2 . The KAM theorem shows that the tori most easily
destroyed are those with rational winding numbers, while almost all other
orbits (those with irrational winding numbers) are preserved. What happens
if we increase the perturbation? The KAM theorem itself doesn’t explicitly
say, but derivative work does [91]. After the rational winding number tori
go chaotic, the irrational tori eventually break up also, even though they are
significantly more robust under perturbation. As the perturbation grows,
more irrational tori go unstable. Most interestingly, the tori go unstable in
order of the degree of irrationality of their the winding numbers. This is
where golden ratio shows up as a physical limit. Mathematicians consider
the golden ratio to be the most irrational of numbers because the rank of a
number’s irrationality is based on the speed of convergence of its continued
fraction expression. The continued fraction expression for the golden ratio
uses only the integer 1, making it the most “difficult” or slowest number to
approximate with rational numbers. It approaches the limit slower than any
other continued fraction expression.
It is at this golden ratio based winding number where physical vortices
are most stable under increasing perturbations. This is the special limit
related to how small denominators correspond to the growth of ε, allowing it
to be minimal. It is the state where the conditions for the KAM theorem are
most easily satisfied. In summary, golden ratio based gravitational, fluidic
and other dynamical vortices are the most stable and therefore the most
statistically probable physically abundant in nature.
4.4.5 Black hole physics
Black hole physics is a good laboratory for testing the limits of general rela-
tivity and quantum mechanics and for putting the two limits together in the
study of quantum gravity theories. Accordingly, let us look for clues about
Planck scale golden ratio physics in black hole equations. The first result is
within the context of non-arbitrarily limiting and manipulating the equations
of general relativity [92, 93]. This is a classic general relativity result with no
relation to quantum mechanics. A rotating black hole can have a transition
of phase between positive specific heat capacity and negative. The transition
is based on the golden ratio when the ratio of angular momentum J and mass
M is kept constant in the equation. In this case, φ is the point where a black
hole’s modified specific heat changes from positive to negative
M4
J2
= φ.
(58)
This non-arbitrary manipulation is analogous to the artificial but non-arbitrary
setting of the velocity of the electron to zero in order to derive the electron
rest mass. That is, although there is no such thing as an electron at rest, the
non-arbitrary but non-physically realistic manipulation of the electron equa-
tion provides a deep fundamental understanding of a limit in nature. Another
result in classical general relativity which indicate special role of golden ratio
in underlying geometry of spacetime was reported recently in [94]. In this
29
case the golden ratio appears in the rather simple field of Schwarzschild black
holes with a cosmological constant. Let us now combine quantum mechanics
with general relativity by considering the loop quantum gravity approach to
quantum gravity [1]. Using this work done on the microstructure of space-
time, we compute black hole entropy. In a simplified framework, the isolated
quantum horizons formulation of loop quantum gravity, we can derive the
lower bound of black hole entropy [28]
e
8π~GS
kA ≥ φ,
(59)
where S is the entropy, A the black hole area and k the Boltzmann constant.
In section (3.1), we go further, using arguments from information theory to
fix the loop quantum gravity parameter, which we can wright as
2πγ = φ.
(60)
5 Discussion
The (1) principle of least action and Noether’s second theorem, (2) gauge
symmetry, (3) general relativity and (4) the implication of spacetime dis-
cretization at the Planck scale strongly suggest an underlying new principle
based on the efficiency of ratio, where, mathematically, the golden ratio is
a special limit in the universe of all ratios and where, experimentally, it is
observed in the fundamental physical equations and limits of nature.
Assuming that reality is discrete at the Planck scale, a logically satisfying
explanation for the observed quantization of charge leads us to ask, “What
is the most efficient organizing principle for how dynamical charge space is
constructed?”. A discrete reality is ultimately numerical, although not neces-
sarily digital. It may be based on self-referential volumetric shape-numbers,
as 3-simplex integers forming a quasicrystalline graph drawing based descrip-
tion of quantum gravity.
This multifaceted overall argument strongly implies that reality is code
theoretic. The natural geometric languages that we know, such as DNA and
other quasicrystals, are defined by the golden ratio, serving as experimental
cases where the golden ratio is used by nature in geometric codes and inspiring
the axiomatic theory that the physical universe behave with optimal digital
efficiency by exploiting our foundational dimensionless constant ultimately
residing at the Planck volume of space. This golden ratio based running
constant starts at the quantum gravity scale and goes upwards fractally (all
quasicrystals are fractal). The well-known self-similarity properties of the
golden ratio and quasicrystals explain the observed fractality of nature at
all scales [62, 96, 97, 98, 99, 100, 95], and provide a posteriori credit to the
Dodecahedron-Icosahedron doctrine [101].
Note that, assuming the discreteness of spacetime at the Planck scale,
we face a problem similar to one at the atomic scale: How do these build-
ing blocks self-organize to build more complex structures? How does order
emerge? A crystalline structure would be a naive answer because crystals are
30
not codes. Conversely, quasicrystalline structures are codes with non-local
long range order. Syntactical freedom in their static and dynamic ordering
rules provide the freedom necessary for quantum non-determinism and the
ability for physicists to recover the gauge symmetry unification equations of
particles and forces that express in the code due to the mapping of Lie alge-
bras to the generating hyper-lattices from which quasicrystals are created.
6 Conclusion
We have presented a compelling idea that we can apply the results and
tools from quantum information and quantum computation to a quantum
spacetime code theoretic view using algebraic graph drawing formalism. We
considered a DQW of a quantum particle on a quantum gravitational field
and studied applications of related entanglement entropy. This memory time
based entanglement entropy drives an entropic force, suggesting a unified pic-
ture of gravity and matter. Following this, we proposed a model for walker
positions topologically encoded on a spin network, which can easily be re-
expressed using twistors. This results in anomaly cancellation because the
particles are no longer points but Planck scale voxels, as tetrahedral units of
spacetime.
Later in this document, we presented a review of results that serve as
compelling clues about an underlying organizing principle related to charge
quantization and the golden ratio. The appearance of the golden ratio in
black hole physics, the indication from the research of physics beyond the
standard model and other evidences suggest there is quasicrystalline geome-
try at the unification scale and allow for a logical argument that nature has a
limit of non-locality in a two particle quantum system. Again, an element of
evidence for quasicrystalline structure at the quantum gravity scale. Due to
the inherent fractal scale invariance of golden ratio based quasicrystals, this
new efficiency and organizing principle shows up at other scales, such as the
atomic scale, where, for example, quasicrystalline structure is present in the
hydrogen atom and DNA. This unifying principle connecting the discussed
points correlates nicely with derivative works from the KAM theorem, where
it is trivially true that literally all systems in nature are both propagating
and rotating simultaneously−forming propagating gravitational and electro-
magnetic vortices or extruded tori. The quasicrystalline phase is a special
limit in thermal dynamics, where spatiotemporal degrees of freedom reach
the maximum non- zero limit.
7 Acknowledgements
We also would like to acknowledge Carlos Castro Perelman for useful sugges-
tions of references.
31
A Binary Matrices
We show that all n-dimensional structures used in relativity and quantum
mechanics reduce to products of binary matrices. The quaternionic and com-
plex structures, and the Pauli and Dirac algebra are illustrated below. The
process [13] involves replacing i by a 2x2 anti-diagonal anti-symmetric “trit”
(-1,0,1) matrix and then further reducing trit matrices by substituting -1 by
2x2 anti-diagonal symmetric non-null binary matrices.
A.1 Negative numbers and complex numbers
Algebraic structures emerge when a set of elements which, (along with op-
erations) constitute our algebra, obey fundamental symmetries, eventually
expressed as algebraic rules or constraints. For example, from the set of
positive numbers, 0,1,2,3,4..., we define a new element m, outside of this
set, satisfying (m)2 = 1. So m is not 1. Indeed m = −1, and we have
defined the negative unit, which we call −1. We extend our set and de-
fine Z = N1 + Nm by using linear combinations of our old unit (1) and
our new unit (m=-1) with coefficients in the old set N. We can also use
2x2 matrices 1 =
(
1 0
0 1
)
and m = −1 =
(
0 1
1 0
)
and check that
(m)2 = 1. Therefore the eigenvalues satisfy the same constraint. The com-
plex numbers have a similar representation z = a1 + bi = a
(
1 0
0 1
)
+
b
(
0 −1
1
0
)
=
(
a −b
b
a
)
, where i satisfies (i)2 = −1. The complex struc-
ture is the multiplicative group of the Gaussian integer units {1,−1, i,−i}.
The trit to bit map µ from T = {−1, 0, 1} to M2(F2) where F2 is the bi-
nary field {0,1} is defined by µ(1) =
(
1 0
0 1
)
, µ(−1) =
(
0 1
1 0
)
, µ(0) =
(
0 0
0 0
)
and can be applied to each matrix element obtained from the
complex units by the map ν from the fourth roots of 1 toM2(T) such that
ν(i) =
(
0 −1
1
0
)
, ν(−i) =
(
0
1
−1 0
)
, ν(0) = µ(0), ν(1) = µ(1), ν(−1) =
µ(−1) to derive 4-dimensional binary matrices inM4(F2) from the combined
map ρ = µ ◦ ν.
For example, ρ(i) =
(
µ(0) µ(−1)
µ(1)
µ(0)
)
=

0 0 0 1
0 0 1 0
1 0 0 0
0 1 0 0
.
Note that in the two following subsections, the algebra of these matrices
is not the standard algebra on Mn(R).
It is also not the standard alge-
bra on Mn(F2). The operations have to be computed in R and then pro-
jected to Mn
2
({02, I2,m}) by a process described in [13], where
(
1 1
1 1
)
projects to 02.
In the alternative representation proposed on Mn(T), the
32
projection is simply realized by the sign operator: ∀ x ∈ R, sign(x) =
 −1
for x < 0
0
for x = 0
1
for x > 0
.
A.2 Quaternions and Pauli matrices
It was shown in [13] how to use 4D trit maps in M4(T) or 8D bitmaps in
M8(F2) to represent quaternions and implement the quaternion group (finite
group of degree 8), the D4 root vectors group of degree 24 and the F4 root
vector group of degree 48. We repeat here the results for quaternions to
deduce the Pauli matrices. We can define as κ the following map from the
positive unit quaternions {1, i, j, k} toM4(T) :
κ(i) =
(
ν(−i) ν(0)
ν(0)
ν(i)
)
=

0
1 0
0
−1 0 0
0
0
0 0 −1
0
0 1
0

κ(j) =
(
ν(0)
ν(1)
−ν(1) ν(0)
)
=

0
0
1 0
0
0
0 1
−1
0
0 0
0 −1 0 0

κ(i) is named yin and κ(j) is named yang in [13] because they generate
the ring of 24 Hurwitz integers [102] with the modified trit matrix algebra
projecting back the result by applying the sign function and its geometric
dual. Naturally, the map κ is completed because the matrices satisfy Hamil-
ton’s quaternion relations ijk = i2 = j2 = k2 = −1. κ(k) = κ(i)κ(j) =
(
ν(0)
ν(−i)
ν(−i)
ν(0)
)
=

0
0
0
1
0
0 −1 0
0
1
0
0
−1 0
0
0
.
Quaternions are defined as binary matrices by applying µ to each element.
This generates the map λ = µ ◦ κ from {1, i, j, k} toM8(F2) and
λ(i) =

0
µ(1)
0
0
µ(−1)
0
0
0
0
0
0
µ(−1)
0
0
µ(1)
0
 =

0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
 ,
λ(j) =

0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 1 0 0 0 0 0
 , λ(k) =

0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 1 0 0 0 0 0 0
1 0 0 0 0 0 0 0
 .
(61)
Pauli matrices σ1 =
(
0 1
1 0
)
, σ2 =
(
0 −i
i
0
)
, σ3 =
(
1
0
0 −1
)
can
be built by applying the map ρ to each of their elements {0, 1,−1, i,−i}:
33
Λ(σ1) = ρ
(
0 1
1 0
)
=
(
ρ(0) ρ(1)
ρ(1) ρ(0)
)
=

0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
 ,
Λ(σ2) =

0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0
0 0 0 0 1 0 0 0
0 0 0 1 0 0 0 0
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
 ,Λ(σ3) =

1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 0 1 0
 .
(62)
We define a new imaginary:
ı = ρ(σ1σ2σ3) = ρ
(
i 0
0
i
)
=
(
ρ(i) ρ(0)
ρ(0) ρ(i)
)
=

0 0 0 1 0 0 0 0
0 0 1 0 0 0 0 0
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0

(63)
The quaternions λ(i), λ(j), λ(k) and the Pauli matrices Λ(σ1),Λ(σ2),Λ(σ3)
are related by: ıλ(i) = Λ(σ3), λ(j) = ıΛ(σ2), ıλ(k) = Λ(σ1). Together with
18 = ρ(12) and ı, they form the bi-quaternion units.
The Clifford algebra Cl3 also admits for basis inM8(F2) : 18,Λ(σ1),Λ(σ2),
Λ(σ3), λ(i), λ(j), λ(k) and ı.
They are illustrated below, where 0 is gray and 1 is white:
A.3 Dirac matrices
The 16 Dirac matrices γa,b, where (a, b) ∈ {0, 1, 2, 3}, γ0,0 = I16, γ0,b =
I2⊗σb, γa,0 = σa⊗ I2 and γa,b = γa,0γ0,b are given in the below table, where
0 is gray and 1 is white:
34
The same process can be extended based on λ(i), λ(j), λ(k) or by com-
bining the Dirac matrices with γ0,ı = I2 ⊗ ı, γı,0 = ı⊗ I2.
A.4 Binary matrix eigenvalues
These binary matrices have eigenvalues which are fourth roots of unity, the
Gaussian integer units. We now systematically study all eigenvalues of n-
dimensional binary matrices Mn(F2). We begin with n=2. There are 16
matrices. Their eigenvalues are, with φ = 1+
√
5
2
and ϕ = φ−1 = −1+
√
5
2
:
{0,0}, {1,0}, {0,0}, {1,0}, {0,0}, {1,0},{-1,1},{φ,-ϕ}, {1,0}, {1,1}, {1,0}, {1,1}, {1,0}, {1,1},{φ,-ϕ},{2,0}
Unexpectedly, the only non-integer eigenvalues of Mn(F2) are φ and
−φ−1. This is one of the deepest mathematical reasons for our overall the-
sis that the unique qualities of golden ratio serve as an ultimately efficient
fundamental constant of physics, expressing a ratio that is code-theoretically
more powerful than 1.
From the eigenvalues of the 512 matrices of M3(F2), the most probable
are by decreasing order 0 (32%), 1 (28%), φ (6%), −φ−1 (6%), 2 (5%), -1
(5%)... From the eigenvalues of the 65,536 matrices of M4(F2), the most
probable are by decreasing order 0 (25%), 1 (18%), -1 (5%), 2 (4%), φ (4%),
−φ−1 (4%), 1−
√
2 (1%), ... From the eigenvalues of the 33 554 432 matrices
ofM5(F2), the most probable are by decreasing order 0 (19%), 1 (11%), -1
(4%), 2 (2%), φ (1%), −φ−1 (1%), 1 −
√
2 (1%), ... They are the roots of
8,927 characteristic polynomials of degree 5. When the matrix dimension
n is grows, the most probable eigenvalues have decreasing probabilities. φ
always remains in the group of the most probable because its characteristic
polynomial has small coefficients (1 or -1). At the other end, the less probable
35
eigenvalue is n and appears only once, for the matrix made of only ones and
polynomials nxn−1 − xn. These most probable eigenvalues are the centers
of exclusion circles with no other values and with radii proportional to their
probabilities. We also see a smaller exclusion interval on the real line as
illustrated in figure 6. This repelling behavior between eigenvalues is a known
indicator of universality [103].
-2
-1
1
2
3
4
5
-2
-1
1
2
Figure 6: M5(F2) spectrum
All of the above binary structure matrices representing the quaternions,
biquaternions, Clifford algebra, Pauli matrices and Dirac matrices haveMn(F2)
sub-blocks which are in {µ(0), µ(1) and µ(−1)}. By using the nilpotent sub-
block  =
(
0 1
0 0
)
, which has 0 as double-eigenvalue, we can implement
dual integers. If we use the “golden” sub-block Φ =
(
0 1
1 1
)
, which has
φ and −φ−1 as eigenvalues, we obtain an interesting structure. Note that
in [77], the neutrino Majorana mass matrix mv =
 0 m
0
m m
0
0
0 matm
 =
(
mΦ
0
0 matm
)
, which is block-diagonal with the sub-block Φ and there-
fore the mass eigenvalues make the neutrino-mixing angle θ12 = arctan
(
φ−1
)
.
If a, b, c and d are integers, (aI2 +bΦ)(cI2 +dΦ) = (ac+bd)I2 +(ad+bc+
bd)Φ) and the Dirichlet integer (a+ bφ) can be implemented as (aµ(1) + bΦ).
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