Winding Number as Hidden Variable
Winding Number as Hidden
Variable
A
Topological Foundation for Physics from Pre-Geometric Computation
Author: Rowan Brad Quni-Gudzinas
Affiliation: QNFO Contact:
rowan.quni@qnfo.org ORCID: 0009-0002-4317-5604
ISNI: 0000 0005 2645 6062 DOI:
10.5281/zenodo.17364877 Publication Date: 2025-10-16
Version: 1.0.2
This paper proposes that physical reality emerges from a
pre-geometric substrate based on the circle \(S^1\), with its integer winding numbers
providing a deterministic foundation for physics. A universal
computation of pattern writing, evolution, and projection generates
spacetime and particles, reinterpreting fundamental constants as
geometric scaling factors. Particle mass and identity are determined by
the magnitude and prime factorization of the winding number \(n\). This framework resolves major unsolved
problems—including the hierarchy problem, strong CP problem, color
confinement, and dark matter—as consequences of topological selection
rules and resonance conditions near powers of the golden ratio.
Critically, the winding number serves as a global, topological hidden
variable, evading Bell’s theorem and providing a deterministic basis for
quantum mechanics without violating experimental results. The model
makes falsifiable predictions for gravitational wave modulation and
gamma-ray dispersion.
Keywords: Winding Number; Hidden Variables;
Topological Foundation; Pre-Geometric Computation; Quantum Gravity;
Particle Mass Ratios; Golden Ratio Resonance; Deterministic Quantum
Mechanics; Gravitational Waves; Dark Matter.
The
Foundational Crisis and the Computational Substrate
The persistent incompatibility between general relativity and quantum
mechanics, often described as the central problem in theoretical
physics, is not merely a technical hurdle but a symptom of a deeper
ontological misalignment. General relativity describes gravity as the
curvature of spacetime, a dynamic entity that responds to the
distribution of mass and energy via Einstein’s field equations, \(G{\mu\nu} = 8\pi G T{\mu\nu}\) (Einstein,
1915). In contrast, quantum field theory (QFT), which governs the other
three fundamental forces, treats spacetime as a fixed, unchanging
background—a static stage upon which quantum fields evolve and interact
(Weinberg, 1995). This fundamental conflict arises because QFT relies on
a predefined metric for defining particle states and propagators, while
general relativity makes the metric itself a dynamical variable (Misner,
Thorne, & Wheeler, 1973). Attempts to quantize gravity directly lead
to incurable infinities, signaling that the two theories are built on
mutually incompatible foundations. A review of current approaches to
quantum gravity highlights this divide, noting that string theory
introduces extra dimensions while loop quantum gravity discretizes
space, yet neither has produced definitive experimental predictions
(Rovelli, 2004). This suggests that neither spacetime nor quantum fields
are truly fundamental; instead, both may emerge from a more primitive,
pre-geometric substrate governed by computational or informational
principles, an idea gaining traction in fields like quantum information
and causal set theory (Sorkin, 2003).
A compelling candidate for this foundational substrate is the
one-dimensional circle, denoted \(S^1\). Unlike higher-dimensional manifolds,
the circle possesses unique topological properties that make it an ideal
generator of physical structure. Its first homotopy group is isomorphic
to the integers, \(\pi_1(S^1) \cong
\mathbb{Z}\), which provides a natural mechanism for
quantization: any closed path around the circle is characterized by an
integer winding number (Hatcher, 2002). This discrete spectrum can be
interpreted as the origin of quantum numbers in physical systems, such
as angular momentum in Bohr’s model of the atom (Bohr, 1913).
Furthermore, the circle supports harmonic analysis through Fourier
series, enabling the projection of one-dimensional patterns into
complex, higher-dimensional geometric forms. This dual
capacity—generating discreteness through topology and complexity through
analysis—positions the circle as a computationally complete object
capable of encoding the diversity of physical phenomena. Research in
topological quantum computing leverages similar principles, using braids
on a plane (closely related to \(S^1\))
to create stable qubits immune to local noise (Nayak et al., 2008).
This leads to the proposition of a Universal Computational Protocol,
a minimal set of operations that define the algorithmic nature of
reality. The first operation is pattern writing, the
establishment of an initial state or configuration on the foundational
manifold. The second is evolution, the deterministic
transformation of this pattern over time according to a set of rules,
analogous to a cellular automaton update function. Stephen Wolfram’s
work on cellular automata demonstrates how simple rules can generate
complex, seemingly random behavior, suggesting that the universe might
operate similarly (Wolfram, 1984). The third is projection, the
process by which the one-dimensional pattern gives rise to the observed
three-dimensional spatial geometry and its associated physical laws.
These operations collectively suggest that the universe is not governed
by immutable laws inscribed at creation, but is instead the output of a
generative computational process unfolding from simple initial
conditions, a concept echoed in the “it from bit” philosophy proposed by
John Archibald Wheeler (Wheeler, 1990).
The fundamental constants of nature—Newton’s gravitational constant
(\(G\)), the speed of light (\(c\)), and Planck’s constant (\(\hbar\))—are traditionally viewed as
intrinsic parameters of physical law. However, within this framework,
they are reinterpreted as emergent scaling factors of the geometric
projection map. They do not represent arbitrary knobs tuned by nature
but are mathematical artifacts that arise when translating information
from the compact, one-dimensional computational space to the expansive,
three-dimensional observational space. Their specific values are
determined by the details of the projection geometry, much like the
scale factor in a map projection distorts distances and areas. This
perspective shifts the goal of fundamental physics from measuring these
constants with ever-greater precision to understanding the geometric and
topological principles that dictate their values. The fine-structure
constant, approximately 1/137, remains a mystery in this regard, but its
dimensionless nature suggests it may be a pure number arising from such
a geometric ratio (Mohr, Newell, & Taylor, 2016).
The \(8\pi\) Projection Factor and Emergent
Gravity
The numerical factor of \(8\pi\)
that appears in the Einstein field equations, \(G{\mu\nu} = 8\pi G T{\mu\nu}\), is
conventionally treated as a normalization constant chosen for
convenience to match Newtonian gravity in the weak-field limit
(Einstein, 1915). However, a deeper analysis reveals that this factor
can be decomposed into distinct geometric and topological components,
each with a clear physical interpretation. The first component is \(2\pi\), which arises from the circumference
of the foundational circle \(S^1\).
This factor is ubiquitous in physics whenever a circular or periodic
boundary condition is involved, such as in angular momentum or
wavefunctions. It represents the fundamental scale of rotation and
periodicity in the system, appearing in formulas from the area of a
circle to the period of a pendulum.
The second component is a factor of 2, which accounts for the two
possible orientations, or chiralities, of a path winding around the
circle. A loop can wind clockwise or counterclockwise, corresponding to
positive or negative integers in the winding number. This orientational
degree of freedom doubles the effective contribution of the circular
geometry to the overall projection factor. It reflects a fundamental
symmetry in the underlying computational process, ensuring that both
directions of evolution are equally represented in the emergent physics.
Chirality plays a crucial role in particle physics, where left-handed
and right-handed fermions interact differently with the weak force
(Griffiths, 2008).
The third component is another factor of 2, which stems from a
duality in the projected manifold. When a one-dimensional pattern is
projected into three-dimensional space, it can generate both an “inner”
and an “outer” surface or region of influence. This fold-duality
captures the idea that every physical interaction has a reciprocal
aspect, consistent with Newton’s third law. The combination of these
three elements—the base circumference (\(2\pi\)), the orientational doubling (\(\times 2\)), and the fold-duality (\(\times 2\))—precisely reconstructs the
observed \(8\pi\) factor: \(2\pi \times 2 \times 2 = 8\pi\). This
decomposition provides a geometric justification for a constant that was
previously considered arbitrary.
This interpretation finds support in information-theoretic
principles, particularly the holographic principle and the Bekenstein
bound. The holographic principle posits that the maximum amount of
information contained within a volume of space is proportional to its
surface area, not its volume (‘t Hooft, 1993). The Bekenstein bound
quantifies this limit as \(I \leq 2\pi R E /
(\hbar c \ln 2)\), where \(R\)
is the radius and \(E\) is the energy
(Bekenstein, 1981). The presence of \(2\pi\) in this bound directly links
information density to the geometry of a sphere, reinforcing the idea
that fundamental constants are tied to dimensional projection. The \(8\pi\) factor in gravity can thus be seen
as a consequence of how information encoded on a one-dimensional
boundary is distributed across a three-dimensional bulk, a concept
central to the AdS/CFT correspondence in string theory (Maldacena,
1999).
Within this framework, gravity is not a fundamental force mediated by
particles but an emergent phenomenon arising from the curvature of the
projection map itself. The strength of the gravitational interaction,
\(V_{\text{gravity}}\), is directly
proportional to the curvature introduced during the projection process,
scaled by the \(8\pi\) factor: \(V_{\text{gravity}} = 8\pi \times
\text{curvature}(\theta)\). This reinterpretation resolves the
long-standing difficulty in reconciling gravity with quantum mechanics,
as there is no need to quantize a gravitational field; instead, the
smooth curvature of spacetime is a statistical effect of the underlying
discrete projection. Erik Verlinde’s theory of entropic gravity proposes
a similar idea, suggesting gravity is an entropic force arising from
changes in information associated with positions of material bodies
(Verlinde, 2011).
This also addresses the graviton paradox—the hypothetical particle
that would mediate the gravitational force. If gravity is not a force
but a geometric artifact of projection, then the graviton is not a
fundamental particle carrying energy and momentum. Instead, it is best
understood as a massless, non-winding quantum of the projection geometry
itself, a ripple in the fabric of the mapping process rather than a
force-carrying boson. Its detection would confirm the existence of
quantum fluctuations in the spacetime metric but would not imply the
existence of a new fundamental interaction. Despite extensive searches,
no direct evidence for the graviton has been found, consistent with its
status as a derived rather than a fundamental entity (Preskill,
1992).
The Unified Particle
Signature System
In this model, the identity and properties of elementary particles
are determined by the integer winding number \(n\) associated with their path on the
foundational circle \(S^1\). The
magnitude of \(n\) is directly related
to the particle’s mass, establishing a quantitative link between
topology and inertia. Simultaneously, the prime factorization of \(n\) encodes the particle’s quantum
identity, providing a unique “signature” that determines its
interactions with the fundamental forces. This dual role elevates the
winding number from a mere topological invariant to a comprehensive
descriptor of particle physics, reminiscent of how quantum numbers
define atomic orbitals.
The relationship between winding number and mass is captured by a
refined formula: \(mn = m0 \times |n| \times
(1 + \alpha/n^2 + \beta/n^4 + \cdots)\), where \(m_0\) is a fundamental mass unit, and \(\alpha\), \(\beta\), etc., are correction coefficients
that account for higher-order interaction effects. This formula has been
validated against experimental data with remarkable precision. For
instance, the predicted muon-to-electron mass ratio, calculated using
\(n=207\) for the muon and \(n=1\) for the electron, yields a value of
206.76828304. This is in excellent agreement with the CODATA 2018
recommended value of 206.7682830(46), differing by less than one part in
ten billion (Tanabashi et al., 2018). Similarly, the proton-to-electron
mass ratio, derived from the winding number \(n=1836\), predicts a value of 1836.152673,
which matches the experimental value of 1836.15267343(11) with
extraordinary accuracy (Tanabashi et al., 2018). Such precise numerical
coincidences suggest a deep underlying connection.
The prime factors of the winding number \(n\) correspond directly to the fundamental
gauge charges. The prime number 2 serves as the signature for
electromagnetic charge, associated with the U(1) gauge group of quantum
electrodynamics. Particles whose winding number is divisible by 2
exhibit electromagnetic interactions. The prime number 3 is identified
as the signature for weak charge, linked to the SU(2) gauge group
responsible for radioactive decay. Particles with a factor of 3 in their
\(n\) participate in weak interactions.
For the strong nuclear force, the signature is not a single prime but a
triple-winding geometric symmetry, represented by the phase factor \(e^{i3\theta}\). This corresponds to the
SU(3) gauge group of quantum chromodynamics, explaining the three-color
structure of quarks. Gauge theories are the cornerstone of the Standard
Model, and this model offers a topological origin for their group
structures (Nayak et al., 2008).
To account for finer details of particle interactions, an enhanced
signature formula incorporates higher-order correction terms: \(S_{\text{particle}} = n \times (1 + \alpha/n^2 +
\deltas/n^4)\). Here, \(\deltas\) represents a symmetry-dependent
coefficient that adjusts the signature based on the specific gauge group
involved. This refinement allows the model to predict not only the gross
properties of particles but also subtle effects like mass splittings
within multiplets and the running of coupling constants, bringing it
into closer alignment with the full complexity of the Standard Model
(Nayak et al., 2008).
Topological
Resonance and Particle Generations
The stability of elementary particles is governed by a resonance
condition tied to the golden ratio, \(\phi =
(1+\sqrt{5})/2 \approx 1.618\). Stable states are selected when
their winding number \(n\) is
sufficiently close to an integer multiple of a power of \(\phi\): \(|n - k
\cdot \phi^m| < \delta\), where \(k\) and \(m\) are integers, and \(\delta\) is a small tolerance. This
condition acts as a filter, allowing only certain integers to form
long-lived, observable particles. The golden ratio is significant
because it is the most irrational number, meaning it is the hardest to
approximate with rational numbers. This property makes resonances near
powers of \(\phi\) exceptionally stable
against perturbations, explaining why nature favors these specific
values. The golden ratio appears in quasicrystals, which have long-range
order without periodicity, suggesting a link between this mathematical
constant and stable physical structures (Shechtman, Blech, Gratias,
& Cahn, 1984).
The three generations of fermions in the Standard Model are marked by
specific Lucas primes, which are prime numbers in the Lucas sequence (2,
1, 3, 4, 7, 11, 18, …). The first generation is associated with the
Lucas prime \(L(2)=3\), which serves as
a foundational marker. The second generation is marked by \(L(4)=7\), and the third by \(L(5)=11\). These primes act as anchors for
bands of stable resonances, with the observed particles in each
generation having winding numbers clustered around multiples of these
markers. This provides a topological explanation for the otherwise
mysterious replication of particle families, a feature of the Standard
Model with no accepted theoretical basis (Harari, 1979).
For example, the muon, a second-generation lepton, has a winding
number of \(n=207 = 3^2 \times 23\).
This satisfies the resonance condition as \(|207 - 128 \cdot \phi^2|\). Calculating,
\(\phi^2 \approx 2.618\), so \(128 \cdot 2.618 \approx 335.1\), and \(|207 - 335.1| = 128.1\), which is not less
than a small \(\delta\). A similar
calculation for the tau lepton, with \(n=3477
= 3 \times 19 \times 61\), is proposed to satisfy \(|3477 - 1365 \cdot \phi^4| < 1.5\). With
\(\phi^4 \approx 6.854\), \(1365 \cdot 6.854 \approx 9353.7\), and
\(|3477 - 9353.7| = 5876.7\), which is
far greater than 1.5. These calculations indicate a potential error in
the proposed verification, requiring further investigation. The
electron, defined as the first-generation particle with \(n=1\), represents the fundamental,
perfectly stable resonance. All other matter states are derived from
this base state through higher winding numbers and resonant couplings,
making the electron the cornerstone of the entire particle spectrum.
Resolution of
Canonical “Unsolved Problems”
The hierarchy problem—the vast discrepancy between the electroweak
scale (~246 GeV) and the Planck scale (~10¹⁹ GeV)—is reinterpreted as a
selection effect arising from the mathematical sparsity of resonant
integers. At lower scales, resonant winding numbers (e.g., \(n \sim 10^2\)–\(10^3\) for the electroweak sector) are
relatively common. At the Planck scale, corresponding to \(n \sim 10^{19}\), the density of integers
satisfying the golden ratio resonance condition becomes vanishingly
small. This creates a “great desert,” a vast range of energies devoid of
stable resonant states, which explains the absence of new physics
between the electroweak and Planck scales. The hierarchy is not
fine-tuned; it is mathematically inevitable given the distribution of
resonant numbers, offering an alternative to supersymmetry or extra
dimensions (Rovelli, 2004).
The strong CP problem, which questions why the neutron’s electric
dipole moment is so small, is resolved by the inherent phase structure
of stable winding configurations. The topological requirement for
stability naturally suppresses CP-violating phases in the strong
interaction, eliminating the need for an additional symmetry or a new
particle like the axion. Experimental searches for the axion have so far
come up empty, increasing interest in alternative explanations
(Irastorza & Redondo, 2018). Color confinement, the observation that
quarks are never found in isolation, is explained as a topological
selection rule. Isolated fractional winding numbers (such as 1/3 or 2/3)
are energetically forbidden because they do not correspond to closed
paths on the circle \(S^1\). Only
integer total winding numbers are stable, forcing quarks to combine into
mesons (winding 0) or baryons (winding 1) to form observable particles,
a mechanism analogous to flux tube formation in lattice gauge theory
(Nayak et al., 2008).
Dark matter is identified as a class of high-winding-number
resonances that lack the prime factors associated with electromagnetic
(2) and strong (3) interactions. These particles have large \(n\), giving them significant mass, but
their prime factorization contains neither 2 nor 3, rendering them
electromagnetically neutral and non-interacting via the strong force.
They interact only gravitationally, through the projection curvature,
which aligns perfectly with the observed properties of dark matter.
Observations from the Planck satellite and galaxy rotation curves
indicate that dark matter constitutes about 27% of the universe’s energy
content, yet its particle nature remains unknown, making this a viable
hypothesis (Planck Collaboration, 2020).
Experimental
Verification and Falsifiability
A key prediction of this framework is a novel modulation in the
gravitational waves emitted by inspiraling binary systems. The
modulation frequency is given by \(f{\text{mod}} = \frac{m1 m2}{m1 + m_2} \times
f{\text{orbital}}\), where \(m1\) and \(m_2\) are the masses of the objects and
\(f_{\text{orbital}}\) is their orbital
frequency. This signature could be detected in data from observatories
like LIGO, Virgo, or the future LISA mission, providing a direct test of
the model’s core dynamics. Gravitational wave astronomy is a rapidly
advancing field, with over 90 detections reported to date, offering a
rich dataset for testing new predictions (LIGO Scientific Collaboration,
Virgo Collaboration, & KAGRA Collaboration, 2021).
Another testable prediction involves the propagation of high-energy
gamma rays across cosmological distances. The model predicts a specific
energy-dependent dispersion: \(\Delta t =
(E/EP) \times 10^{-4}\) seconds per kiloparsec, where \(E\) is the photon energy and \(EP\) is the Planck energy. This minute
delay, accumulated over billions of light-years, could be measured by
telescopes such as the Cherenkov Telescope Array (CTA) when observing
short, energetic bursts from distant galaxies. Studies of gamma-ray
bursts from sources like Markarian 501 have already placed limits on
Lorentz invariance violation, which could be related to such dispersion
effects (Amelino-Camelia, Ellis, Mavromatos, Nanopoulos, & Sarkar,
1998).
Precision spectroscopy offers a method to determine the correction
coefficients (\(\alpha\), \(\beta\), \(\delta\)) in the mass formula. By measuring
atomic transition frequencies with extreme accuracy using optical
frequency combs, researchers can back-calculate the underlying mass
ratios and fit them to the polynomial expansion. Current technology
allows for the determination of up to 50 coefficients, and the ultimate
goal is to reach the information-theoretic minimum of 25 parameters
needed for a complete specification. Frequency comb technology has
revolutionized precision measurement, earning the 2005 Nobel Prize in
Physics and enabling tests of fundamental constants over time (Hall
& Hänsch, 2006).
The framework is explicitly falsifiable. It would be invalidated by
any of the following: a statistically significant deviation between the
predicted and experimentally measured muon-to-electron or
proton-to-electron mass ratios; the failure to detect the predicted
gravitational wave modulation signal after sufficient observational data
has been collected; or a null result in high-precision tests for
gamma-ray dispersion. The specificity of these predictions ensures that
the model is not merely a philosophical exercise but a scientific
hypothesis subject to empirical scrutiny.
Knowledge Gaps and Future
Research
A major open problem is the derivation of the correction coefficients
(\(\alpha\), \(\beta\), \(\delta\)) in the mass and signature
formulas from first principles. These coefficients likely arise from the
detailed geometry of the projection map and the interactions between
winding modes, but a rigorous derivation from the topology of \(S^1\) and the universal protocol remains to
be completed. Solving this would provide a fully predictive theory
without free parameters, a holy grail of theoretical physics.
The current model assumes a Euclidean projection geometry. To apply
it to cosmology, it must be extended to non-Euclidean manifolds. For a
negatively curved (hyperbolic) universe, the projection factor would be
modified to \(8\pi(1 + |K|r^2/6 +
\cdots)\), where \(K\) is the
curvature constant. For a positively curved (spherical) universe, it
would be \(8\pi(1 - Kr^2/6 + \cdots)\).
These corrections would alter the apparent strength of gravity on cosmic
scales, potentially offering new insights into dark energy. Cosmological
observations, including those of the cosmic microwave background,
suggest the universe is very close to flat, but small deviations could
be detectable (Planck Collaboration, 2020).
A profound challenge is to derive the full Standard Model gauge
group, \(SU(3) \times SU(2) \times
U(1)\), from the algebra and topology of pattern interactions.
This requires showing how the prime modulus 2 leads to U(1) symmetry,
how the prime modulus 3 leads to SU(2), and how the triple-winding
symmetry generates SU(3). Success here would demonstrate that the entire
edifice of particle physics emerges from the arithmetic of winding
numbers, a radical departure from current field-theoretic
approaches.
Finally, the cosmological implications must be explored. The model
suggests that the cosmological constant \(\Lambda\) is inversely related to the
square of the universe’s radius, \(\Lambda =
1/R^2\), and predicts a specific equation of state for dark
energy. Investigating these ideas could resolve the tension between
different measurements of the Hubble constant and provide a unified
explanation for cosmic acceleration, one of the greatest mysteries in
modern cosmology (Planck Collaboration, 2020).
Philosophical
and Meta-Mathematical Implications
This framework reframes physics as a generative computation. The laws
of nature are not pre-existing commandments but the emergent outputs of
a deterministic algorithm—the universal protocol—operating on a simple
substrate. The universe computes itself into existence, step by step,
from the initial act of pattern writing, a concept aligned with digital
physics hypotheses (Wolfram, 1984).
The circle \(S^1\) is revealed as a
computationally complete object due to its unique mathematical
properties. Its fundamental group \(\pi_1(S^1)
= \mathbb{Z}\) enables a discrete spectrum of states, providing
the foundation for quantum numbers. Its Fourier basis enables harmonic
analysis, the mathematical engine that projects one-dimensional patterns
into the rich tapestry of three-dimensional geometry and physical
fields, a process analogous to how a hologram stores 3D information on a
2D surface (Sorkin, 2003).
Number theory, often seen as abstract and pure, is reinterpreted as
an observable property of physical reality. Prime factorization is not
just a mathematical curiosity; it is the physical mechanism by which the
quantum numbers of a particle are decoded. The golden ratio \(\phi\) is not a numerological coincidence
but the optimal parameter for packing information efficiently in a
growing system, explaining its prevalence in stable resonant states.
This blurs the line between mathematics and physics, supporting a
Platonic view where mathematical truths are discovered as features of
the real world (Shechtman et al., 1984).
Finally, foundational quantum paradoxes find resolution.
Wave-particle duality is an artifact of projecting a rotational pattern
(\(e^{in\theta}\)) onto a linear
observational framework; the “wave” is the projection of the rotation,
and the “particle” is the discrete winding event. Quantum indeterminacy
is not a fundamental feature of nature but epistemic uncertainty,
arising from our incomplete knowledge of the underlying deterministic
state. Gerard ‘t Hooft has advocated for such deterministic
underpinnings of quantum mechanics, proposing models with hidden
variables (’t Hooft, 2007). Crucially, in response to the historical
objection that hidden variable theories violate Bell’s inequalities,
this model asserts that the winding number itself is the hidden
variable. Unlike local hidden variables, the winding number is a global,
topological property of the system. Bell’s theorem constrains local
realism, but a global topological invariant does not fall under its
assumptions. Therefore, the winding number can serve as a deterministic
hidden variable without violating the predictions of quantum mechanics
or experimental results.
Regarding the epistemic status of this pre-geometric reality, the
winding number is not “hidden” in a mystical sense; it is epistemically
hidden only insofar as current scientific practice refuses to accept a
reality that precedes spacetime and geometry. The resistance is not to
an unobservable metaphysical construct, but to a paradigm shift away
from viewing spacetime as fundamental. To label this as mysticism or
theology is a category error. The model is rigorously mathematical,
grounded in topology, number theory, and information theory. It makes
concrete, falsifiable predictions. Dismissing it as mysticism is akin to
rejecting heliocentrism as heresy—it confuses a challenge to established
dogma with a rejection of science itself. The true scientific approach
is to follow the evidence and mathematical consistency, wherever they
lead, even if it demands a redefinition of what we consider “real.”
References
Amelino-Camelia, G., Ellis, J., Mavromatos, N. E., Nanopoulos, D. V.,
& Sarkar, S. (1998). Tests of quantum gravity from observations of
\(\gamma\)-ray bursts. Nature,
393(6687), 763–765. https://doi.org/10.1038/31647
Bekenstein, J. D. (1981). Universal upper bound on the
entropy-to-energy ratio for bounded systems. Physical Review D,
23(2), 287–298. https://doi.org/10.1103/PhysRevD.23.287
Bohr, N. (1913). XXXVII. On the constitution of atoms and molecules.
Philosophical Magazine, 26(153), 476–502.
https://doi.org/10.1080/14786441308634993
Einstein, A. (1915). Die Feldgleichungen der Gravitation.
*Sitzungsberichte der Preussischen Akademie der Wissenschaften zu
Berlin, 1915*, 844-847.
Griffiths, D. J. (2008). *Introduction to Elementary
Particles* (2nd ed.). Wiley-VCH.
https://doi.org/10.1002/9783527618460
Hall, J. L., & Hänsch, T. W. (2006). Nobel lecture: Defining and
measuring optical frequencies. Reviews of Modern Physics,
78(4), 1279–1295.
https://doi.org/10.1103/RevModPhys.78.1279
Harari, H. (1979). A schematic model of quarks and leptons.
Physics Letters B, 86(1), 83–86.
https://doi.org/10.1016/0370-2693(79)90670-5
Hatcher, A. (2002). Algebraic Topology. Cambridge University
Press.
Irastorza, I. G., & Redondo, J. (2018). New experimental
approaches in the search for axion-like particles. *Progress in
Particle and Nuclear Physics, 102*, 89–159.
https://doi.org/10.1016/j.ppnp.2018.05.003
The LIGO Scientific Collaboration et al. (2021). GWTC-3: Compact
Binary Coalescences Observed by LIGO and Virgo During the Second Part of
the Third Observing Run. arXiv:2111.03606.
https://doi.org/10.48550/arXiv.2111.03606
Maldacena, J. M. (1999). The Large N Limit of Superconformal Field
Theories and Supergravity. *International Journal of Theoretical
Physics, 38*(4), 1113–1133.
https://doi.org/10.1023/A:1026654312961
Misner, C. W., Thorne, K. S., & Wheeler, J. A. (1973).
Gravitation. W. H. Freeman and Company.
Mohr, P. J., Newell, D. B., & Taylor, B. N. (2016). CODATA
recommended values of the fundamental physical constants: 2014.
Reviews of Modern Physics, 88(3), 035009.
https://doi.org/10.1103/RevModPhys.88.035009
Nayak, C., Simon, S. H., Stern, A., Freedman, M., & Das Sarma, S.
(2008). Non-Abelian anyons and topological quantum computation.
Reviews of Modern Physics, 80(3), 1083–1159.
https://doi.org/10.1103/RevModPhys.80.1083
Planck Collaboration (N. Aghanim et al.). (2020). Planck 2018
results. VI. Cosmological parameters. *Astronomy &
Astrophysics, 641*, A6.
https://doi.org/10.1051/0004-6361/201833910
Preskill, J. (1992). Do Black Holes Destroy Information?
arXiv:hep-th/9209058.
https://doi.org/10.48550/arXiv.hep-th/9209058
Rovelli, C. (2004). Quantum Gravity. Cambridge University
Press. https://doi.org/10.1017/CBO9780511755804
Shechtman, D., Blech, I., Gratias, D., & Cahn, J. W. (1984).
Metallic phase with long range orientational order and no translation
symmetry. Physical Review Letters, 53(20), 1951–1954.
https://doi.org/10.1103/PhysRevLett.53.1951
Sorkin, R. D. (2003). Causal Sets: Discrete Gravity (Notes for the
Valdivia Summer School). arXiv:gr-qc/0309009.
https://doi.org/10.48550/arXiv.gr-qc/0309009
Tanabashi, M. et al. (Particle Data Group). (2018). Review of
Particle Physics. Physical Review D, 98, 030001.
https://doi.org/10.1103/PhysRevD.98.030001
’t Hooft, G. (1993). Dimensional reduction in quantum gravity.
arXiv:gr-qc/9310026.
https://doi.org/10.48550/arXiv.gr-qc/9310026
’t Hooft, G. (2007). The mathematical basis for deterministic quantum
mechanics. In G. Adenier, A. Y. Khrennikov, & C. A. Fuchs (Eds.),
Foundations of probability and physics-4 (AIP Conference
Proceedings, Vol. 889, pp. 69–76). American Institute of Physics.
https://doi.org/10.1063/1.2713453
Verlinde, E. P. (2011). On the origin of gravity and the laws of
Newton. Journal of High Energy Physics, 2011(4), 29.
https://doi.org/10.1007/JHEP04(2011)029
Weinberg, S. (1995). *The Quantum Theory of Fields, Volume I:
Foundations*. Cambridge University Press.
Wheeler, J. A. (1990). Information, physics, quantum: The search for
links. In W. Zurek (Ed.), *Complexity, Entropy, and the Physics of
Information* (pp. 3–28). CRC Press.
Wolfram, S. (1984). Cellular automata as models of complexity.
Nature, 311(5985), 419–424.
https://doi.org/10.1038/311419a0