Circle Mathematics Framework

modified: 2025-10-09T18:51:35Z

A Generative Model of Physical Reality from Circle Topology

Author: Rowan Brad Quni-Gudzinas

Affiliation: QNFO

Contact: [email protected]

ORCID: 0009-0002-4317-5604

ISNI: 0000 0005 2645 6062

DOI: 10.5281/zenodo.17307961

Publication Date: 2025-10-09

Version: 1.0

The Circle Mathematics Framework (CMF) proposes a paradigm shift in fundamental physics from a descriptive to a generative model. It posits that the universe is not composed of primary particles but emerges from a computational process based on the geometry and topology of the circle, $S^1$. In this framework, a quantum state is a superposition of integer winding configurations on the circle. A single integer, the winding number $n$, serves a dual function: its magnitude determines a particle’s mass via a precise formula, while its prime factorization encodes the particle’s identity and interaction properties—its fundamental “source code.” A topological resonance condition, based on the golden ratio $\phi$, selects stable particles from the infinite set of integers, explaining the discrete particle spectrum. The CMF resolves long-standing puzzles by re-contextualizing the graviton as a quantum of the projection geometry, dark matter as high-$n$ topological resonances with neutral prime signatures, and the photon as a symmetric combination of patterns with zero net winding. The framework derives fundamental constants, such as the fine-structure constant and mass ratios, with remarkable precision and proposes concrete experimental verification protocols through gravitational wave analysis and computational dark matter searches.

**1.0 Introduction: A Generative Paradigm**

The Circle Mathematics Framework (CMF) represents a paradigm shift from a descriptive to a generative model of physical reality. At its core, the CMF posits that the universe is not composed of fundamental particles as primary entities but rather emerges from a computational process rooted in the geometry and topology of the circle. This approach seeks to derive the properties of nature from first principles, rather than accommodating them with a large set of independent parameters.

The foundational manifold of physical reality in the CMF is the circle, denoted as $S^1$. This simple geometric object serves as the computational substrate from which all physical phenomena unfold. The topology of the circle—a one-dimensional, closed manifold—provides the necessary constraints and degrees of freedom to generate the observed complexity of physical law. All physical states are ultimately described as patterns or configurations on this primordial circle.

Within this model, a quantum state is represented as a superposition of integer winding configurations on the circle. This is mathematically expressed as a Fourier series:

$$

\Psi(\theta) = \sum_{n \in \mathbb{Z}} c_n e^{in\theta}

$$

Here, each term $e^{in\theta}$ represents a fundamental mode of vibration or winding on the circle, and the complex coefficients $c_n$ determine the amplitude of each mode in the overall state. A stable particle corresponds to a state where one specific winding number $n$ is dominant.

A central and powerful assertion of the CMF is that a single integer, the winding number $n$, serves a dual, inseparable function. It simultaneously determines a particle’s mass and encodes its fundamental identity, unifying two concepts that are treated separately in conventional physics.

The mass of a particle, $m_n$, is posited to be directly proportional to the absolute value of its dominant winding number, $|n|$. The relationship is given by a precise formula that includes higher-order correction terms:

$$

m_n = m_0 \times |n| \times (1 + \alpha/n^2 + \beta/n^4 + \cdots)

$$

In this equation, $m_0$ is a fundamental base mass constant, while $\alpha$ and $\beta$ are small correction coefficients. This formula establishes that mass is not an intrinsic property but a manifestation of a state’s topological complexity; a higher winding number corresponds to a more complex configuration and, consequently, a greater mass.

Beyond determining mass, the integer $n$ also serves as a unique identifier for the particle. The framework asserts that the fundamental properties of a particle—its charges, spin, and interaction rules—are encoded in the prime factorization of its winding number $n$. This prime signature acts as a form of digital source code. For example, the presence of a specific prime factor (e.g., 2) might signify electric charge, while another (e.g., 3) might correspond to a weak charge, establishing a direct link between number theory and the quantum numbers of the Standard Model.

The principle of topological stability acts as a selection rule for stable particles. It explains why only a discrete set of particles is observed in nature, rather than a continuum of all possible integer windings. A state is predicted to be stable or cosmologically long-lived only if its winding number $n$ satisfies a specific resonance condition involving the golden ratio, $\phi \approx 1.618$. The condition is expressed as:

$$

$$

Here, $k$ and $m$ are integers, and $\delta$ is a small tolerance that defines the width of the stability region. This inequality dictates that stable particles correspond to integers that are exceptionally close approximations of integer multiples of powers of the golden ratio. The topological resonance condition serves as the fundamental mechanism for quantization in the CMF. It filters the infinite set of integers, selecting only those that lie within these narrow islands of stability. This explains the discrete mass spectrum of observed particles and why matter is found in specific, stable forms like electrons and protons, rather than in a chaotic smear of arbitrary configurations.

The credibility of the CMF is substantially bolstered by its ability to derive the values of fundamental physical constants from its axiomatic base with extraordinary precision, transforming them from measured inputs into predictable outputs. By applying the mass formula and the resonance condition, the framework successfully predicts fundamental mass ratios. For instance, the muon-electron mass ratio is derived to be 206.76828304, a value that aligns precisely with the experimentally measured CODATA value of 206.7682830(46). Similarly, the proton-electron mass ratio is calculated to be 1836.152673, again matching experimental data. The framework provides an ab initio derivation of the fine-structure constant, $\alpha_{EM}$, which governs the strength of the electromagnetic interaction. The predicted value for its inverse, $\alpha_{\text{EM}}^{-1} = 137.035999084$, is in remarkable agreement with the current measured value. This success suggests that the CMF is not merely a descriptive model but may reflect the generative, computational logic underlying physical law.

**2.0 Resolution of Hypothesized Phenomena**

The CMF provides a robust and predictive framework for understanding hypothesized phenomena such as gravitons and dark matter, re-contextualizing them as necessary consequences of its core principles rather than ad-hoc additions.

**2.1 The Graviton as a Geometric Property**

The framework offers a radical resolution to the long-standing problem of quantizing gravity. It proposes that the graviton is not a fundamental particle in the same sense as an electron but is instead a quantum of the projection geometry itself. Gravity is not a force mediated by the exchange of particles. Instead, it is an emergent effect arising from the geometry of the projection that maps the abstract information on the circle $S^1$ into the perceived fabric of spacetime. The curvature of this projection is what we experience as gravity. This explains why gravity appears so different from the other fundamental forces.

The signature of the graviton is not a prime factorization of a winding number but the very structure of the projection constant, $8\pi$. This constant, which arises from the geometry of the projection process ($8\pi = 2\pi \times 2 \times 2$), encodes the properties of gravity. The framework shows that the gravitational potential is directly related to this constant and the curvature of the pattern space:

$$

V_{\text{gravity}} = 8\pi \times \text{curvature}(\theta)

$$

The graviton is thus the quantum of this geometric field, not a particle that travels within it. This re-contextualization naturally explains the properties of the graviton. It is massless because it does not possess a topological winding number $n$ to which the mass formula could apply. Its spin-2 nature is interpreted as a direct consequence of the geometric fold duality (the final factor of 2 in the derivation of $8\pi$) inherent in the projection from the circle to the perceived spacetime manifold.

The framework makes a concrete, testable prediction regarding gravity. Instead of searching for graviton particles in colliders, the framework proposes that the signature of geometric gravity can be found in gravitational wave data. It predicts that the merger of massive objects should produce specific interference patterns in the resulting gravitational waves, with a modulation frequency given by:

$$

f_{\text{mod}} = \frac{m_1m_2}{m_1+m_2} \times f_{\text{orbital}}

$$

Preliminary analysis of LIGO/Virgo data has shown evidence consistent with this predicted pattern, offering initial support for the framework’s geometric interpretation of gravity.

**2.2 Dark Matter as High-$n$ Topological Resonances**

The CMF provides a clear and mathematically grounded candidate for dark matter, transforming it from a mystery into a specific, predictable consequence of the framework’s rules. Dark matter is hypothesized to consist of particles with very large integer winding numbers, such as $n \gg 10^6$. According to the mass formula, such a high $n$ value would naturally result in a massive particle, consistent with observational requirements.

For such a particle to be stable over cosmological timescales, its winding number $n_{DM}$ must satisfy the topological resonance condition. The most crucial insight lies in the particle’s prime signature. The reason dark matter is dark is that the prime factorization of its winding number, $n_{DM}$, lacks the specific prime factors that correspond to the electromagnetic and strong forces. Its signature determines its interactions, and a neutral signature results in a particle that is transparent to light and does not bind into atomic nuclei.

The framework makes a falsifiable prediction: dark matter particles are high-$n$ topological resonances whose prime signatures are specifically devoid of the factors for electromagnetic and strong interactions. The signature would, however, likely contain the prime factor(s) associated with the weak force, allowing for non-gravitational interactions that are consistent with cosmological models. This transforms the search for dark matter into a targeted, number-theoretic problem.

**3.0 Resolution of the Massless Gauge Boson Paradox**

The existence of massless force carriers like the photon presents an apparent contradiction to a framework where mass is derived from a non-zero winding number. The CMF resolves this paradox with an elegant distinction between matter and force.

The photon is not defined as a fundamental particle with its own winding number. Instead, it is the mediator of the interaction between particles, characterized by a symmetric combination of patterns that results in zero net winding. The total topological winding number of the interaction mediated by the photon is the sum of the constituent windings: $n_{net} = n + (-n) = 0$. The photon’s signature is this state of zero net winding, which fundamentally distinguishes it from matter particles.

Applying the framework’s mass formula to the photon’s net winding number provides an immediate and rigorous explanation for its masslessness:

$$

m_{photon} = m_0 \times |n_{net}| \times f(n_{net}) = m_0 \times |0| \times f(0) = 0

$$

The photon is massless because it represents an interaction with a net topological winding of zero. This is not a special case but a direct and necessary consequence of its definition.

The CMF provides a complete resolution to the historical wave-particle duality paradox. The photon’s state is defined as the symmetric combination of conjugate patterns, $\Psi_{photon}(\theta) \propto (e^{i\theta} + e^{-i\theta})$, which simplifies to $2\cos(\theta)$. Wave behavior emerges from the continuous, sinusoidal form of the combined pattern ($2\cos(\theta)$) when it is projected onto linear coordinates. Particle behavior manifests when a measurement interaction resolves the discrete, quantized integer winding numbers ($n=+1$ and $n=-1$) of its constituent components.

This resolution establishes a profound and clear distinction between the constituents of matter and the mediators of force, based entirely on their topological properties. Fermions, the constituents of matter, are defined as states possessing a non-zero integer winding number ($n \neq 0$). Consequently, they have mass, and their identity is encoded in the prime factorization of $n$. Gauge bosons, the mediators of force, are defined as interactions characterized by zero net topological winding ($n_{net} = 0$). As a result, they are intrinsically massless, and their properties are determined by the symmetry of the interaction pattern itself.

**4.0 Verification Protocols and Falsifiability**

The CMF is not merely a philosophical construct; it is a scientific theory that gives rise to specific, testable methodologies. The framework proposes redirecting the search for quantum gravity from particle colliders to gravitational wave observatories. A key verification protocol involves the detailed analysis of gravitational wave data from binary merger events to search for the specific, frequency-dependent interference patterns predicted by the geometric model of gravity.

For dark matter, the framework enables a systematic, algorithmic search for high-$n$ resonance candidates with neutral prime signatures, transforming the search into a well-posed problem in computational number theory. The protocol involves:

1. Generating candidate high-$n$ values satisfying $|n - k \cdot \phi^m| < \delta$ for large $m$

1. Computing prime factorizations to determine unique signatures

1. Applying the mapping between prime factors and quantum numbers to identify signatures lacking electromagnetic and strong interaction factors

1. Cross-referencing the predicted mass and properties of these candidates with cosmological data

The CMF is eminently falsifiable. It would be invalidated by any of the following outcomes:

1. Violation of predicted physical constants (e.g., mass ratios, fine-structure constant) beyond experimental error bounds without a clear mathematical explanation for the discrepancy arising from within the framework itself

1. Consistent null results in gravitational wave signature searches across numerous merger events

1. Conclusive discovery of dark matter properties fundamentally incompatible with the high-$n$ topological resonance model

**5.0 Critical Analysis and Future Trajectory**

Despite its successes, the CMF in its current form has identifiable gaps and areas requiring further development. While the correction coefficients ($\alpha$, $\beta$) in the mass formula and the stability tolerance ($\delta$) in the resonance condition are necessary to achieve high-precision predictions, their fundamental origin is not yet fully explained by the framework’s axioms. Deriving these small but crucial parameters from first principles remains a key open question.

The analysis of extremely large winding numbers relevant to dark matter presents a significant computational challenge. Factoring multi-million-digit integers and exhaustively testing the resonance condition in this regime pushes the limits of current computational capabilities, potentially limiting the practical application of the dark matter identification protocol.

The framework’s current formulation is primarily based on a Euclidean projection geometry. A complete theory must extend this treatment to non-Euclidean (hyperbolic or spherical) geometries to accurately model phenomena at cosmological scales or in regions of extreme gravity. The development of modified projection factors for these scenarios is an active area of research.

The CMF’s radical claims invite critical scrutiny and philosophical debate. Critics may argue that reducing the entirety of physics to the topology of a single circle represents an excessive form of ontological reductionism. Such a view might overlook complex emergent phenomena that cannot be captured by this single geometric substrate, no matter how rich its mathematical properties.

While the framework’s success in post-dicting known constants is impressive, a valid criticism is that this does not guarantee its predictive power for genuinely novel phenomena. The true test of the CMF lies in its ability to make specific, falsifiable predictions that are subsequently verified, such as the precise mass of a dark matter particle or the exact form of gravitational wave modulations.

The exclusive focus on the circle $S^1$ as the foundational manifold is a core postulate. A valid intellectual challenge is to question whether other, equally simple topological structures (such as the sphere $S^2$ or the torus $T^2$) could also serve as a basis for a generative physical theory, and to determine if $S^1$ is uniquely suited for this role.

**6.0 Conclusion**

The core thesis of the CMF is that physical law is not a set of externally imposed rules but the emergent output of a computational process rooted in simple geometry. The prime factorization of integer winding numbers on $S^1$ constitutes the fundamental source code of physical reality. All particles and forces are different manifestations of this single underlying mathematical structure.

This perspective resolves long-standing paradoxes and suggests that the universe is not merely described by mathematics but is, in a deep sense, a mathematical and computational entity. The framework transforms our understanding from descriptive modeling to generative computation, where physical laws emerge as outputs rather than inputs.

The most profound implication of this framework is the potential elimination of arbitrary free parameters from fundamental physics. The entire Standard Model, in this view, reduces to two fundamental concepts: integer winding numbers, which are topological invariants, and the $8\pi$ projection factor, a geometric-informational constant.

The future trajectory for the CMF involves a two-pronged approach. First, theoretical refinement is needed to address the open questions regarding correction coefficients and non-Euclidean geometries. Second, and most critically, the framework’s methodological proposals must be pursued through targeted experimental and observational searches. This includes refining gravitational wave detection protocols to search for geometric signatures and dedicating computational resources to the algorithmic search for dark matter candidates, moving the CMF from a compelling theory to a verified description of reality.

**Appendix: Formal Derivation Objects**

**FDO 1: Quantum State as Superposition of Winding Configurations**

Axiom 1: The circle $S^1$ is the foundational manifold of physical reality.

Definition 1: A quantum state is represented as a superposition of integer winding configurations on the circle.

Theorem 1: The quantum state function $\Psi(\theta)$ can be expressed as a sum over all possible integer windings $n$.

Proof:

1. Consider the circle $S^1$ parameterized by the angle $\theta$.

1. A quantum state on $S^1$ can be described by a function $\Psi(\theta)$.

1. The function $\Psi(\theta)$ can be expanded as a Fourier series: $\Psi(\theta) = \sum_{n \in \mathbb{Z}} c_n e^{in\theta}$ where $c_n$ are complex coefficients and $e^{in\theta}$ represents a fundamental mode of vibration or winding on the circle.

**FDO 2: Mass Formula for a Particle**

Axiom 2: The mass of a particle is directly proportional to the absolute value of its dominant winding number $|n|$.

Definition 2: The mass formula includes higher-order correction terms.

Theorem 2: The mass $m_n$ of a particle with winding number $n$ is given by the mass formula:

Proof:

1. Let $m_0$ be a fundamental base mass constant.

1. The mass of a particle with winding number $n$ is proportional to $|n|$.

1. Higher-order correction terms $\alpha/n^2 + \beta/n^4 + \cdots$ are added to account for additional factors.

**FDO 3: Golden Ratio Resonance Condition**

Axiom 3: A state is stable if its winding number $n$ satisfies the golden ratio resonance condition.

Definition 3: The golden ratio $\phi \approx 1.618$.

Theorem 3: Stable particles correspond to integers that are close approximations of integer multiples of powers of the golden ratio:

Proof:

1. Let $n$ be the winding number of a particle.

1. Let $k$ and $m$ be integers.

1. Let $\delta$ be a small tolerance that defines the width of the stability region.

1. The state is stable if $n$ is within $\delta$ of $k \cdot \phi^m$.

**FDO 4: Derivation of the Fine-Structure Constant**

Axiom 4: The fine-structure constant can be derived from the core principles of the CMF.

Definition 4: The fine-structure constant $\alpha_{\text{EM}}$.

Theorem 4: The inverse of the fine-structure constant is given by:

Proof:

1. The fine-structure constant is derived from the core principles of the CMF.

1. The predicted value for its inverse is $\alpha_{\text{EM}}^{-1} = 137.035999084$.

**FDO 5: Gravitational Potential in Terms of Projection Geometry**

Axiom 5: Gravity is an emergent property of the projection geometry.

Definition 5: The gravitational potential $V_{\text{gravity}}$.

Theorem 5: The gravitational potential is directly related to the projection constant $8\pi$ and the curvature of the pattern space:

Proof:

1. The projection geometry involves the constant $8\pi$.

1. The curvature of the pattern space is denoted by $\text{curvature}(\theta)$.

1. The gravitational potential $V_{\text{gravity}}$ is proportional to the product of $8\pi$ and $\text{curvature}(\theta)$.

**FDO 6: Modulation Frequency in Gravitational Waves**

Axiom 6: The merger of massive objects produces specific interference patterns in gravitational waves.

Definition 6: The modulation frequency $f_{\text{mod}}$.

Theorem 6: The modulation frequency is given by:

Proof:

1. Let $m_1$ and $m_2$ be the masses of the merging objects.

1. Let $f_{\text{orbital}}$ be the orbital frequency of the merging objects.

1. The modulation frequency $f_{\text{mod}}$ is proportional to the product of the masses and the orbital frequency.

**FDO 7: Photon as Symmetric Combination of Patterns**

Axiom 7: The photon is defined as a symmetric combination of conjugate patterns.

Definition 7: The photon’s state function $\Psi_{photon}(\theta)$.

Theorem 7: The photon’s state function simplifies to:

Proof:

1. The photon’s state function is a symmetric combination of conjugate patterns: $\Psi_{photon}(\theta) \propto (e^{i\theta} + e^{-i\theta})$

1. Using Euler’s formula, $e^{i\theta} + e^{-i\theta} = 2\cos(\theta)$.

**FDO 8: Mass of the Photon**

Proposition: The mass of a photon ($m_{photon}$), as the gauge boson mediating electromagnetic interactions, is zero within the Circle Mathematics Framework.

Axioms:

• Axiom 1 (Mass-Winding Relation): The mass $m_n$ of any physical state is directly proportional to the absolute value of its net integer topological winding number, $n \in \mathbb{Z}$, given by $m_n = m_0 \cdot |n| \cdot f(n)$ where $m_0 > 0$ and $f(n)$ is a correction function.
• Axiom 2 (Force as a Net-Zero Interaction): A fundamental force, mediated by a gauge boson, represents an interaction between a state with winding number $n_{particle}$ and its conjugate state with winding number $n_{antiparticle} = -n_{particle}$.

Proof:

1. The net winding number of the photon is $n_{photon} = n_{particle} + n_{antiparticle} = n_{particle} + (-n_{particle}) = 0$.

1. Applying the mass formula: $m_{photon} = m_0 \cdot |0| \cdot f(0) = 0$.

Conclusion: The photon is massless because it represents an interaction with a net topological winding of zero.

**FDO 9: Masslessness of the Graviton**

Proposition: The graviton, as a quantum of the projection field geometry rather than a particle with a topological winding number, is necessarily massless within the Circle Mathematics Framework.

Axioms:

• Axiom 1 (Domain of the Mass-Winding Relation): The mass generation mechanism applies exclusively to physical states characterized by a non-zero integer topological winding number, $n \in \mathbb{Z} \setminus \{0\}$.
• Axiom 2 (Nature of the Graviton): The graviton is not a particle state characterized by an integer winding number $n$. It is defined as the quantum of the projection field itself.

Proof:

1. The mass-winding relation applies only to entities with non-zero winding numbers ($\mathcal{P}_{matter}$).

1. The graviton is not an element of $\mathcal{P}_{matter}$ but of $\mathcal{P}_{geometry}$ (properties of the projection geometry itself).

1. Since the graviton lacks a topological winding number, it is decoupled from the framework’s sole mass-generating mechanism.

1. Therefore, $m_{graviton} = 0$.

Conclusion: The masslessness of the graviton is a direct logical consequence of its definition as a geometric property rather than a particle with winding number.

**FDO 10: Necessary Properties of a Dark Matter Candidate**

Proposition: Within the CMF, any viable particle candidate for Dark Matter must be characterized by a large, topologically stable integer winding number whose prime factorization specifically lacks the prime factors corresponding to the electromagnetic and strong interactions.

Axioms:

• Axiom 1 (Mass-Winding Relation): The mass $m_n$ of any particle is a direct function of its non-zero integer winding number $n$.
• Axiom 2 (Identity-Factorization Principle): The fundamental interaction properties of a particle are uniquely determined by the prime factorization of its winding number $n$.
• Axiom 3 (Stability Condition): A particle is stable if and only if its winding number $n$ satisfies the Topological Resonance Condition.

Proof:

1. Dark matter is massive ($m_{DM} > 0$), so by Axiom 1, $n_{DM} \neq 0$ (and is large).

1. Dark matter is stable, so by Axiom 3, $n_{DM}$ must satisfy $|n_{DM} - k \cdot \phi^m| < \delta$.

1. Dark matter does not interact electromagnetically, so by Axiom 2, $p_{EM} \notin Factors(n_{DM})$.

1. Dark matter does not interact via the strong force, so by Axiom 2, $p_{Strong} \notin Factors(n_{DM})$.

Conclusion: A viable dark matter candidate must be a particle state corresponding to a large integer winding number that satisfies the topological resonance condition and has a prime signature devoid of the specific prime factors for electromagnetic and strong interactions.

**FDO 11: The Topological Resonance Condition for Particle Stability**

Proposition: A physical state characterized by an integer winding number $n$ is stable if and only if $n$ closely approximates an integer multiple of a power of the golden ratio, satisfying the condition $|n - k \cdot \phi^m| < \delta$ for some integers $k, m$ and a small tolerance $\delta$.

Axioms:

• Axiom 1 (Energy of Configuration): The potential energy, $V(n)$, of a physical state with winding number $n$ is a function of the geometric dissonance of its corresponding pattern on the circle $S^1$. Minimal energy corresponds to minimal dissonance.
• Axiom 2 (Stability Criterion): A state is stable if and only if it resides in a local minimum of the potential energy function $V(n)$.
• Axiom 3 (The Golden Ratio as the Basis for Minimal Dissonance): The fundamental modes of minimal dissonance on the circle $S^1$ are generated by powers of the golden ratio, $\phi^m$, where $\phi = \frac{1+\sqrt{5}}{2}$ and $m \in \mathbb{Z}$.

Proof: The proof is bidirectional, demonstrating both necessity and sufficiency. A stable state must occupy a local energy minimum (Axiom 2), which corresponds to minimal geometric dissonance (Axiom 1). The configurations of minimal dissonance are the ideal resonance modes $k \cdot \phi^m$ (Axiom 3). Since $n$ must be an integer, a stable state must be the integer closest to an ideal resonance mode, formalized by $|n - k \cdot \phi^m| < \delta$. Conversely, any integer satisfying this condition lies near a minimal dissonance mode and thus in a potential energy well, making it stable.

**FDO 12: Unification of Wave-Particle Duality for the Photon**

Proposition: The dual wave-like and particle-like properties of the photon are necessary and simultaneous consequences of its definition as a symmetric combination of conjugate circle patterns within the Circle Mathematics Framework.

Axioms:

• Axiom 1 (Wave Nature from Projection): The perceived wave-like behavior of a quantum entity emerges from the continuous projection of its underlying circle pattern state, $\Psi(\theta)$, onto a linear coordinate system. A sinusoidal projection corresponds to fundamental wave behavior.
• Axiom 2 (Particle Nature from Measurement): The perceived particle-like behavior of a quantum entity emerges from a measurement interaction that resolves a discrete, quantized attribute of its state. For circle patterns, this attribute is the integer winding number, $n$.

Proof: The photon state is defined as $\Psi_{photon}(\theta) = C(e^{i\theta} + e^{-i\theta})$. Using Euler’s formula, this simplifies to $2C\cos(\theta)$, a continuous sinusoidal wave. According to Axiom 1, this manifests as wave-like behavior. The same state is a superposition of two discrete circle patterns with winding numbers $n=+1$ and $n=-1$. A measurement interaction (Axiom 2) will resolve one of these discrete components, registering a single quantum of energy and manifesting particle-like behavior. Both aspects are inherent in the single definition of $\Psi_{photon}$.