Mass as Structural Inconsistency: A Fractal Derivation of the Standard Model Mass Spectrum
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Abstract
This research paper presents a structured and analytical summary of the Fractal Consistency Law (FCL), a theoretical framework aiming to explain the origin of particle masses through the fractal geometry of spacetime. According to this hypothesis, mass emerges as a quantitative measure of “structural inconsistency” between a particle’s intrinsic configuration and the self-similar, non-differentiable texture of spacetime. Using Particle Data Group (PDG 2024) mass values and a hybrid Maximum Likelihood Estimation (MLE) + Bayesian fitting methodology, the model identifies scale-invariant relationships among leptons, quarks, and bosons. The fit reproduces the Standard Model mass spectrum with an average relative error of ≈1.9%, while the framework remains theoretical and unverified experimentally.
Background and Context
The Fractal Consistency Law (FCL) was first proposed as a theoretical model seeking to derive the Standard Model mass spectrum from a deeper geometric structure of spacetime. The key idea is that each particle’s mass depends on how harmoniously its field configuration aligns with the fractal geometry of spacetime. This paper reformulates the original theory into a formal academic framework, clarifying assumptions, mathematical structure, and implications. It must be noted that the FCL is a theoretical hypothesis only and has not been validated experimentally.
Introduction: The Mass Generation Problem
Within the Standard Model (SM), mass originates from interactions with the Higgs field. While the Higgs mechanism successfully explains how mass arises dynamically, it does not explain why each particle has its particular numerical value. The FCL addresses this gap by proposing that mass is not intrinsic but an emergent property arising from geometric compatibility — or structural inconsistency — with a fractal spacetime texture.
The Fractal Consistency Law (FCL)
Conceptual Definition
The Fractal Consistency Law asserts that spacetime is continuous but non-differentiable, exhibiting self-similarity across scales. Every fundamental particle interacts with this underlying fractal structure; the degree of its geometric compatibility defines its effective mass. Formally:
mi ∝ δi = f(DF, ε)
where DF is the effective fractal dimension and ε the observational (scale) parameter, and δi quantifies the structural inconsistency — the extent a particle’s internal field deviates from perfect fractal symmetry.
Relation to Relativity
The FCL is presented as compatible with both Special and General Relativity. Spacetime remains continuous, but non-differentiability at microscopic scales introduces scale dependence. In this view, relativity emerges as the smooth macroscopic limit of a fundamentally fractal spacetime, consistent with the ideas of Scale Relativity (Nottale).
Methodology and Data Analysis
Dataset
Mass values were taken from Particle Data Group (PDG 2024), including asymmetric uncertainties for leptons, quarks, and gauge bosons.
Computational Approach
A hybrid MLE–Bayesian fitting approach was used. MLE (Maximum Likelihood Estimation) ensured numerical stability across wide mass ranges (notably for up-type quarks), while Bayesian fitting handled uncertainty propagation and probabilistic consistency. The fit minimized the overall error under fractal scaling constraints without introducing new adjustable parameters.
Results and Interpretation
The derived scaling law reproduces the observed mass hierarchy with high precision. Masses plotted on a logarithmic scale align along discrete self-similar intervals, implying a deep fractal regularity in spacetime structure. This coherence suggests the mass spectrum follows universal geometric constraints.
| Particle Sector | Mean Relative Error |
|---|---|
| Leptons | 1.8% |
| Quarks | 2.5% |
| Bosons | 1.1% |
| Overall Mean | 1.9% |
The fits imply that the Standard Model’s mass values are not arbitrary but follow fractal scaling relations determined by spacetime geometry. These results are promising yet require reproducible code and further theoretical justification to move beyond hypothesis.
Comparison with the Standard Model
| Aspect | Standard Model | Fractal Consistency Law (FCL) |
|---|---|---|
| Source of Mass | Interaction with the Higgs field | Geometric inconsistency with fractal spacetime |
| Parameters | Empirical Yukawa couplings | Derived from scale relations |
| Theoretical Basis | Quantum field theory (smooth spacetime) | Scale-dependent fractal geometry |
| Experimental Status | Verified (Higgs discovery) | Theoretical, unverified |
The FCL is intended to complement — not contradict — the Higgs mechanism by explaining why the empirical Yukawa couplings take their observed numerical values.
Implications and Future Directions
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Neutrino Mass Predictions
The FCL scaling pattern provides predictive estimates for neutrino masses that could be tested by next-generation neutrino experiments.
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Connection to Scale Relativity
The model aligns conceptually with Nottale’s Scale Relativity, where non-differentiable spacetime yields quantization and fractal scaling.
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Beyond the Standard Model Applications
FCL could offer insight into mass distributions of dark matter or hypothetical beyond-SM particles if they follow the same scaling laws.
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Future Theoretical Development
Future work should aim to derive FCL from first principles within quantum field theory or quantum gravity and connect it with renormalization group frameworks.
Conclusion
The Fractal Consistency Law frames mass as an emergent property arising from the self-similar, non-differentiable fabric of spacetime. Statistical analysis of PDG 2024 data reveals a coherent fractal hierarchy across known particle masses. Although theoretical and unconfirmed, the FCL is mathematically consistent, testable, and offers a geometrically intuitive bridge between quantum field theory and spacetime geometry.
References and Acknowledgments
- Reyna Ugarriza, C. D. (2025). Mass as Structural Inconsistency: A Fractal Derivation of the Standard Model Mass Spectrum.
- Particle Data Group (2024). Review of Particle Physics.
- Nottale, L. (1993). Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity. World Scientific.
Acknowledgment: This paper reformulates and analyzes concepts originally proposed by Cesar Daniel Reyna Ugarriza, with critical review and structured presentation in this manuscript by the candidate author for academic and research discussion.
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