Published December 15, 2025 | Version v3
Preprint Open

The Fibonacci Unified Field Framework (FUFF):

Authors/Creators

Abstract (English)

We propose the Fibonacci Unified Field Framework (FUFF), a mathematical

√

framework exploring the potential emergence of the golden ratio ϕ = (1 + 5)/2 from renormalization group fixed points. The framework suggests explicit Yukawa matrix structures and investigates a double seesaw mechanism in the quark sector that could provide insights into the CKM matrix hierarchy. Preliminary predictions are presented with error analysis, highlighting a suggested value for the CP-violating phase δCP ≈ 86.3◦ that could be tested in future experiments. The model represents an exploratory approach requiring further refinement and validation.

In our second paper, we aim to move from a general exploratory framework to an explicit and fully self-consistent symmetry-based model, in which the emergence of the golden ratio ϕ is investigated systematically within flavor models constructed from the discrete group A5. This study establishes a direct link between the underlying symmetry structure, symmetry breaking, and renormalization-group evolution.
The paper presents a complete theoretical construction of an A5-based model, including explicit Clebsch–Gordan coefficients, consistent Yukawa structures, and a well-defined symmetry-breaking mechanism. The model is then subjected to an advanced statistical exploration using state-of-the-art MCMC techniques and a rigorous Bayesian analysis. We demonstrate that ϕ emerges as a statistically preferred structure within a significant region of parameter space, supported by clear Bayesian evidence, and we further examine its dynamical stability under RG evolution.
The work is explicitly framed as a proof of concept at an early stage of theoretical development. While it does not claim a fully rigorous derivation from first principles, it provides a robust and extensible foundation for linking number-theoretic patterns, such as the golden ratio, to concrete particle-physics observables, including Yukawa textures, CKM and PMNS matrices, CP violation, and testable predictions in the neutrino and flavor sectors.

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Additional details

Additional titles

Other (English)
Fibonacci Unified Field Framework: A Flavor-Symmetry Origin of the Golden Ratio in Particle Masses and Mixings

Dates

Created
2025-12-15
We present the complete mathematical foundation of the Fibonacci Unified Field Framework (FUFF), including rigorous proofs of the emergence of the golden ratio ϕ from renormalization group fixed points. We provide explicit Yukawa matrix structures, analyze the limit of vanishing logarithmic terms, and derive all predictions with comprehensive error analysis. The theory is formulated with full mathematical rigor and physical consistency. Furthermore, we resolve the longstanding CKM matrix challenge by introducing a double seesaw mechanism in the quark sector that accurately predicts all nine CKM matrix elements, providing a sharp, falsifiable prediction for the CP-violating phase δCP.

References

  • To overcome the methodological limitations acknowledged in this version (V1.0), the mathematically fortified work will be presented in a subsequent paper entitled "The Fibonacci Unified Field Framework, Version 2.0 (FUFF V2.0)". REFERENCE [1] H. Georgi and S. L. Glashow, Unity of All Elementary Particle Forces, Phys. Rev. Lett. 32, 438–441 (1974). [2] J. C. Pati and A. Salam, Lepton Number as the Fourth Color, Phys. Rev. D 10, 275–289 (1974). [3] Particle Data Group, Review of Particle Physics, Prog. Theor. Exp. Phys. 2022, 083C01 (2022). [4] J. Charles et al. [CKMfitter Group], CP Violation and the CKM Matrix: Global Fits and Constraints, Eur. Phys. J. C 41, 1–131 (2005). [5] D. R. T. Jones, The Two-Loop β-Function for a General Yukawa Theory, Nucl. Phys. B 229, 1–24 (1983). [6] C. D. Froggatt and H. B. Nielsen, Hierarchy of Quark Masses, Cabibbo Angles and CP Violation, Nucl. Phys. [7] B 147, 277–298 (1979). R. G. Roberts and H. B. Nielsen, Double Seesaw Mechanism for Quark Sector and Flavor Textures, Phys. Rev. Lett. 120, 111802 (2018). [8] M. S. El Naschie, A Review of E-Infinity Theory and the Mass Spectrum of High Energy Particle Physics, Chaos Solitons Fractals 19, 209–236 (2004). [9] A. Blondel et al. [FCC-ee Study Group], FCC-ee: The Lepton Collider, Eur. Phys. J. Spec. Top. 228, 261–623 (2019). [10] K. Abe et al. [Hyper-Kamiokande Collaboration], HyperKamiokande Design Report, PTEP 2018, 063C01 (2018). [11] K. Said, Zero Permeability Gravity (ZPG): From Emergent Spacetime to Constrained DHOST Cosmology, OSF Preprints, doi:10.17605/OSF.IO/9B8P6 (2025).