Grand Unification via the Fibonacci-Prime Basis

From Neutrino Texture to Proton Decay 📄 Overview This research paper presents a novel extension of the "squared-primes neutrino texture" into a full Grand Unified Theory (GUT). The framework demonstrates that the denominators governing neutrino mixing angles—$\{9, 10, 45\}$—are not arbitrary numbers but correspond precisely to the dimensions of fundamental representations of the $SO(10)$ gauge group. By utilizing a basis of consecutive Fibonacci primes $\{2, 3, 5\}$, the theory derives the GUT scale, the unified coupling constant, the Weinberg angle, and the proton decay lifetime with zero free parameters. 🗝️ Key Theoretical Insights 1. The Fibonacci-Prime Basis The theory rests on the unique properties of the three consecutive Fibonacci primes: $F(3) = 2$ $F(4) = 3$ $F(5) = 5$ These primes form the building blocks for the dimensions of the gauge groups in the unification chain ($SU(5)$, $SO(10)$, $E_6$). 2. Neutrino Denominators as SO(10) Representations The paper maps the mixing angle denominators derived in previous works to specific Lie algebra representations: Mixing Angle Fraction Denominator SO(10) Representation Physical Interpretation Reactor ($\theta_{13}$) $1/45$ 45 Adjoint ($45$) Smallest mixing; couples to full gauge sector. Solar ($\theta_{12}$) $3/10$ 10 Fundamental ($10$) Large mixing; fundamental vector direction. Atmospheric ($\theta_{23}$) $4/9$ 9 Traceless Fund. ($10-1$) Largest mixing; concentrated in the traceless sector. 3. The Phi-Power Energy Ladder Energy scales are determined by the Golden Ratio ($\varphi$) raised to the power of the group's adjoint dimension ($n$): $$\mu_G = m_p \times \varphi^n$$ Standard Model ($n=12$): $\varphi^{12} \approx 322$ GeV (Electroweak scale vicinity). GUT Scale ($E_6, n=78$): $M_{GUT} = m_p \times \varphi^{78} \approx 1.88 \times 10^{16}$ GeV. Note: This result is within 6.2% of the standard SUSY $SO(10)$ prediction. 🧮 Major Derivations Unified Coupling Constant ($\alpha_{GUT}$) The coupling constant at the unification scale is derived from the largest Fibonacci prime in the basis: $$\alpha_{GUT} = \frac{1}{F(5)^2} = \frac{1}{25} = 0.04$$ Verification: One-loop MSSM running from $M_Z$ yields an inverse coupling of $\approx 24.32$, matching the prediction within 2.7%. Weinberg Angle ($\theta_W$) The tree-level Weinberg angle at unification is a ratio of alternating Fibonacci numbers: $$\sin^2 \theta_W(M_{GUT}) = \frac{F(4)}{F(6)} = \frac{3}{8} = 0.375$$ This reproduces the canonical $SU(5)$/$SO(10)$ prediction exactly. Proton Decay Lifetime The framework predicts proton decay via the $p \to e^+ \pi^0$ channel. The lifetime is structurally linked to the $E_6$ dimension ($78$) and the coupling prime ($5$): $$\tau_p \sim \frac{M_{GUT}^4}{\alpha_{GUT}^2 m_p^5} \sim 10^{36} \text{ years}$$ 🧪 Predictions vs. Experiment The framework makes 11 specific predictions. Below is the falsifiability scorecard presented in the paper: Prediction Value Status Relevant Experiment Proton Lifetime $\tau_p \sim 10^{36}$ years Allowed (> Current Bound) Hyper-Kamiokande (2028+) Solar Angle $\sin^2 \theta_{12} \approx 0.304$ Consistent JUNO Reactor Angle $\sin^2 \theta_{13} \approx 0.022$ Consistent JUNO Atmospheric Octant First Octant ($