Topological Geometry of Neutrino Mass and Mixing

Abstract

This work presents an analytical, parameter-free derivation of the lepton mixing matrix and active neutrino mass spectrum from an icosahedral $A_5$ flavor manifold coupled to the three-generation Iwahori-Hecke algebra $H_3(q)$. Partitioning the icosahedral vertex curvature across the six fundamental Weyl chambers of $H_3(q)$ fixes an elementary angular defect ${\Delta }_{AD}=\pi /18$, unifying ultra-high-energy mass generation and low-energy flavor mixing as dual realizations of a single geometric invariant. At the topological braiding threshold $\mu \sim M_0$, the right-handed Majorana scale $M_0$ breaks scale invariance and drives the five-dimensional Temperley-Lieb quotient into its hyperbolic regime with loop modulus $d_{AD}^{\prime }=2\cosh {\Delta }_{AD}$, where Quantum Yang-Baxter Baxterization projects the algebraic roots onto a hierarchical heavy spectrum $M_1>M_2>M_3>0$. Integrating out and inverting these states via a flavor-isotropic Seesaw mechanism Type-I locks in the normal mass hierarchy with $m_1\approx 3.64$ meV, ${\sum }_i{m}_i\approx 63.5$ meV, and $m_{\beta }\approx 9.45$ meV. Simultaneously, probability conservation requires analytic continuation onto the compact unitary circle, factorizing the mixing matrix into the $A_5$ Golden Ratio baseline, a unitary boundary twist generated by the longest Coxeter involution, and a discrete Majorana parity matrix. This geometry yields ${\theta }_{13}=8.49^{\circ }$, ${\theta }_{12}=32.11^{\circ }$, ${\theta }_{23}=45^{\circ }$, and maximal Dirac CP violation, ${\delta }_{CP}=\pm \pi /2$, while discrete $SO(3)$ parities quantize the relative Majorana phases, ${\alpha }_{13}=-\pi ,{\alpha }_{23}=0$, preventing $0\nu \beta \beta$ decay cancellation, $m_{\beta \beta }\approx 1.14$ meV, and suppressing renormalization group drift to ${\dot{\theta }}_{ij}\sim 10^{-7}$ rad.

arXiv comment29 pages (main text) plus four Appendices. Improves the underlying algebra, updates the main results accordingly, and corrects typos from the previous version