Quantum Spacetime Imprints: The 24-Cell, Standard Model Symmetry and Its Flavor Mixing

TL;DR

This work proposes that the SM flavor structure emerges from the quantum geometry of spacetime encoded by the 24-cell and governed by the exceptional group F4. A two-tier construction uses a matter 24-cell for kinematic labeling and the F4 root system for gauge dynamics, yielding a unified hypercharge functional and anomaly cancellation, with A4 and T' flavor symmetries arising from tetrahedral substructures. Neutrino mixing is derived from an ideal TBM pattern perturbed by a minimal distortion principle that generates a realistic theta_13 and mass splittings, while quark mixing is addressed via a geometric T' framework yielding a Cabibbo angle around 13 degrees through spinor overlaps and MD projection. The approach is linked to UV completions via flux stabilization and Gaussian Yukawas, implying that SM flavor may be a residual imprint of quantum spacetime geometry with testable TeV-scale signatures and connections to spinfoam and string-theoretic constructions. Overall, the paper provides a coherent geometric route from 24-cell combinatorics to the full SM flavor phenomenology, tying hypercharge, mixing, and mass hierarchies to the underlying quantum geometry of spacetime.

Abstract

In our previous work \cite{FaragAli:2024jpo} we outlined an exploratory framework in which the 24-cell acts both as the quantum of spacetime and as a geometric representation of elementary particles. In this paper we provide comprehensive mathematical and phenomenological evidence that deepens and refines this primary model. The symmetry of the 24-cell yields a unified hypercharge functional that reproduces Standard Model hypercharge assignments while ensuring anomaly cancellation. By projecting the vertices of the 24-cell onto a three-dimensional flavor subspace using a Minimal Distortion Principle, an emergent tetrahedral structure appears that gives rise to an effective $A_4$ symmetry in the neutrino sector. Extending this discrete symmetry to its binary double cover $T^\prime$ provides the spinorial representations and intrinsic complex phases needed to generate realistic quark Yukawa textures. Guided by the tenet that spacetime and matter are inextricably linked, our analysis suggests that the intricate flavor mixing of the Standard Model may be a residual imprint of the underlying quantum-geometric nature of spacetime.

Figures (6)

• Figure 1: Outer ring (blue): eight short–root colour placeholders on the matter scaffold; these are distinct from the $SO(8)$ triality $8_v$ used later for generation counting and are inert under triality.
• Figure 2: Branching of the $F_4$ adjoint ($\mathbf{52}$) under the SM gauge group. This diagram represents only the gauge sector; fermions reside in other representations (e.g., the $\mathbf{26}$). The coset dimension (28 or 40) depends on the specific hypercharge embedding.
• Figure 3: Triality breaking in the $D_4$ subalgebra of $F_4$. A $\mathbf{28}$ Higgs acquires a nonzero VEV $\langle \phi\rangle\neq0$, breaking the $\mathbb{Z}_3$ symmetry that permutes $\mathbf{8_v}$, $\mathbf{8_s}$, and $\mathbf{8_c}$. Each 8-dimensional representation then corresponds to one SM generation, with right-handed states embedded via conjugation.
• Figure 4: Subgroup relations: $F_4$ contains $T'$, which projects onto $A_4$ with kernel $\mathbb{Z}_2$.
• Figure 5: Schematic 2D analogy of the tetrahedron's MDP projection. Residual strains $\epsilon_{ij}$ deviate from $-\tfrac{1}{3}$.