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

TL;DR

This paper proposes a quantum-geometric model where the 24-cell encodes spacetime and particle symmetries, providing a unified framework that reproduces Standard Model features and explains flavor mixing through geometric projections and discrete symmetries.

Contribution

It introduces a novel geometric approach using the 24-cell to unify spacetime and particle symmetries, deriving Standard Model hypercharges and flavor structures from geometric projections and discrete groups.

Findings

Reproduces Standard Model hypercharge assignments

Derives an $A_4$ symmetry in the neutrino sector

Explains flavor mixing as a geometric imprint

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.