# Unification based on the mysterious cubic-structure grouping of quarks and leptons

## TL;DR

This paper proposes a unification model linking quarks and leptons through a cubic-structure symmetry, extending the Pati-Salam model and suggesting a spacetime with an eleven-dimensional structure involving the seven-sphere.

## Contribution

It introduces a novel unification framework based on cubic-structure symmetry, connecting gauge symmetry, flavor symmetry, and higher-dimensional spacetime.

## Key findings

- The model extends the Pati-Salam framework with two sixteen-dimensional fermion representations.
- It derives the discrete cubic flavor symmetry from gauge symmetry.
- The gauge algebra indicates an eleven-dimensional spacetime with a seven-sphere manifold.

## Abstract

We present a unification model based on the well-known but mysterious cubic-structure grouping of quarks and leptons that suggests an underlying symmetry connection deemed explainable by a unified theory. It results in an extension of the Pati-Salam model that consolidates the fermions into two sixteen-dimensional chiral representations of the gauge group. Moreover, the discrete cubic flavor symmetry arbitrarily conjectured in the literature is found here to be implied by the gauge symmetry. Furthermore, the gauge algebra also describes an eleven-dimensional spacetime decomposed into the usual spacetime and the seven-sphere, which is the manifold of unit octonions. This suggests such a spacetime is apt for embracing all elementary particles and their interactions.