A 4D 24-Cell Quantum Cellular Automaton with Emergent Fermions, Gauge Fields, and Regge Gravity

A 4D 24-Cell Quantum Cellular Automaton with Emergent Fermions, Gauge Fields, and Regge Gravity

Abstract

We investigate a discrete model of spacetime based on a four-dimensional lattice with 24-
cell (icositetrachoron) adjacency, motivated by the search for a maximally symmetric and 
geometrically space-filling substrate without voids at the lattice scale underlying relativistic 
quantum field theory. We emphasize that this terminology refers strictly to the complete 
Euclidean tessellation of the spatial manifold and does not imply the absence of a spectral 
gap in the Hamiltonian.

Unlike hypercubic lattices, the 24-cell lattice possesses self-duality, a kissing number of 24, 
and an associated 𝐹4 symmetry group, making it a natural candidate for suppressing 
anisotropies and Lorentz-violating artifacts in discrete formulations.

Local degrees of freedom are taken to be quaternionic and evolve according to a discrete, 
self-dual update rule, defining a quantum cellular automaton. We show that, in the long
wavelength limit, this construction admits effective Dirac fermions, with mass emerging from 
a local symmetry reduction of the lattice dynamics. Gauge fields arise naturally as 
holonomies associated with non-trivial loops around dual lattice centers, reproducing 
Maxwell electrodynamics in the continuum approximation. Gravitational dynamics are 
interpreted as a thermodynamic response of the lattice to defect density, closely related to 
Regge calculus.

We further discuss cosmological implications, including the possibility of a parameterdependent phase transition beyond which the continuum spacetime description ceases to 
be valid. No claims of inevitability or specific cosmological timelines are made. The model 
provides a unified discrete framework in which matter, gauge interactions, and gravity 
emerge from lattice geometry and symmetry