4D Discrete Lattice Proof of Concept: Emergent Relativistic Kinematics, Low-Energy Quantum Dynamics, and Regge-Based Gravitational Dynamics

Abstract

We present a proof of concept in which spacetime is modelled as a four-dimensional discrete lattice with a fundamental update step. The goal is not a complete unification scheme but a controlled derivation of four mutually linked emergent mechanisms. First, relativistic kinematics is obtained from structural transition constraints: the speed limit $c$ and time dilation arise kinematically rather than as postulates, and inertia emerges directly from the proper temporal advance $v_t = c\sqrt{1-v_s^2/c^2}$, kinematically reproducing the exact sign and form of relativistic kinetic energy at low energies without appealing to a static scalar mass. Second, the geometric bivector $\mathcal{M}$ does not act as a pure scalar mass degree of freedom; instead, it is formally reinterpreted as the spin and magnetic-moment degree of freedom through its coupling to the electromagnetic Wilson plaquettes $\mathcal{F}^{\mu\nu}$; the resulting term mimics the Pauli equation for spin and dipolar magnetic moment, resolving the fermionic nature of lattice excitations. Third, the low-energy limit of a nearest-neighbour tight-binding Hamiltonian recovers the Schr\"odinger equation with an effective inertial mass $m \propto (J\ell^2)^{-1}$. Fourth, smooth gradients of the link-tension field $\mathcal{T}_{xy}$ define a Gordon-type effective metric in the weak-field regime, while the full Einstein dynamics is recovered via Regge calculus by extremising a simplicial action built directly from $\mathcal{T}_{xy}$. The framework is falsifiable: the geometric inversion symmetry of the 4D lattice links exactly cancels all odd-order (linear) Lorentz-invariance-violation dispersive corrections, so that the first surviving modified dispersion relation (MDR) is quadratic in $E^2\ell^2$; this dramatically raises the tolerance bounds and shields the model against linear falsification, in full consistency with the timing analysis of the gamma-ray burst GRB~090510 measured by the Fermi-LAT observatory. The present paper acts as the direct foundation that anticipates the developments of Paper~2, where the bivector dynamics and the complete Lorentz-invariance-violation analysis are worked out in detail.