Condensed Matter Formulation of the MMU Model: Hamiltonian Dynamics of a Tetrahedral Elastic Lattice

This work studies the emergence of relativistic excitations in a discrete elastic lattice with tetrahedral geometry. The model is based on a lattice whose sites carry three internal oscillatory degrees of freedom denoted w2, w3 and w4. Starting from a local Hamiltonian of coupled internal oscillators we construct the full lattice Hamiltonian and derive the corresponding dynamical matrix. In the continuum limit the system reproduces the canonical MMU master equation describing elastic spacetime dynamics. When the stiffness hierarchy satisfies the condition k2 equals k3 two internal modes become degenerate and a massless collective excitation with linear dispersion appears. The resulting excitation possesses two transverse polarizations and exhibits a Dirac like low energy structure. The analysis demonstrates how relativistic Dirac type dynamics can emerge from purely geometric lattice dynamics without introducing quantum postulates. The work connects tetrahedral lattice geometry, internal elastic response and relativistic field behavior within the Methane Metauniverse framework.