The Angular Tetravalent Lattice: A Geometric Proposal for the Emergence of Mass and E = mc²

We previously demonstrated that a discrete angular tetravalent lattice natively generates massless Weyl dynamics and robust chiral edge currents. In this paper, we propose a geometric mechanism for the emergence of mass. By introducing the S₂ (stable manifestation) state as a discrete elastic restoring force, we model geometric inertia as a probabilistic resistance to spatial translation. Through a systematic Taylor expansion, we show how this discrete inertia yields a candidate mass-like term in the continuum limit. By analyzing the dispersion relation, we demonstrate that this resistance effectively scales the spatial propagation speed relative to the internal rest-frame rotational frequency. Within this framework, we propose a geometric ansatz in which rest energy is interpreted as the cross-product of orthogonal lattice velocities, offering a conceptual motivation for Einstein’s E = mc². Quantitative scaling tests and linear regression analysis confirm that while the absolute information light-cone (c = 1) remains strictly preserved, the macroscopic wave speed drops proportionally to the geometric inertia. We emphasize that the present continuum equations describe an open system; the recovery of a conservative massive Dirac/Weyl operator remains an open problem. This record contains the full PDF of the paper together with the complete Python simulation code used to generate the numerical results and figures.