Part 20: Emergent Lorentz Symmetry and the Macroscopic Continuum Limit of Discrete H₄ Geometry

Overview A central challenge faced by every discrete spacetime framework concerns the emergence of Lorentz symmetry. If physical reality is fundamentally discrete, why does nature appear continuous at observable scales? Why do relativistic phenomena emerge with such extraordinary precision? And why does a universal limiting velocity exist? The present work proposes that Lorentz symmetry is not fundamental. Instead, it emerges as a large-scale effective property of discrete H4-derived geometry. Building upon previous Parts of the Origin Geometry program, we investigate how statistical isotropy, multiscale screening, elastic signal propagation, and coarse-grained continuum behavior collectively generate an effective Lorentzian spacetime description [12–20, 24–39]. Emergent Causality and the Limiting Velocity Within this framework, the observed velocity scale c is interpreted at the infrared level as the screened limiting propagation speed of geometric disturbances supported by the substrate [30–39]. At sufficiently large scales, microscopic anisotropies and directional irregularities become progressively suppressed, yielding an effectively isotropic continuum limit whose governing wave equations exhibit Lorentz-like invariance. The resulting picture suggests that relativistic spacetime may arise naturally from the collective behavior of an underlying discrete geometric medium. The framework remains intentionally conservative. No derivation of General Relativity is attempted. No complete quantum gravity model is introduced. No claim is made that exact Lorentz symmetry holds at all scales [15, 43, 44]. Instead, the present Part investigates whether Lorentz symmetry itself may be understood as an emergent infrared geometric phenomenon. Structural Bridge in the OG Program This Part also plays a structural role within the Origin Geometry program. Previous Parts developed the microphysical layer of the framework: mass-like hierarchy, spin-like topology, winding-based family organization, charge-like topology, projection triality, confinement-like behavior, and dynamical localization. The present Part completes that first layer by establishing the effective Lorentzian arena in which such structures become macroscopically observable [1–11]. It therefore serves as the bridge from the microphysical branch of Origin Geometry to the subsequent macroscopic and cosmological branch.