Derivation of Lorentz Invariance from Euclidean Simplicial Geometry (Universe Engine v13.3.4)

AbstractThis module (v13.3.4) addresses the "Holy Grail" of discrete physics. It does not merely preserve Lorentz symmetry; it explains its geometric origin. Standard lattice theories often suffer from anisotropy (the speed of light varies with direction). The Universe Engine solves this by introducing a Fundamental Invariant based on 4D Euclidean geometry, where time is treated as a physical path length on the lattice. Key Theoretical Breakthroughs: Euclidean Relativity: We demonstrate that the Minkowski metric (ds² = c²dt² - dx²) is not a fundamental axiom but a derived consequence of a strictly Euclidean conservation law:L_total² = L_space² + L_time²The "minus sign" in relativity emerges naturally when coordinate time (L_total) is the hypotenuse of the spacetime triangle. Geometric Time Dilation: Time dilation is derived as a geometric necessity:γ = 1 / √(1 - v²/c²)As an object moves faster through space (L_space increases), it must move slower through its internal dimension (L_time decreases) to conserve the total path length. Resolution of Anisotropy: By defining the speed of light as the maximum propagation rate on the lattice (L_time → 0), the theory recovers perfect isotropy in the macroscopic limit, making the discrete lattice compatible with Einstein's relativity. The Simulation (lorentz_invariance_simulation.py)The included Python code provides a numerical verification of the theory: Visualizes the "Spacetime Trade-off" (Fundamental Invariant circle). Simulates particle trajectories on a 2D simplicial lattice (simulates particle trajectories on a simplified 2D spacetime slice (1 space + 1 time) to demonstrate the principle). Calculates the emergent Gamma factor and compares it with the theoretical prediction, showing near-perfect agreement. Demonstrates Length Contraction and Time Dilation as geometric projection effects. ContextThis paper proves that the Universe Engine is not just a "toy model" but a serious candidate for a Unified Field Theory, capable of reproducing the pillars of modern physics (Relativity) from simpler, discrete ingredients. Files Included: Lorentz_Invariance_Euclidean_Derivation.pdf (Mathematical Proof) lorentz_invariance_simulation.py (Numerical Verification) Author InformationJulian Zoria (Independent Researcher)ORCID: 0009-0002-2424-5291Email: julian.zoria@proton.me