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: Lₜotal² = Lₛpace² + Lₜime²The "minus sign" in relativity emerges naturally when coordinate time (Lₜotal) 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 (Lₛpace increases), it must move slower through its internal dimension (Lₜime decreases) to conserve the total path length. Resolution of Anisotropy: By defining the speed of light as the maximum propagation rate on the lattice (Lₜime → 0), the theory recovers perfect isotropy in the macroscopic limit, making the discrete lattice compatible with Einstein's relativity. The Simulation (lorentzᵢnvarianceₛimulation. 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: LorentzInvarianceEuclideanDerivation. pdf (Mathematical Proof) lorentzᵢnvarianceₛimulation. py (Numerical Verification) Author InformationJulian Zoria (Independent Researcher) ORCID: 0009-0002-2424-5291Email: julian. zoria@proton. me
Julian Zoria (Thu,) studied this question.