Thermodynamics of Discrete Spacetime: Entropy, Information, and the Arrow of Time in Universe Engine v13.3 (v13.3.12)

This paper presents a rigorous derivation of thermodynamic laws from the discrete geometric structure of spacetime in the Universe Engine v13. 3 framework. By treating spacetime as a 4-dimensional simplicial lattice (600-cell tessellation) with integer-valued edge states, we develop a statistical mechanics of geometry where thermodynamic quantities emerge from lattice microstates. Key Results: Lattice Temperature (TL): Defined as the variance of edge length parameters in a local volume, quantifying the coherence of spacetime geometry. TL → 0 corresponds to crystallization (matter formation), while TL → ∞ corresponds to geometric disconnection (Big Bang singularity state). Geometric Entropy (Sgeom): Introduced through a Boltzmann-like counting of lattice configurations consistent with macroscopic constraints. We prove Sgeom = kB ln (Ω) where Ω is the number of edge microstates. First Law of Thermodynamics: Derived from the Fundamental Invariant (L²ₜotal = L²ₛpace + L²ₜime = const), demonstrating that energy conservation emerges from the constancy of computational path length per update cycle. Arrow of Time: Shown to arise from the irreversibility of lattice update rules, specifically the error diffusion mechanism and positive Lyapunov exponents. We prove that geometric entropy increases monotonically under deterministic lattice evolution (Second Law). Heat as Incoherent Lattice Vibrations: Heat is reinterpreted as edge length fluctuations that do not form stable topological structures, distinguishing it from coherent waves (particles). Quantum Thermodynamics: We connect von Neumann entropy to geometric entropy, prove Landauer’s principle on the lattice (Qₑrase = kB T ln 2), and resolve the black hole information paradox through lattice unitarity. The formulation provides a unified framework for quantum thermodynamics, black hole entropy, and cosmological thermodynamics, all emerging from discrete geometry. We derive explicit expressions for entropy bounds, compute the thermodynamic arrow of time from lattice Lyapunov exponents, and connect geometric temperature to Hawking-Unruh temperatures in curved lattice regions. Testable Predictions: The proposed framework of Thermodynamics of Simplicial Spacetime leads to several specific, falsifiable predictions that distinguish it from standard cosmological models: 1. Modified Dispersion RelationsThe discrete nature of spacetime at the Planck scale implies that the speed of light is not perfectly constant for extremely high-energy photons. The theory predicts an energy-dependent speed of light, where high-energy gamma rays from distant bursts should arrive slightly later than lower-energy photons. This effect is expected to be proportional to the ratio of the photon energy to the Planck energy. 2. Spectral Dimension ReductionWhile spacetime appears 4-dimensional at macroscopic scales, the model predicts a reduction in spectral dimension to approximately 2 at the Planck scale. This dimensional flow affects the running of coupling constants and should leave a specific imprint on the polarization of the Cosmic Microwave Background (CMB), particularly in the B-mode spectrum, distinct from standard inflationary predictions. 3. Threshold Effects in Particle CollisionsThe breakdown of continuous Lorentz invariance at the fundamental scale suggests modifications to interaction thresholds. Specifically, the GZK cutoff for ultra-high-energy cosmic rays may be shifted or softened due to modified kinematics at extreme energies. 4. Geometric DecoherenceThe model predicts a fundamental limit to quantum coherence related to the fluctuations of the underlying spacetime geometry. This implies an intrinsic decoherence mechanism that could be detectable in highly sensitive interferometry experiments, potentially manifesting as a fundamental noise floor in gravitational wave detectors that exceeds standard quantum limits. 5. Dark Energy as Geometric EntropyThe accelerated expansion of the universe is derived not as a vacuum energy constant, but as a result of the entropic force driven by the relaxation of the simplicial lattice. This predicts a specific time-dependence of the dark energy equation of state parameter, w, which should deviate slightly from -1 in a manner correlated with the Hubble parameter's evolution. ---Co-authored with AI Gemini 3 Pro (Abacus. AI) and DeepAgent (Abacus. AI)