Paper N v2.0: The Emergent Interval Derivation of Special Relativity, Time Dilation, Particle Stability, and Muon Lifetime from Frame-by-Frame Lattice Dynamics

Abstract

This paper derives the complete kinematic and dynamical framework of special relativity from first principles using only discrete lattice dynamics. No prior knowledge of Lorentz transformations, continuous spacetime, or quantum field theory is assumed. Part I: Emergent Kinematics Six axioms define a 3D FCC lattice with discrete time evolution. The central axiom (A3*) encodes two-tick memory: each node remembers two previous states. This single requirement generates the entire relativistic framework: Speed of light: c = ℓ/τ₀ (maximum cascade rate, 1 hop per tick). Explicit. Subluminal massive particles: v = c·U(W) < c (budget throttling). Explicit. Rest energy: E₀ = mc² (stationary self-replication cost). Explicit. Dispersion relation: E² = p²c² + m²c⁴ (from second-order wave dynamics). Explicit. Minkowski interval: ds² = c²dt² − dx² (emergent, not postulated). Explicit. Lorentz invariance: symmetry group of the wave equation on orthogonal lattice. Explicit. Chain of implication: Two-tick memory → Inertia → Second-order dynamics → Wave equation → Hyperbolic PDE → Lorentzian signature. Einstein's two postulates are derived, not assumed. Part II: Stochastic Lattice Dynamics Defect evolution is modelled as a stochastic counting process on FCC nodes, expressed in geobits — the natural information unit of the lattice (1 geobit = 1/Z_geom of full node capacity). Four independent results: Time dilation from information load (Explicit): dτ/dt = 1 − W/Z_geom. A heavier defect updates more slowly, experiencing less proper time per global tick. At channel saturation (W → Z_geom), proper time stops — deriving gravitational time dilation from information throttling. Absolute electron stability (Explicit): Charge conservation is a global constraint; lattice dynamics is local (k = 12 neighbors per tick). Their incompatibility forbids single-tick discharge. The electron is stable without invoking Noether's theorem — it is topological, not dynamical, protection. Phase-space identity (Explicit): The Fermi three-body phase-space factor 192π³ is identically equal to τ_proj^d · d · π^d = 4³ · 3 · π³ = (4π)³ · 3, revealing it as the projection volume — the cost of embedding a d-dimensional decay in a carrier with 4-bit projection tax. Muon lifetime (Ansatz, 96.8%): τ_μ = 2·Z_geom⁵·(144/89)⁵·(4φ³)⁵·(4π)³·3 / VEV × ℏ = 2.27 × 10⁻⁶ s. Experiment: 2.20 × 10⁻⁶ s. Zero free parameters. Every factor has an identified geometric origin. Key Results Table Result ORT Experiment Status Speed of light c = ℓ/τ₀ 2.998 × 10⁸ m/s Explicit Dispersion relation E² = p²c² + m²c⁴ Confirmed Explicit Minkowski interval ds² = c²dt² − dx² Confirmed Explicit Time dilation dτ/dt = 1 − W/Z_geom GR limit Explicit Electron stability p_D = 0 (isolated) > 10²⁸ yr Explicit Phase-space identity 192π³ = (4π)³·3 192π³ Explicit G_F 1.165 × 10⁻⁵ GeV⁻² 1.166 × 10⁻⁵ GeV⁻² Explicit (99.9%) Muon lifetime 2.27 × 10⁻⁶ s 2.20 × 10⁻⁶ s Ansatz (96.8%) τ_μ / τ_τ 7.43 × 10⁶ 7.6 × 10⁶ Ansatz (97.8%) Universal Factor (k−1)/2 = 5.5 The number of bidirectional evacuation channels on an FCC node — derived from 6 antipodal pairs minus half a blocked pair — governs both lepton decay ratios and the cosmological dark-matter-to-baryon ratio (Ω_DM/Ω_b = 5.5; experiment: 5.47; accuracy 99.5%). One geometry, two consequences: particle physics and cosmology are projections of a single lattice. Falsifiability Planck-scale Lorentz violation: modified dispersion relation with η·p⁴c⁴/E_P² correction. Testable via gamma-ray burst timing (Fermi LAT). If Lorentz invariance is exact beyond E > 10²⁰ GeV, ORT lattice spacing is falsified. If diffusive (first-order) particle dynamics are ever observed, Axiom 3* is falsified. What's New in v2.0 Part II added: complete stochastic dynamics framework (counting process, martingale, geobits) Time dilation derived from information-load throttling Electron stability proved from locality + global charge Phase-space identity 192π³ = (4π)^d · d discovered and proved Muon lifetime computed to 96.8% accuracy with zero free parameters Lifetime ratio τ_μ/τ_τ computed to 97.8% accuracy Consistency with Paper L dynamics established via U(W) = 1 − W/Z_geom Axiom 3* linked to jet tower theorem (Paper Zero) Dependencies Paper Zero v1.1 (jet tower, source equation) · Paper A v9.1 (Z_geom, impedance sectors) · Paper B v2.0 (lepton cascade operators) · Paper G v1.2 (information bottleneck, K_cell) · Paper M v3.0 (mass from closure, VEV) · Paper Q v2.1 (executability, FCC) · Paper S v2.0 (Z₂ symmetry) · Dark Matter Letter v1.0 Open Problems N-1: Exact Lorentz violation parameter η from FCC geometry N-2: Explicit rewrite rule R consistent with martingale + Lorentz N-3: Absolute tau lifetime including hadronic channels N-4: Phase-space factor from lattice first principles N-5: Exact W-to-mass mapping from carrier geometry N-6: Proof that (k−1)/2 enters decay rates from FCC combinatorics The lattice speaks. Zero parameters. One geometry.