Fractal Entropic Geometrodynamics: Emergent Gravity and Three Particle Generations from Discrete Topological Constraints

We present a strictly discrete framework in which General Relativityand the three-generation structure of the Standard Model emerge fromtwo axioms and one principle. An O (N) simulation engine---implemented in Rust and freelyavailable---Poisson-sprinkles events into a 4D causal diamond, builds the Hasse diagram, and detects topological defects calledCausal Prisms (bipartite subgraphs). A discrete phase invariant based on the sign trichotomy of bulk momentumclassifies prisms into exactly three generations (g 3), producinga mass hierarchy Gen\, 1\,: \, Gen\, 2\,: \, Gen\, 3 = 4. 55: 6. 53: 7. 73and CPT symmetry---from pure combinatorics. The spectral dimension flows from (=1) = 1. 95 0. 002in the UV, crossing = 4 at 3--4. A finite-size scaling analysis across five lattice sizes (N = 10⁵ to 10⁷, M 8 realisations each) reveals thatall boundary-sensitive observables follow the 4D scaling lawO (N) = O_ + a\, N^-1/4 (R² = 0. 9996 for Qₓ₎₏₎), whiletopological invariants---mass ratios and generationfractions---are N-independent. The emergent fine structure constant = Qₓ₎₏₎/ (8) and self-energy ratio = 1/Qₓ₎₏₎ - 1are locked by the exact identity (1+) = 1/ (8) ;the bare topological coupling ₀^-1 = 165. 1 1. 0 setsthe UV-cutoff value for renormalization-group studies. Gravity is the Fisher information cost of macroscopic deformation;the Einstein equations are thermodynamic. Bounded Hasse degree (D 15) makes the computation O (N), providing a proof---not an assertion---that the physics is local.