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 $= 4.55 : 6.53 : 7.73$and CPT symmetry---from pure combinatorics.The spectral dimension flows from (σ=1) = 1.95 ± 0.002in the UV, crossing = 4 at σ ≈ 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 Qₜₒₚₒ), whiletopological invariants---mass ratios and generationfractions---are N-independent.The emergent fine structure constant α = Qₜₒₚₒ/(8π)and self-energy ratio Ω = 1/Qₜₒₚₒ - 1are locked by the exact identity α(1+Ω) = 1/(8π);the bare topological coupling α₀⁻¹ = 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.