Emergence of Newtonian Gravity and Lepton Mass Hierarchy from String Rewriting Systems on Power-Law Graphs

We present a computational framework demonstrating that fundamental physical laws can emerge from discrete string rewriting systems defined on graphs with power-law connectivity. By constructing onedimensional lattices augmented with long-range connections following a probability distribution P (d) ∼ d −α , we solve the Poisson equation on the resulting graph Laplacian to derive an effective gravitational potential. Through systematic grid search over the parameter space (α, N), we identify α = 2.2 as producing a gravitational force law F ∼ r −2.003 , deviating from Newtonian gravity by merely 0.34%. Furthermore, we model three generations of leptons using string rewriting rules of lengths L = 3, 5, 7, deriving a two-parameter mass formula that reproduces the experimental mass ratios m µ /m e = 206.8 and m τ /m µ = 16.8 with zero error. The framework suggests a unified geometric origin for both gravitational and particle physics phenomena through a single structural parameter α ≈ 2.