Number-Theoretic Graphs with Small-World Navigability: Percolation, Structure, and Greedy Routing in MirrorDNA

We study a family of number-theoretic graphs Gn defined on vertex set {1, 2, …, n} where edges are determined by arithmetic predicates on vertex pairs. We systematically vary the edge predicate from strict (Sum=Prime ∧ Diff=Pow2) to relaxed (Sum=Prime ∨ Diff=Pow2) and map the resulting percolation phase transition. Our empirical analysis on a real-world knowledge vault of n = 5,090 nodes reveals five key findings: The critical percolation threshold occurs at k* = 19 ≈ 1.54·log₂n, producing a connected graph at just 0.10% density The power-of-2 difference graph Dn exhibits strong small-world structure (σ = 19.96) with clustering 28× higher than equivalent random graphs Greedy routing on Dn succeeds with 100% reliability, achieving 97.4% BFS-optimal paths with mean stretch 1.008 The prime-sum graph Pn is provably triangle-free and bipartite We conjecture k*/log₂n ∈ [1.0, 2.0] for all n, supported by data across seven scales All results validated against stored vault data (100.00% edge match, 10-seed ER baselines) with 221 passing tests. The supplementary zip contains the full mirrorgraph Python framework (10 modules), analysis scripts, result JSONs, publication figures including screenshots of the deployed vault on mobile, and SHA256 checksums.