Prime Power-of-Two Graphs: The Hidden Topology of the Primes

Open-source computational framework for the Prime Power-of-Two Graph Author: Tanmay Singh (Independent Researcher) Year: 2026 The Prime Power-of-Two Graph (G(N)) has: Vertices: odd primes (p \le N) Edges: pairs of primes that differ by a power of two ((|p-q| = 2^k), (k \ge 1)) This engine builds and analyses (G(N)) efficiently up to (N = 10^9) (on machines with ≥ 32 GB RAM) using: Memory-efficient odd-only segmented sieve Streaming edge generation (no full adjacency list) Dense NumPy Union-Find (path compression + union-by-rank) Degree statistics, connected components, persistence (T(p)) Isolated-prime analysis with Miller–Rabin escape tests Heuristic covering-certificate search Automatic figure generation and markdown report All conclusions produced by the software are computational observations, never proofs.