This paper introduces RGE-256, a new 256-bit ARX-based pseudorandom number generator (PRNG) designed to promote strong internal state diffusion through a structured combination of modular addition, rotation, XOR operations, and cross-coupled mixing flows. RGE-256 employs a dual-quad state update mechanism with cascaded nonlinear mixing, incorporates a geometric rotation schedule derived from Recursive Division Tree (RDT) entropy constants, and supports an optional BLAKE3-based whitening layer applied to output blocks. We evaluate RGE-256 using the full Dieharder test suite (114 tests), along with avalanche analysis, autocorrelation measurements, and bit-distribution analysis. Comprehensive streaming validation shows that all RGE-256 variants complete Dieharder without failures, pass the full SmokeRand default battery (42 tests) with Quality 4.0 ratings, and process approximately 145 GB per core. Internal measurements indicate near-maximal entropy (7.999999 bits per byte), an ideal avalanche effect (15.97 bits), and near-zero autocorrelation. Independent testing further reports successful completion of TestU01 BigCrush and PractRand testing up to 1 TiB of output.