Diagnostics and Stability Analysis of PRG-XOR Sequences: Pseudo-deterministic Isolation and Rank Stability

This work presents a systematic diagnostic framework for evaluating pseudo-random generator XOR sequences (PRG-XOR) in comparison with purely random XOR sequences. The study employs rank-based metrics, deficiency rates, and Pseudo-deterministic Stability Tests (PST) to quantify the structural integrity and reproducibility of PRG-XOR constructions in Unique-SAT isolation tasks. Results demonstrate full-rank stability, low variance, and robust pseudo-deterministic behavior, providing empirical evidence toward the derandomization of the Isolation Lemma and potential support for the $P = BPP$ hypothesis. The paper includes computational tools, visualizations, and code appendices for reproducibility.