Statistical quantum mechanics of the random permutation sorting system: A self-stabilizing true uniform random number generator.

Abstract

We introduce the random permutation sorting system (rpss), a software-defined framework that generates high-quality, uniform randomness by leveraging a statistical model with properties analogous to quantum ensembles. The RPSS operates as a self-contained entropy engine based on a fundamental pair of conjugate observables: the permutation count (N[over ̂]_{p}) and the elapsed sorting time (T[over ̂]). While the raw distributions of these observables are highly right-skewed and heavy-tailed, we provide a mathematical proof of their synchronous convergence to uniformity. The permutation count distributions follow a negative binomial model, and the observed modes for different repetition parameters m closely match the theoretical predictions, confirming the validity of the model. This convergence arises from the compounded effect of permutation counts on elapsed time and a process of modular uniformization, in which a vast number of microstates-stemming from combinatorial complexity and intrinsic system-level jitter-map onto a finite set of uniform symbols. This mechanism enables the RPSS to function as a true uniform random number generator. We implemented the RPSS as the qpp-rng, a practical software realization that autonomously harvests entropy from a platform's intrinsic microarchitectural dynamics. Unlike prior benchmark-focused studies, this work provides a deep theoretical and experimental investigation of the underlying physical mechanism. Empirical validation confirms our predictions, demonstrating rapid entropy convergence and statistically uniform outputs, as assessed by NIST SP 800-90B min-entropy and χ-squared statistics, consistent with central-limit-theorem expectations. The RPSS establishes a class of quantum-inspired ecocryptosystems, in which randomness is simultaneously harvested from unpredictable hardware behavior and amplified through a combinatorial process. This dual-source approach provides a compact, self-stabilizing entropy engine suitable for foundational applications in cryptography, blockchain protocols, and digital currency systems, offering a robust, platform-agnostic alternative to conventional entropy sources.