Statistical quantum mechanics of the random permutation sorting system: A self-stabilizing true uniform random number generator
Phys. Rev. E 113, 024117 – Published 12 February, 2026
DOI: https://doi.org/10.1103/24hx-9y3l
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 and the elapsed sorting time . 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 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.
