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Statistical Quantum Mechanics of the Random Permutation Sorting System (RPSS): A Self-Stabilizing True Uniform RNG

Randy Kuang

TL;DR

RPSS addresses the need for a robust, platform-agnostic source of true uniform randomness by combining combinatorial permutation entropy with physical system jitter within a quantum-inspired statistical framework. The core idea is to treat the permutation count $N_p$ and elapsed time $T$ as conjugate observables whose joint distribution, when projected modulo $2^n$, yields uniformly distributed $n$-bit outputs. The paper provides a formal model, derives distributional properties (negative binomial for $N_p$, compound for $T$), and proves that modular reduction produces near- to essentially-uniform outputs under entropy-convergence conditions, validated by min-entropy, $\chi^2$, and CLT-based tests. Practically, RPSS offers a software-defined TURNG that is self-stabilizing, cross-platform, and applicable to cryptography, blockchains, and post-quantum cryptography contexts.

Abstract

We present the Random Permutation Sorting System (RPSS), a novel framework for true uniform randomness generation grounded in statistical quantum mechanics. RPSS is built on a pair of conjugate observables, the permutation count and the elapsed sorting time, whose heavy-tailed raw distributions synchronously converge to uniformity through modular reduction. This mathematically proven convergence establishes RPSS as a True Uniform Random Number Generator (TURNG). A practical implementation, QPP-RNG, demonstrates how intrinsic system jitter, arising from microarchitectural noise, memory latency, and scheduling dynamics, interacts with combinatorial complexity to yield a compact, self-stabilizing entropy source. Empirical validation under the NIST SP 800-90B framework confirms rapid entropy convergence and statistically uniform outputs. RPSS thus defines a new class of quantum-inspired entropy engines, where randomness is simultaneously harvested from unpredictable system jitter and amplified by combinatorial processes, offering a robust, platform-independent alternative to conventional entropy sources.

Statistical Quantum Mechanics of the Random Permutation Sorting System (RPSS): A Self-Stabilizing True Uniform RNG

TL;DR

RPSS addresses the need for a robust, platform-agnostic source of true uniform randomness by combining combinatorial permutation entropy with physical system jitter within a quantum-inspired statistical framework. The core idea is to treat the permutation count and elapsed time as conjugate observables whose joint distribution, when projected modulo , yields uniformly distributed -bit outputs. The paper provides a formal model, derives distributional properties (negative binomial for , compound for ), and proves that modular reduction produces near- to essentially-uniform outputs under entropy-convergence conditions, validated by min-entropy, , and CLT-based tests. Practically, RPSS offers a software-defined TURNG that is self-stabilizing, cross-platform, and applicable to cryptography, blockchains, and post-quantum cryptography contexts.

Abstract

We present the Random Permutation Sorting System (RPSS), a novel framework for true uniform randomness generation grounded in statistical quantum mechanics. RPSS is built on a pair of conjugate observables, the permutation count and the elapsed sorting time, whose heavy-tailed raw distributions synchronously converge to uniformity through modular reduction. This mathematically proven convergence establishes RPSS as a True Uniform Random Number Generator (TURNG). A practical implementation, QPP-RNG, demonstrates how intrinsic system jitter, arising from microarchitectural noise, memory latency, and scheduling dynamics, interacts with combinatorial complexity to yield a compact, self-stabilizing entropy source. Empirical validation under the NIST SP 800-90B framework confirms rapid entropy convergence and statistically uniform outputs. RPSS thus defines a new class of quantum-inspired entropy engines, where randomness is simultaneously harvested from unpredictable system jitter and amplified by combinatorial processes, offering a robust, platform-independent alternative to conventional entropy sources.

Paper Structure

This paper contains 24 sections, 2 theorems, 45 equations, 4 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Theory and Model
  3. The RPSS System
  4. Statistical Mechanism of the RPSS System
  5. Statistical Laws of RPSS Observables
  6. Permutation Count
  7. Elapsed-Time Observable $\hat{T}$
  8. Modular Uniformization of RPSS Observables
  9. Entropy Analysis and Convergence of the RPSS
  10. Self-Converged TURNG
  11. Experiments and Discussions
  12. Measurements of the Conjugate Observables
  13. Raw Distributions of the Conjugate Observables
  14. Modular-Reduced Distributions of the Conjugate Observables
  15. Clarification of Near-Uniform and Essentially Uniform Behavior
  16. ...and 9 more sections

Key Result

Theorem A.1

Let $\hat{N}_p \sim \text{NegativeBinomial}(m, p)$ with mean $M = m/p$, and let $R = 2^n$ be a modulus. For any fixed integer $m \ge 1$ and fixed $R$, the modular reduction $\tilde{n}_p = \hat{N}_p \bmod R$ is asymptotically uniform as $M \to \infty$ (or equivalently, as $p \to 0$). That is, for any

Figures (4)

  • Figure 1: Raw permutation count distributions for sorting the disordered array $\{3, 2, 0, 1\}$ at repetition values $m = 1, 2, 3$.
  • Figure 2: Raw elapsed-time distributions (in ticks) for sorting the disordered array $\{3, 2, 0, 1\}$ at repetition values $m = 1, 2, 3$.
  • Figure 3: Modular-reduced permutation count distributions for sorting the disordered array $\{3, 2, 0, 1\}$ at repetition value $m = 3$.
  • Figure 4: Modular-reduced elapsed-time distributions (in ticks) for sorting the disordered array $\{3, 2, 0, 1\}$ at repetition value $m = 3$.

Theorems & Definitions (4)

  • Theorem A.1: Uniformity of $\tilde{n}_p$
  • proof
  • Theorem A.2: Uniformity of $\tilde{T}$
  • proof