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Composition Law of Conjugate Observables in Random Permutation Sorting Systems

Yurang R. Kuang

TL;DR

This work addresses generating cryptographic-grade randomness from general-purpose computation by formulating a composition law for RPSS, a framework with two conjugate observables: the discrete permutation count $\hat{N}_p$ and the continuous elapsed time $\hat{T}$. The central result, $\varphi_{\hat{T}}(\omega) = G_{\hat{N}_p}(\varphi_X(\omega))$, links the characteristic function of timing to the probability generating function of permutation counts, enabling entropy purification via geometric convergence under modular reduction. The authors prove convergence bounds and validate them empirically, achieving Shannon entropy $H > 7.9998$ bits per byte with near-uniform distributions across diverse platforms, culminating in a self-contained TURNG (QPP-RNG) that adapts to environment while remaining provably uniform. This software-defined, hardware-agnostic approach provides a practical path to cryptographic-grade randomness suitable for post-quantum security, with strong resistance to timing-based attacks and straightforward integration into cryptographic modules.

Abstract

We present the discovery of a fundamental composition law governing conjugate observables in the Random Permutation Sorting System (RPSS). The law links the discrete permutation count Np and the continuous elapsed time T through a functional relation connecting the characteristic function of timing distributions to the probability generating function of permutation counts. This framework enables entropy purification, transforming microarchitectural timing fluctuations into uniform randomness via geometric convergence. We establish convergence theorems with explicit bounds and validate the results experimentally, achieving Shannon entropy above 7.9998 bits per byte and chi-square uniformity across diverse platforms. The composition law provides a universal foundation for generating provably uniform randomness from general-purpose computation, securing cryptographic purity from emergent computational dynamics.

Composition Law of Conjugate Observables in Random Permutation Sorting Systems

TL;DR

This work addresses generating cryptographic-grade randomness from general-purpose computation by formulating a composition law for RPSS, a framework with two conjugate observables: the discrete permutation count and the continuous elapsed time . The central result, , links the characteristic function of timing to the probability generating function of permutation counts, enabling entropy purification via geometric convergence under modular reduction. The authors prove convergence bounds and validate them empirically, achieving Shannon entropy bits per byte with near-uniform distributions across diverse platforms, culminating in a self-contained TURNG (QPP-RNG) that adapts to environment while remaining provably uniform. This software-defined, hardware-agnostic approach provides a practical path to cryptographic-grade randomness suitable for post-quantum security, with strong resistance to timing-based attacks and straightforward integration into cryptographic modules.

Abstract

We present the discovery of a fundamental composition law governing conjugate observables in the Random Permutation Sorting System (RPSS). The law links the discrete permutation count Np and the continuous elapsed time T through a functional relation connecting the characteristic function of timing distributions to the probability generating function of permutation counts. This framework enables entropy purification, transforming microarchitectural timing fluctuations into uniform randomness via geometric convergence. We establish convergence theorems with explicit bounds and validate the results experimentally, achieving Shannon entropy above 7.9998 bits per byte and chi-square uniformity across diverse platforms. The composition law provides a universal foundation for generating provably uniform randomness from general-purpose computation, securing cryptographic purity from emergent computational dynamics.

Paper Structure

This paper contains 49 sections, 1 theorem, 29 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Results
  3. Definition and Compound Structure
  4. Composition Law of Conjugate Observables
  5. Derivation via Characteristic Functions
  6. Moment Structure of the Compound Distribution
  7. Moments of $\hat{N}_p$.
  8. Mean and Variance of $\hat{T}$.
  9. Higher Cumulants of $\hat{T}$.
  10. Statistical Implications.
  11. Synchronous Convergence to Uniformity
  12. Key Implication.
  13. Empirical Runtime Distribution of Individual Permutations
  14. Cryptographic Entropy Strength and Attack Resistance
  15. Statistical Variability as Entropy Source
  16. ...and 34 more sections

Key Result

Theorem 1

Let $\hat{N}_p \sim \mathrm{NB}(m, p)$ with $p = 1/N!$, and let where $X_j$ are i.i.d. positive random variables with finite mean $\mu_X$ and variance $\sigma_X^2$. Under reduction modulo $R$, as $M = m N! \to \infty$, both observables converge to the discrete uniform distribution with exponential convergence rates: for $k = 0,1,\dots,R-1$, where $0 < \rho_N, \rho_T < 1$ are geometric decay cons

Figures (3)

  • Figure 1: Empirical distribution of permutation count $\hat{N}_p$ for ($N=4, m=4$).
  • Figure 2: Empirical raw elapsed time $\hat{T}$ distributions exhibiting fat, skinny, and ultra-skinny morphological phenotypes under identical parameters ($N=4, m=4$). Dramatic shape variations demonstrate system sensitivity to microarchitectural conditions.
  • Figure 3: Empirical modular-reduced permutation count and elapsed time distribution ($\hat{T} \bmod R$). Histogram flatness demonstrates TURNG output uniformity across morphological phenotypes.

Theorems & Definitions (2)

  • Theorem 1: Synchronous Convergence of RPSS Observables
  • proof