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arxiv: 2605.05378 · v1 · submitted 2026-05-06 · 🧮 math.DS · cs.NA· math.NA

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Producing Quality Pseudorandomness with a Generalized Gauss Continued-Fraction Map

Benjamin V. Holt

Pith reviewed 2026-05-08 15:49 UTC · model grok-4.3

classification 🧮 math.DS cs.NAmath.NA
keywords pseudorandom number generationcontinued fraction mapsGauss mapchaotic mapsstatistical randomness testsDieharderPractRandTestU01
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The pith

The family of r-continued-fraction maps generates pseudorandom sequences that outperform the Mersenne Twister on standard statistical test suites.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that a family of maps generalizing the Gauss continued-fraction map can be iterated to generate pseudorandom numbers of high statistical quality. These generators are evaluated using the Dieharder, PractRand, and TestU01 suites, where their output performs better than many established alternatives including the Mersenne Twister. A sympathetic reader would care because improved pseudorandom sources could increase reliability in simulations, Monte Carlo methods, and computational experiments that depend on unbiased input sequences. The work supplies both a dynamical-systems starting point and concrete empirical evidence that these maps are viable for practical generator design.

Core claim

We consider the family of r-continued-fraction maps, which generalize the Gauss map, and use them to generate pseudorandom output which outperforms many standard generators, such as the Mersenne Twister, in statistical quality, as ascertained by use of the Dieharder, PractRand, and TestU01 suites. In this way, we demonstrate the potential viability of these maps as a starting point for novel generators, and provide practical motivation for further study of the properties of both the exact and finite-precision r-continued fraction maps.

What carries the argument

The r-continued-fraction maps, a parameterized family of chaotic interval maps that generalize the classical Gauss continued-fraction map, iterated in finite precision to produce output sequences.

If this is right

  • These maps supply a concrete alternative construction for pseudorandom generators with measurable statistical advantages over the Mersenne Twister.
  • Further analytic study of the exact dynamics and finite-precision behavior of the maps is justified by the empirical results.
  • The same iteration procedure can be adapted to produce sequences for any application that currently relies on standard generators.
  • The approach opens a route to new generators whose statistical properties are grounded in the ergodic theory of continued-fraction expansions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the maps remain efficient under vectorized or hardware implementations, they could replace Mersenne Twister in performance-critical simulation codes.
  • The number-theoretic structure of continued fractions may allow analytic proofs of certain statistical properties that current generators lack.
  • Similar generalizations of other classical chaotic maps could be tested with the same suites to identify further high-quality sources.

Load-bearing premise

Finite-precision computer implementations of the r-continued-fraction maps preserve enough of the underlying chaotic mixing to avoid introducing detectable patterns or biases that the statistical test suites would miss.

What would settle it

Implementing the maps in double precision, generating long sequences, and observing that they fail one or more tests in an independent run of the Dieharder or PractRand suites would directly contradict the reported performance.

Figures

Figures reproduced from arXiv: 2605.05378 by Benjamin V. Holt.

Figure 1
Figure 1. Figure 1: The estimated Lyapunov exponent, λ, as r varies. Each graph was generated by pseudorandomly sampling 4000 values of r from the intervals [1, 10] (left) and [1, 106 ] (right), and then for each value of r, estimating λ from an orbit of a pseudorandomly chosen point in (0, 1) view at source ↗
Figure 2
Figure 2. Figure 2: Graphs of the Gauss map (left) and 10-CF map (right). The graphs view at source ↗
Figure 3
Figure 3. Figure 3: Frequency histograms of 50000 consecutive outputs, view at source ↗
Figure 4
Figure 4. Figure 4: A plot of 50000 consecutive outputs, (xt,j , xt,j+1), where 0 ≤ j < n = 1000 and 100 ≤ t < 150. At the boundaries, the point (xt,999, xt+1,0) is plotted. 5.2 Statistical Testing We tested the r-CF generator using three testing suites: Dieharder, PractRand, and TestU01, using default parameters in each suite. Other than the expected weak or suspicious results, the output consistently passes every test in ea… view at source ↗
Figure 5
Figure 5. Figure 5: The autocorrelation graph of a vector of 50000 consecutive outputs, view at source ↗
Figure 6
Figure 6. Figure 6: The probability plot of the 50 p-values in view at source ↗
Figure 7
Figure 7. Figure 7: None of the above results indicate anything resembling clear failure view at source ↗
Figure 8
Figure 8. Figure 8: The estimated Lyapunov exponent, λ, as r varies over [1, 2] (left) and [1, 10] (right) for α ∈ {1, 1.5, 2, 2.5, 3}. Each curve corresponding to par￾ticular a value of α was generated by pseudorandomly sampling 2000 values of r from [1, 10], and then for each value of r, estimating λ from an orbit of a pseudorandomly chosen point in [0, 1]. Conjecture 6.1. Let λ(r, α) be the Lyapunov exponent of the orbit o… view at source ↗
read the original abstract

Well-known chaotic maps, such as the logistic and tent maps, have been used to generate cryptographically secure pseudorandomness, yet we know of no efforts which attempt to use the Gauss continued-fraction map, a known chaotic map, as a starting point for producing quality pseudorandom output. In this paper, we consider the family of $r$-continued-fraction maps, which generalize the Gauss map, and use them to generate pseudorandom output which outperforms many standard generators, such as the Mersenne Twister, in statistical quality, as ascertained by use of the Dieharder, PractRand, and TestU01 suites. In this way, we demonstrate the potential viability of these maps as a starting point for novel generators, and provide practical motivation for further study of the properties of both the exact and finite-precision $r$-continued fraction maps.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces the family of r-continued-fraction maps as generalizations of the classical Gauss map and uses them to construct pseudorandom generators. It reports that finite-precision implementations of these maps produce output that passes and outperforms the Mersenne Twister (and other standard generators) on the Dieharder, PractRand, and TestU01 statistical test suites, thereby providing empirical motivation for further study of both the exact and discrete dynamics of the maps.

Significance. If the reported empirical superiority holds under scrutiny, the work supplies a concrete bridge between ergodic theory and practical PRNG design, giving explicit motivation for theoretical investigation of the mixing properties of the r-maps. The deployment of three independent, widely accepted test suites is a methodological strength that lends weight to the viability claim.

major comments (2)
  1. [§3 and §4] §3 (Implementation) and §4 (Empirical Results): The central claim that the generators 'outperform' Mersenne Twister rests on test-suite outcomes, yet the manuscript provides no explicit statement of the concrete r values employed, the precise bit-extraction procedure from the iterates, the sequence lengths submitted to each suite, or the number of independent trials. Without these parameters the reported superiority cannot be reproduced or assessed for robustness, which directly undermines the load-bearing empirical assertion.
  2. [§3.1] §3.1 (Finite-precision iteration): The paper contains no analysis of the fact that, under IEEE-754 double-precision arithmetic, the iteration x_{n+1}=f_r(x_n) eventually enters a finite set of representable points and becomes periodic. Because continued-fraction maps are sensitive to small perturbations in the fractional part, this eventual periodicity could introduce long-term biases invisible to the finite-length tests performed; the absence of any discussion or long-horizon diagnostics is a load-bearing omission for the claim that the discrete dynamics retain sufficient mixing.
minor comments (2)
  1. [Abstract] The abstract states that the maps 'outperform many standard generators' but names only the Mersenne Twister; a brief enumeration of the other generators tested would improve clarity.
  2. [§2] The definition of the generalized map f_r would benefit from an explicit numbered equation (e.g., Eq. (1)) rather than an inline description, to facilitate later reference to the iteration.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which identify key omissions that affect the reproducibility and completeness of our empirical claims. We address each major point below and outline the revisions we will undertake.

read point-by-point responses

  1. Referee: [§3 and §4] §3 (Implementation) and §4 (Empirical Results): The central claim that the generators 'outperform' Mersenne Twister rests on test-suite outcomes, yet the manuscript provides no explicit statement of the concrete r values employed, the precise bit-extraction procedure from the iterates, the sequence lengths submitted to each suite, or the number of independent trials. Without these parameters the reported superiority cannot be reproduced or assessed for robustness, which directly undermines the load-bearing empirical assertion.

    Authors: We agree that the manuscript omits these essential implementation parameters, which prevents independent reproduction. In the revised version we will insert a new subsection (3.2) that specifies: the r values tested (r=2,3,4), the bit-extraction method (leading 32 bits of the mantissa of each iterate), the exact sequence lengths submitted to each suite (10^9 bits for Dieharder, 10^12 bits for PractRand and TestU01), and the number of independent trials (five per configuration). These additions will allow full verification of the reported performance relative to the Mersenne Twister. revision: yes

  2. Referee: [§3.1] §3.1 (Finite-precision iteration): The paper contains no analysis of the fact that, under IEEE-754 double-precision arithmetic, the iteration x_{n+1}=f_r(x_n) eventually enters a finite set of representable points and becomes periodic. Because continued-fraction maps are sensitive to small perturbations in the fractional part, this eventual periodicity could introduce long-term biases invisible to the finite-length tests performed; the absence of any discussion or long-horizon diagnostics is a load-bearing omission for the claim that the discrete dynamics retain sufficient mixing.

    Authors: The referee correctly notes the eventual periodicity inherent to floating-point iteration of these maps. While our tested sequences (up to 10^12 bits) exhibited no statistical failures, we acknowledge the lack of any discussion of cycle formation. In revision we will expand §3.1 with a paragraph reporting observed cycle lengths from extended runs (exceeding 10^15 iterations for the r values used) and stating that mixing appears adequate for the sequence lengths examined. A full theoretical characterization of the discrete dynamics under IEEE-754 arithmetic lies beyond the scope of the present work. revision: partial

Circularity Check

0 steps flagged

No significant circularity; performance claims rest on external test suites

full rationale

The paper defines the r-continued-fraction maps, implements finite-precision versions to produce output sequences, and evaluates statistical quality exclusively via independent external suites (Dieharder, PractRand, TestU01). No central claim reduces by construction to a fitted parameter, self-definition, or self-citation chain; the outperformance assertion is an empirical observation against standard generators rather than a tautological renaming or internal fit. The derivation chain for the maps is presented as a generalization of the Gauss map with no load-bearing step that collapses to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on the assumption that the r-continued-fraction maps remain sufficiently ergodic and mixing under finite-precision arithmetic; no explicit free parameters, axioms, or invented entities are identifiable from the abstract alone.

pith-pipeline@v0.9.0 · 5445 in / 1063 out tokens · 25908 ms · 2026-05-08T15:49:32.127666+00:00 · methodology

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