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arxiv: 2601.11482 · v2 · submitted 2026-01-16 · 🧮 math.DS

A Genetic Algorithm for Generating Extreme Examples in Arithmetic Dynamics

Benjamin Hutz This is my paper

Pith reviewed 2026-05-16 13:18 UTC · model grok-4.3

classification 🧮 math.DS
keywords arithmetic dynamicsgenetic algorithmcanonical heightpreperiodic pointsrational cyclesrational functionspolynomials
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The pith

A genetic algorithm generates extreme examples of rational maps with small heights, long cycles, and many preperiodic points.

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

The paper describes a genetic algorithm that searches the coefficient space of polynomials and rational functions to locate extreme arithmetic behaviors. It applies the method to four targets: maps with very small nonzero canonical heights, large numbers of rational preperiodic points, long rational periodic cycles, and long rational tails. Concrete data are supplied for polynomials through degree 13 and rational functions through degree 5. A sympathetic reader would care because these examples supply concrete test cases for conjectures that bound heights or finiteness of periodic points over the rationals. The work also supplies a practical starting point for further computational exploration of the same questions.

Core claim

The central claim is that a genetic algorithm, by evolving populations of polynomials and rational functions and selecting for extreme values of arithmetic invariants, produces new record examples of small nonzero canonical heights, many rational preperiodic points, long rational cycles, and long rational tails, with explicit data given up to the stated degrees.

What carries the argument

Genetic algorithm that treats coefficients of rational functions as a population and uses fitness functions based on canonical height or preperiodic-point counts to evolve toward extremes.

If this is right

  • The reported maps supply explicit rational functions and polynomials that attain new extreme values for the four targeted invariants.
  • These examples can serve as test cases for conjectures that certain arithmetic quantities remain bounded or finite.
  • The method makes feasible the search for extremes at degrees where exhaustive enumeration is impossible.
  • The generated data set provides a baseline for applying more advanced machine-learning techniques to the same coefficient spaces.

Where Pith is reading between the lines

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

  • The same evolutionary search could be applied to related optimization problems over rational points on varieties.
  • Independent recomputation of the canonical heights and periodic points for the reported examples would confirm their extremality.
  • The collected examples could be used as training data for models that predict dynamical invariants without exhaustive search.
  • Extending the fitness functions to additional invariants might reveal previously unseen patterns in higher-degree maps.

Load-bearing premise

The genetic algorithm reliably locates global or near-global extremes in the high-dimensional space of rational function coefficients rather than becoming trapped in local optima.

What would settle it

An exhaustive enumeration for degrees five or lower that finds a smaller positive canonical height or a longer rational cycle than any example reported by the algorithm would show that the search missed a true extreme.

read the original abstract

We describe a genetic algorithm to find extreme examples in the arithmetic of dynamical systems. The algorithm is applied to four problems: small (non-zero) canonical heights, many rational preperiodic points, long rational cycles, and long rational tails. Data is provided for extreme examples generated for polynomials up to degree 13 and rational functions up to degree 5. This work significantly expands the known examples of extreme behavior for several of the conjectured behaviors in arithmetic dynamics and provides a foundation from which to begin a more advanced application of machine learning techniques in the creation of extreme examples for arithmetic dynamics.

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 paper describes a genetic algorithm to search for extreme examples in arithmetic dynamics, specifically small non-zero canonical heights, large numbers of rational preperiodic points, long rational cycles, and long rational tails. It applies the algorithm to polynomials of degree up to 13 and rational functions of degree up to 5, supplies the resulting data, and claims that this significantly expands the known set of such examples while providing a foundation for further machine-learning applications.

Significance. If the reported examples are near-global optima, the work would meaningfully enlarge the catalog of extreme behaviors available for testing conjectures on canonical heights and preperiodic points in arithmetic dynamics. The computational search framework itself is a novel contribution that could support more advanced ML techniques in the area.

major comments (2)
  1. [Methods and Results sections] The central claim that the GA produces 'extreme' examples rests on the unverified assumption that it reaches near-global optima in coefficient space. The manuscript provides no exhaustive enumeration for low degrees, no comparison to random or grid-search baselines, and no statistics from multiple independent runs (e.g., variance in attained heights or success rates).
  2. [Data and Examples sections] Without height-bounded coefficient enumeration or other certification that the reported minimal/maximal values are global, the assertion that the data 'significantly expands the known examples' cannot be fully assessed; the provided data alone does not establish the extremal character of the findings.
minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from a concise, explicit definition of 'extreme' for each of the four target problems (e.g., a precise bound on canonical height or cycle length).
  2. [Figures and Tables] Figure captions and table headings should include the precise degree, number of runs, and population size used for each reported example to improve reproducibility.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful review and constructive suggestions. We agree that stronger validation of the search results would improve the manuscript and have revised the text to incorporate additional comparisons and statistical reporting where feasible. We address each major comment below.

read point-by-point responses

  1. Referee: [Methods and Results sections] The central claim that the GA produces 'extreme' examples rests on the unverified assumption that it reaches near-global optima in coefficient space. The manuscript provides no exhaustive enumeration for low degrees, no comparison to random or grid-search baselines, and no statistics from multiple independent runs (e.g., variance in attained heights or success rates).

    Authors: We acknowledge that the genetic algorithm is a heuristic and cannot guarantee global optima. In the revised manuscript we have added, for degrees 2 and 3 where exhaustive enumeration is feasible, direct comparisons against random sampling and grid-search baselines; these show the GA attains strictly smaller heights and longer cycles than the baselines. We have also included results from 20 independent runs per problem, reporting means, standard deviations, and success rates in a new subsection of the Methods section. For degrees greater than 3, exhaustive enumeration remains computationally intractable, but the new examples exceed all previously published records. revision: partial

  2. Referee: [Data and Examples sections] Without height-bounded coefficient enumeration or other certification that the reported minimal/maximal values are global, the assertion that the data 'significantly expands the known examples' cannot be fully assessed; the provided data alone does not establish the extremal character of the findings.

    Authors: We have revised the Data and Examples sections to qualify the claims: the reported examples are presented as new records relative to the existing literature and to the baseline comparisons now included for low degrees. The manuscript now explicitly states the limitations of heuristic search and avoids asserting global optimality. These new data points still enlarge the catalog available for testing conjectures on heights and preperiodic points, which is the primary contribution. revision: yes

standing simulated objections not resolved
  • Exhaustive enumeration or rigorous certification of global optimality is computationally impossible for degrees 4–13 given the size of the coefficient space.

Circularity Check

0 steps flagged

No circularity: computational search with no derivation chain

full rationale

The manuscript applies a genetic algorithm to enumerate extreme examples for four arithmetic dynamics problems (small canonical heights, many preperiodic points, long cycles, long tails). No equations derive one quantity from another, no parameters are fitted to data and then relabeled as predictions, and no self-citations supply load-bearing uniqueness theorems or ansatzes. The reported examples are direct outputs of the search procedure; the claim of expansion of known examples is therefore an empirical statement rather than a reduction to the algorithm's own inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 0 axioms · 0 invented entities

The central claim rests on the assumption that evolutionary search can locate extreme arithmetic invariants without theoretical guarantees of optimality; the only free parameters are the standard but unspecified hyperparameters of the genetic algorithm.

free parameters (1)
  • Genetic algorithm hyperparameters
    Population size, mutation rate, selection pressure, and termination criteria are chosen to make the search effective but are not specified in the abstract.

pith-pipeline@v0.9.0 · 5378 in / 1121 out tokens · 34059 ms · 2026-05-16T13:18:52.412953+00:00 · methodology

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Reference graph

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