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Counting the number of $1_{m}$-preperiodic $\mathcal{O}_{K}$-points of a discrete dynamical system with applications from arithmetic statistics, VII

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Pith's one-line read The main theorem is false: for φ_{p,c}(z)=z^p+c over F_p with p|c, every point is fixed, so the number of 1_n-preperiodic points is 0, not p.

arxiv 2606.14468 v2 pith:ZUKCDVSL submitted 2026-06-12 math.NT math.DS

Brian Kintu This is my paper
classification math.NTmath.DS
keywords preperiodicarithmeticnumberpointsvarphifixedmathbbmathcal
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies maps of the form f(z)=z^d+c modulo a prime p. For d=p, Fermat's little theorem says z^p=z in F_p, so f(z)=z+c. If c is a multiple of p, then f(z)=z and every point stays fixed forever. The paper's definition of a 1_n-preperiodic point explicitly requires the point to move after the first step, so the true count is zero. The paper instead claims the count is p. A concrete check: with p=3 and c=3, the map is z^3≡z, so all three points are fixed and no 1_1-preperiodic point exists, while Theorem 2.2 predicts 3.

The same mistake appears for d=p−1. With p=5 and c=0, the true number of such points is 3, not the claimed 5; with p=5 and c=1, the true number is 3, not the claimed 4. In every case the proof only shows that a difference polynomial vanishes, but the definition also excludes points that are already periodic. That exclusion is never enforced. Later sections build averages, densities, zeta functions, and number-field statistics on these false counts, so those results are not established.

Extended reading notes

Core claim

Theorem 2.2: For p≥3 and φ_{p,c}(z)=z^p+c, the count N_c^{(1_n)}(p) of 1_n-preperiodic points modulo p equals p if p|c and 0 otherwise. Theorem 3.2 similarly claims M_c^{(1_n)}(p)=p, p−1, or 0 depending on c mod p. These counts are the load-bearing content; the averages in §5, densities in §6-8, and all later arithmetic-statistics corollaries are derived from them.

Load-bearing premise

In both central proofs, the author shows f(z)=φ^{n+1}(z)-φ(z)≡0 mod p for the relevant c and then concludes that the defined count N or M is the number of roots of f. This ignores the explicit exclusion φ^n(z)-z≠0 in equations (1)-(2). Under p|c, φ(z)=z^p≡z, so every z satisfies φ^n(z)-z≡0 and the true count is 0, not p. Under c≡1 mod p in Theorem 3.2, the proof counts eventually-fixed points that are themselves fixed, giving 3 instead of 4 at p=5,n=1.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted; p, n, c are variables. The main hidden postulate is the equivalence between the defining set and a root set, which is false. The later sections import heavy external theorems under conditional nonemptiness assumptions.

assumptions (3)
  • standard math Fermat's Little Theorem: z^p≡z in F_p and z^{p-1}=1 for z≠0.
    Used throughout §2-3 to reduce iterates of z^p+c; the fact is correct, but the paper draws incorrect conclusions from it.
  • ad hoc to paper The zero set of f(z)=φ^{n+1}(z)-φ(z) mod p is the set counted by N or M.
    This is the load-bearing false premise introduced in the proofs of Theorems 2.2 and 3.2; it silently drops the exclusion φ^n(z)-z≠0 from definitions (1) and (2).
  • domain assumption The families of fields K_f and L_g are nonempty and satisfy the hypotheses of [5], [11], [14], [31], [32].
    Sections 10-14 assert density/equidistribution results under 'Assume Cor 8.1/8.2' without verifying that the thin dynamical subfamily inherits the global hypotheses of those theorems.

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Pith. "Pith review of Counting the number of $1_{m}$-preperiodic $\mathcal{O}_{K}$-points of a discrete dynamical system with applications from arithmetic statistics, VII." pith.science (2026). https://pith.science/paper/ZUKCDVSL

@misc{pith2026260614468,
  author       = {Pith},
  title        = {Pith review of: Counting the number of $1_m$-preperiodic $\mathcalO_K$-points of a discrete dynamical system with applications from arithmetic statistics, VII},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZUKCDVSL}},
  note         = {Machine review of arXiv:2606.14468}
}
abstract

In this follow-up article of a multi-part series on (strictly) preperiodic point-counting, we inspect an astonishing relationship between the set of (strictly) $1_{m}$-preperiodic points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ for all $c, z \in \mathcal{O}_{K}$ and the coefficient $c$, where $K$ is any number field of degree $n\geq 1$, $d>2$ is an integer and $m\in \mathbb{Z}_{\geq 1}$ is any fixed (eventual period). As before, we wish to study counting problems that are inspired by torsion point-counting in arithmetic statistics and (strictly) preperiodic point-counting in arithmetic dynamics. In doing so, we then first prove that for any prime $p\geq 3$ and for any fixed $\ell \in \mathbb{Z}_{\geq 1}$ and fixed (eventual period) $m\in \mathbb{Z}_{\geq 1}$, the average number of distinct $1_{m}$-preperiodic integral points of any odd degree map $\varphi_{p^{\ell}, c}$ modulo prime ideal $p\mathcal{O}_{K}$ is unbounded or zero as $c$ tends to infinity. Inspired further by work of Doyle-Poonen, along with conjectural work of Hutz and $\textit{abc}(\textit{d})$-conditional work of Panraksa on $K$-rational preperiodic points of any even degree map $\varphi_{(p-1)^{\ell}, c}$ for any prime $p\geq 5$ in arithmetic dynamics, we then also prove that for any fixed (eventual period) $m \in \mathbb{Z}_{ \geq 1}$, the average number of distinct $1_{m}$-preperiodic integral points of any $\varphi_{(p-1)^{\ell}, c}$ modulo prime ideal $p\mathcal{O}_{K}$ is unbounded or zero as $c\to \infty$. Finally, we then apply density, polynomial- and number field-counting, and Sato-Tate equidistribution results from arithmetic statistics, and thereby obtaining further a stream of counting and statistical results on arithmetic objects that arise naturally in our polynomial discrete dynamical settings.

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