# Verification report — AMR-152-0049 (Hénon problem list, Section 11 (P. Ingram), Conjecture 4: rational periodic points of quadratic Hénon maps)

Verification dates: 2026-10-17/18 (two independent verification runs on the first written version of the results);
2026-10-18 (the note as written: the program `check_note.py`, and a third independent run, whose programs are in
`reproducibility/independent_run_2/`).

**Verdict.** The note shows that Conjecture 4 of Section 11 of the list, **as printed, is false for every δ ∈ Q**,
in each reading the text supports; it also describes all rational 3-cycles of the family.
- **The statement.** For f_{c,δ}(x, y) = (y, y² + c − δx) with c, δ ∈ Q, the conjecture asserts that for all but
  finitely many δ, every Q-rational periodic point of every f_{c,δ} has period 1 or 2 ("period dividing 2").
- **Settled.** Theorem 1.1: for every δ ≠ 1 the map with c = −(29δ² − 38δ + 29)/16 has the rational 3-cycle with
  x-coordinates −1/2 + 7t, −1/2 + t, −1/2 − 5t (t = (1 − δ)/4); for δ = 1 and μ ≠ 0 the map with
  c = −1 − (μ − 1/2)² has the 3-cycle with x-coordinates −1/2 − μ, −1/2 + μ, −1/2 + μ; for every δ ≠ 0 an explicit
  c (numerator δ⁶ − 2δ⁵ + 2δ⁴ + 2δ³ + 2δ² − 2δ + 1) gives a 4-cycle with x-coordinates (p, q, r, r); for δ = 1 the
  maps with c = −ν² have the 4-cycles (−ν, −ν, ν, ν). Theorem 1.2: shifted by 1/2, the x-triples of rational 3-cycles
  are exactly the rational points, off the line u = v = w, of the cone over the cubic
  u³ + v³ + w³ − u²v − v²w − w²u = 0, which is absolutely irreducible with a single singular point, a node at (1:1:1)
  with tangents conjugate over Q(√−3) (geometric genus 0), with an explicit bijective parametrisation by
  (h, k) ∈ Q² \ {0} and δ = 1 + (w² − v²)/(v − u). Corollary 1.3: for every δ ≠ 1 an explicit degree-6 function
  c_δ(τ), τ ∈ Q \ {0, 1}, gives all rational 3-cycles (infinitely many c, at most two 3-cycles per map); for δ = 1 a
  rational 3-cycle exists if and only if −(1 + c) is the square of a rational number (zero included: c = −1 has the
  cycle (−1, 0) → (0, 0) → (0, −1)). Corollary 1.4: the conjecture is false in the readings below. Proposition 1.5:
  f_{c,δ}⁻¹ is conjugate over Q to f_{c/δ², 1/δ}.
- **Credit.** The conjecture is Ingram's (Section 11 of the list). The interpolation construction that opens the
  same section already produces rational 4-cycles for every δ ∉ {0, 1} (with the source's own substitution y₃ = y₂,
  δ = (y₀ − y₂)/(y₁ − y₂)), and the same linear algebra gives 3-cycles for every δ ≠ 1; the counterexamples are
  essentially implicit in the source, and the note's 4-cycle family is exactly the normal form of that construction.
  The case δ = −1 is Ingram's (Proc. London Math. Soc. 108 (2014)); the case δ = 0 is the classical map z² + c
  (Morton, Acta Arith. 62 (1992) and 87 (1998); Walde–Russo 1994; Morton–Raianu, J. Number Theory 240 (2022);
  Flynn–Poonen–Schaefer 1997; Stoll 2008; Poonen 1998).
- **What the note adds.** The explicit universal families (3-cycles for δ ≠ 1 and for δ = 1, 4-cycles for every
  δ ≠ 0, the δ = 1 family of 4-cycles), the complete description of the rational 3-cycles with its corollaries, and
  the (elementary) duality. The novelty is modest; no priority is claimed.
- **Probably misstated.** The printed statement contradicts the construction with which its own section begins; the
  note says that it is probably not the intended statement and that the intended one cannot be recovered from the
  text.
- **Not settled.** Which periods occur for a given δ. The corrected statement suggested by the data is only asked
  (Question 5.2: periods at most 4 for all but finitely many δ? period 5 for infinitely many δ?).

The note is unrefereed. Under the v1.8.0 policy of the dataset (solved only if peer-reviewed), the record keeps its
status, with an unrefereed-claim note pointing to this note.

## Statement checked
- **Primary source.** "Hénon Maps: A List of Open Problems", Arnold Mathematical Journal 10 (2024), no. 4, 585–620,
  doi:10.1007/s40598-024-00252-x; Crossref author: the collective pseudonym "Julia Xénelkis de Hénon"; published
  online 2024-08-13. arXiv:2312.03907, a single version (v1, 2023-12-06), which lists sixteen authors (P. Berger,
  E. Bedford, F. Bianchi, X. Buff, S. Crovisier, T.-C. Dinh, R. Dujardin, C. Favre, T. Firsova, P. Ingram, Y. Ishii,
  L. Palmisano, E. Pujals, J. Raissy, S. Štimac, G. Vigny).
  - Section 11, "Hénon maps over number fields (P. Ingram)", was read in the TeX source of v1 (verification runs A
    and B) and in the journal's HTML edition at amj.math.stonybrook.edu, article 23-70 (verification run B, and again
    when the note was written). In both, the item is **Conjecture 4** (a global conjecture counter; the
    HTML edition prints "Conjecture 4."), with the same wording.
  - The statement (paraphrased): for all but finitely many rational δ, the rational periodic points of every map
    (y, y² + c − δx) with rational c have period dividing 2.
  - Context in the section: it opens with an interpolation argument (prescribing a cycle of length deg F + 2 for
    (y, F(y) − δx) is a linear system for the coefficients of F and δ, with determinant not identically zero, checked
    by the substitution y_{d+1} = y_d); then a question on longer cycles, a uniform boundedness conjecture, and two
    "readily falsifiable" conjectures: Conjecture 3 (δ = −1, periods in {1, 2, 3, 4, 6, 8}, citing Ingram 2014) and
    Conjecture 4; then a remark that infinite families would show both to be sharp, if true. No evidence is given for
    Conjecture 4.
- **Corpus record.** ulamai/UnsolvedMath v1.8.0 (snapshot d371ee9), record AMR-152-0049, status `open`, title
  "Henon Maps: Open Problems — Conjecture 49" (a running number of the extraction, not the source's), source
  arXiv:2312.03907; `proposed_by` "Berger, Bedford" is an artefact of the extraction (the section is signed by
  Ingram). The statement of the record is the TeX of Conjecture 4, verbatim.

## Readings
| Reading | Answer | Where |
|---|---|---|
| (R1) literal: a finite exceptional set of δ, all c | false: the exceptional δ form all of Q | Theorem 1.1, Corollary 1.4 |
| (R2) for every δ (or all but finitely many δ), all but finitely many c | false: for every δ infinitely many c carry a rational 3-cycle | Corollary 1.3(a) (δ ≠ 1), Theorem 1.1(b) (δ = 1), Corollary 1.4 |
| (R3) all but finitely many pairs (c, δ) | false | Corollary 1.4 |
| δ = 0 excluded (Hénon maps proper) | false | the families cover every δ ≠ 0 |
| integral c, δ and integral points | (R1) and (R3) false: δ ≡ 3 (mod 4) gives integral c and an integral 3-cycle; (R2) false at δ = 1 (c = −ν²); for a fixed integer δ ≠ 1 only finitely many integers c carry a rational 3-cycle, so the integral form of (R2) is not decided | Remark 3.2, Corollary 1.4 |
| sign convention (y, y² + c + δx) | relabels δ; false | — |
| any quadratic Hénon map (y, Ay² + By + E − δx), A ≠ 0 | conjugate over Q to f_{c,δ} with the same δ; false | Lemma 2.1(b) |
| "period dividing 4" instead of 2 | false for every δ (3-cycles) | Theorem 1.1(a), (b) |

"Period" is the exact (least) period throughout, as in the source; "period dividing 2" means period 1 or 2.

## Results in the paper
- Theorem 1.1 (a)–(d): proved by direct verification of the cyclic recurrence (Section 3), with the exact periods
  from explicit non-vanishing differences (a − b = 6t; a − b = −2μ; r − p = δ² + 1; f²(−ν, −ν) = (ν, ν)).
- Theorem 1.2: Lemma 4.1 (the linear system for (c, δ) and the determinant D, with D(u − 1/2, v − 1/2, w − 1/2)
  = −F), Lemma 4.2 (F(m + h, m + k, m) = 2m(h² − hk + k²) + h³ − h²k + k³); the node and its tangent cone; the
  singular locus by resultants (chart w = 1: F_u is linear in v; the quartics (u − 1)(23u³ + 11u² − 7u − 3) and
  −(u − 1)(9u³ + 17u² − u − 1); the cubic factors have resultant 331776 = 2¹²·3⁴ ≠ 0; line w = 0: no common zero);
  absolute irreducibility (a reducible cubic would have two singular points, a triple point or a tacnode); genus 0.
- Corollary 1.3: the fibre over δ ≠ 1 by τ = h/k; the cyclic shift τ ↦ 1 − 1/τ; deg c_δ = 6 because N_δ has degree 6
  with leading coefficient (δ − 1)² and N_δ(0) = N_δ(1) = (δ − 1)²; δ = 1 gives exactly the family of Theorem 1.1(b).
- Corollary 1.4; Proposition 1.5 (T(x, y) = (δy, δx)); Remarks 3.1 (uniqueness of the 4-cycle through the diagonal
  for δ ∉ {0, 1}), 3.2 (integrality; for a fixed integer δ ≠ 1 only finitely many integers c carry a rational
  3-cycle, since c_δ(p/q) ∈ Z forces p, q and p − q to divide (δ − 1)²), 3.3 (the interpolation construction of the
  source and its normal form).
- Section 5 (computations, labelled as such): Lemma 5.1 (archimedean and p-adic bounds for periodic orbits; the
  parity of v_p(c)), the three searches, Table 1, Question 5.2.

## Computations (programs and outputs in reproducibility/)
- `check_note.py` (standard library, exact arithmetic; 92 checks, 0 failed, about 2 seconds): every identity of the
  proofs as an identity of polynomials or rational functions; the families on 1958 parameter values each by iterating
  the map; every quoted cycle; the complete periodic sets of four maps; a test of Corollary 1.3(b) on 295 values of c.
- Search (i), box search (first version; standard library): 31 values of δ; exhaustive for integral δ within its box
  (cycles of length ≤ 40 with three consecutive x-coordinates of exact common denominator L in the list and
  numerators ≤ min(8L, 96)), heuristic for non-integral δ. Re-run in full when the note was written: identical.
- Search (ii), complete search (verification run B; NumPy): for the same 31 values of δ and all c = N/M with
  |c| ≤ C_δ, M = m²k, m ≤ m_δ, k | den(δ) (6,001 to 116,601 values of c per δ), all rational periodic points, any
  period, by Lemma 5.1. Runs for δ = −1, 1, −4 repeated when the note was written: identical.
- Search (iii), 5-cycles with a repeated x-coordinate (verification run B; SymPy): 1110 values of δ; only δ = −4 and
  −1/4 (at δ = 1 the resultants vanish identically and the method gives nothing). Repeated: identical.
- Results (Table 1 of the note): δ = −1: exactly {1, 2, 3, 4, 6, 8} (Ingram's set; period 8 only at c = −9/16 in (ii));
  δ = 1: {1, …, 7, 9, 10}; δ = −4: {1, 2, 3, 4, 5}, period 5 only at c = −469/16 (the map has exactly this 5-cycle and
  one 2-cycle); δ = −1/4: the dual; δ = 0: {1, 2, 3}; the other 26 values: subsets of {1, 2, 3, 4}. Apart from δ = ±1,
  every 4-cycle found is that of Theorem 1.1(c) except six: δ = −4 (c = −685/16, −365/16, −5749/144), δ = −1/4
  (c = −685/256, −365/256) and δ = 5/2 (c = −2785/256, x-coordinates −69/16, −29/16, 51/16, 61/16; found in the
  recorded output of search (i)).
- These are TESTED data on finite ranges; the note labels them as computations, not theorems.

## Independent verification runs
All three runs were AI-assisted, re-derived the proofs and wrote their own programs; runs A and B examined the first
written version and did not read each other's report; run C examined the note as written.

**Run A (statement and proofs).** Read the TeX source of arXiv:2312.03907v1 (Section 11; numbering of the conjecture
counter) and of arXiv:1111.3609v1; re-derived families A, B, C and the description of the 3-cycles; recomputed all
identities with its own SymPy/Fraction program (no code of the first version reused) and iterated the map itself;
found the δ = 1 family of 4-cycles (−r, −r, r, r), c = −r²; closed the gap in the singular-locus argument by
resultants; proved the p-adic bound used by the box search; ran its own box search for six integral δ; ran a copy of
the Gröbner-basis program for length 4 (see below); checked the bibliography of the first version against Crossref.
Verdict: the disproof is correct and fully certified; eleven fixes required.

**Run B (independent computation, literature, bibliography).** Verified the families by iterating the map on 1958
parameter values each, with negative controls; recomputed the period-3 locus by a Gröbner basis and by a resultant
(both give the cubic), and the singular locus; wrote the complete search (ii) and tested it; tested Corollary
1.3(b) on 6,001 values of c; computed the 5-cycles with a repeated x-coordinate by resultants for 1110 values of δ;
re-ran all programs of the first version with identical outputs; specialised the interpolation construction of the
source to degree 2 and found that it gives rational 4-cycles for infinitely many δ (298 random rational 4-tuples, 275
distinct δ) and 3-cycles for every δ ≠ 1; repeated the literature search (arXiv API, Semantic Scholar, zbMATH Open,
Crossref, two web searches) and checked the bibliography. Verdict: claim confirmed; novelty to be reduced; six fixes
required.

**Run C (the note as written; programs and outputs in `reproducibility/independent_run_2/`).** Re-read Section 11 of
the list in the TeX source of arXiv:2312.03907v1 (global conjecture counter: the item is Conjecture 4; the record's
statement is the source's text character by character) and in the journal's HTML edition (same wording and number),
the introduction of Ingram 2014 in the TeX source of arXiv:1111.3609v1 (shared theorem counter per section:
Conjecture 1.5, Proposition 1.6, the example b = −9/16), the zbMATH reviews of Morton–Raianu and Silverman 1994, and
the Crossref records of all ten DOIs of the bibliography (all correct). Re-derived every proof line by line, and first
the parts added at the writing stage: the parametrisation by (h, k), the fibres c_δ(τ) of Corollary 1.3(a) with the
bound of two rational 3-cycles per map (a degree-6 map takes each value at most six times, the three points of a cycle
have distinct τ), the resultant argument for the singular locus, and the identification of the normal form of the
source's construction with Theorem 1.1(c). Own programs (written from the printed statements): the identities of
Theorems 1.1 and 1.2, Corollary 1.3, Proposition 1.5 and Remarks 3.1–3.3 in SymPy, by iterating the map (91 checks);
the singular locus by Gröbner bases and resultants; the numbers of points of the cubic over F_p for twelve primes
(p for p ≡ 1 (mod 3), p + 2 for p ≡ 2 (mod 3)), which confirm independently that the cubic is absolutely irreducible
with one node, non-split over Q; a map with two rational 3-cycles for δ = −1 (c = −186874/2025; the eliminant of
f³ = id has degree 8 with exactly these six rational roots), so the bound of Corollary 1.3(a) is attained; the exact
periods of every cycle quoted in the note, the complete periodic sets of twelve maps by an own implementation of the
bounds of Lemma 5.1, Corollary 1.3(b) on 591 values of c and Lemma 5.1(c) on 690 maps (41 checks); an own box search
over the ranges of search (i), listing every cycle (and, for non-integral δ, also the denominators not divisible by
den(δ): nothing new); an own complete search over the ranges of search (ii) for all 31 values of δ by a different
algorithm (pruning instead of pointer doubling): the numbers of c with a cycle of each exact length agree with
`complete_search_summary.md` for every parameter set; and search (iii) in a different formulation, with the same
result. Every period of Table 1, the six 4-cycles not of Theorem 1.1(c), the two 5-cycles and every quoted cycle were
reproduced. The package was re-run from an extracted copy of the archive: `check_note.py` (92 checks), the three parts
of search (i), all 34 recorded runs of search (ii) and both runs of search (iii) gave outputs identical to the
recorded ones up to the timing fields. Literature search repeated on 18 October 2026 (arXiv API, zbMATH Open,
Crossref; OpenAlex refused): nothing new. Verdict: no error in the theorems; one required correction (C1 below) and
changes of wording and of the package.

**Corrections required by runs A and B**, all applied in the note:
- A1, B2: Morton's period-4 theorem is in Acta Arith. 87 (1998) 89–102 ([Mor98]); the Compositio Math. 103 (1996)
  paper (dynatomic curves) is not cited.
- A2: the reading "for fixed δ, all but finitely many c" is refuted (Corollaries 1.3(a), 1.4; Theorem 1.1(b)); it is
  nowhere called trivially true.
- A3: the corrected statement appears only as Question 5.2.
- A4, B1: the interpolation paragraph that opens Section 11 of the source is cited (introduction, Remark 3.3, Scope
  and priority, abstract): it gives rational 4-cycles for every δ ∉ {0, 1} (δ = (y₀ − y₂)/(y₁ − y₂) with y₃ = y₂) and
  3-cycles for every δ ≠ 1; the printed statement is probably not the intended one; the novelty claim is reduced to the
  explicit universal formulas, the δ = 1 families and the complete description of the 3-cycles.
- A5, B4: Ingram's items are cited as Conjecture 1.5 and Proposition 1.6 of arXiv:1111.3609v1, with the remark that
  the journal numbering was not checked.
- A6: c of Theorem 1.1(a) is an integer exactly for integers δ ≡ 3 (mod 4) (Remark 3.2), not only δ ≡ 3 (mod 8).
- A7: the δ = 1 family (−ν, −ν, ν, ν), c = −ν², is Theorem 1.1(d).
- A8: the Gröbner-basis search of the first version is not used in the note (only mentioned as not used). For the
  record: its sanity run for length 4 with x₀ = x₁ returned exactly the cycle of Theorem 1.1(c) at the sixteen values
  δ ∈ {−4, −1/4, 2, 3, −2, 5, 1/2, −3, 7, −6, 4, −8, 9, 2/3, −5/2, −1} where it was run (TESTED at these values
  only); at δ = 1 the ideal is not zero-dimensional (Theorem 1.1(d) is a one-parameter family) and the method reports
  nothing, so a "0" from it is meaningful only together with its zero-dimensionality flag.
- A9: the resultant argument that (1:1:1) is the only singular point is in the proof of Theorem 1.2(b).
- A10: status wording: see the verdict above and the suggested label below.
- A11: the wording of the journal edition was re-read (HTML edition, article 23-70, re-read when the note was written) and the page is named
  in the note; the statement about c = −29/16 is attributed to [MR22, Theorems 1 and 4] as stated in the zbMATH review
  (Zbl 1521.11016), and the note says that [MR22] was not read.
- B3: arXiv:2010.16291 is by T. Gauthier and G. Vigny (geometric dynamical Northcott property for regular polynomial
  automorphisms of the plane); the note does not cite it, and its Verification paragraph mentions the corrected
  attribution only generically.
- B5: full reference for Morton–Raianu, J. Number Theory 240 (2022) 685–729, doi:10.1016/j.jnt.2022.01.005.
- B6: "period dividing 4" is refuted for every δ by the 3-cycles (introduction); Silverman, Math. Z. 215 (1994)
  237–250, is mentioned with its zbMATH review (numerical examples of Hénon maps with rational periodic points) and
  the statement that it could not be read.

**Correction required by run C**, applied in the note:
- C1: the clause "and also for integral c and δ and integral points" of Corollary 1.4 overstated the integral form of
  reading (R2): for a fixed integer δ ≠ 1 only finitely many integers c carry a rational 3-cycle (Remark 3.2 now
  proves this: c_δ(p/q) ∈ Z forces p, q and p − q to divide (δ − 1)²; for δ = 3 only c = −11). Corollary 1.4 now says
  that for integral data (R1) and (R3) remain false, (R2) remains false at δ = 1 (c = −ν²), and that the integral
  form of (R2) is otherwise not decided. Wording: Section 1.3 names the source's substitution y₃ = y₂ where the
  normal form is identified with Theorem 1.1(c); the description of search (i) states that, by Lemma 5.1(b), the
  orbit of a periodic point through an enumerated triple stays in (1/L)Z for every δ (so the search is complete
  within the enumerated triples), and that for non-integral δ only denominators divisible by den(δ) are examined.

**Changes made when the note was written** (examined by run C): the parametrisation of the cone by
(h, k) ∈ Q² \ {0} with w₀ = −(h³ − h²k + k³)/(2(h² − hk + k²)) and δ = 1 − hk(h − k)/(h² − hk + k²); the explicit
fibres of Corollary 1.3(a) with c_δ(τ) of degree 6 and the bound of two rational 3-cycles per map (δ ≠ 1); the
identification of the normal form of the source's construction with Theorem 1.1(c) (Remark 3.3, with the
determinant −(y₀ − y₁)(y₀ − y₂)(y₁ − y₂)² and a₂ = −((y₀ − y₂)² + (y₁ − y₂)²)/((y₀ − y₂)(y₁ − y₂)²)); the direct
proof of Theorem 1.1(c) by four relations; the list of the six 4-cycles not in family C (including δ = 5/2, read
from the recorded output of search (i)); the statement that the rational periodic points of f_{−469/16,−4} are
exactly one 5-cycle and one 2-cycle (computed by `check_note.py` with the method of Lemma 5.1, and also found by run
B's search); and the program `check_note.py`. Every identity in these additions is checked by `check_note.py`, and
all of them were re-derived and recomputed independently by run C.

## Relation to the literature, novelty and scope
- **Searched** (all requests anonymous and logged): both editions of the list; Ingram 2014 (TeX source of
  arXiv:1111.3609v1); Kim–Krieger–Postolache–Szeto, arXiv:2412.01668v2 (higher degree, many integral periodic
  points; nothing on the quadratic family); the works citing the list in Semantic Scholar (10) and zbMATH Open (4),
  and those citing Ingram 2014 in zbMATH Open (6), by title and abstract; more than thirty arXiv abstract searches on
  arithmetic properties of Hénon maps (hits include Hsia–Kawaguchi arXiv:1810.03841, Y. Zhang arXiv:2407.15952,
  Abboud–Zhang arXiv:2602.10310, Ji–Xie–Zhang arXiv:2511.13443, Mello arXiv:2607.12668, Gauthier–Vigny
  arXiv:2010.16291, Lee–Silverman arXiv:1907.13247, none of which constructs rational cycles for varying Jacobian);
  zbMATH Open; Crossref; four web searches (2026-10-17); a repeat of the arXiv search when the note was written. OpenAlex
  was not available and was not used; the publisher's page of Silverman 1994 could not be accessed, and the paper
  was not read. Run C repeated the search on 18 October 2026 (arXiv API: five queries on Hénon maps, periodic
  points, rational points, number fields, Jacobians, polynomial automorphisms; zbMATH Open: title and citing-work
  searches for the list and for Ingram 2014; Crossref: two bibliographic searches and the ten DOI records; OpenAlex:
  not available) and found nothing new.
- **Found:** no printed statement that Conjecture 4 is false, and no explicit family of rational 3- or 4-cycles for
  all Jacobians. A search that finds nothing is not a proof of novelty. The counterexamples are, however, implicit in
  the source's own construction (see Credit).
- **Silverman, Math. Z. 215 (1994):** bounded height of periodic points; according to the zbMATH review
  (Zbl 0807.58021) it contains numerical examples of Hénon maps with rational periodic points; not read; these may
  include rational 3- or 4-cycles for particular Jacobians. The note says so.
- **What was read:** Section 11 of the list (TeX of v1 and the journal's HTML edition); the introduction of
  Ingram 2014 (arXiv v1); KKPS by keyword; the zbMATH reviews of Morton–Raianu and Silverman; the abstract of Stoll
  2008 (which states Morton's N = 4 and Flynn–Poonen–Schaefer's N = 5 results). The proofs of cited theorems were not
  checked.
- **Suggested label for the dataset:** AMR-152-0049 solved (the printed conjecture is false for every rational δ),
  with the remarks that the printed statement contradicts the construction that opens its own section and is probably
  misstated, that the counterexamples are essentially implicit in that construction, and that the note is unrefereed
  (under the v1.8.0 policy: status unchanged, unrefereed-claim note).

## Public release and license
The note, its source files, this report and the reproducibility package are released under CC BY 4.0
(`LICENSE.txt`). The package contains `check_note.py` with its recorded output, the box search of the first
version and the two programs of verification run B used in Section 5, with their recorded outputs. The note is an
unrefereed preprint.