# Verification report — AMR-042-0001 (Ward's Problem A: the exact order of mixing)

Verification date: 2026-10-03. Paper: "The Exact Order of Mixing in a Problem of Ward: A Seven-Term Polynomial
with No Sparser Multiple" (14 pages).

**Verdict.** The exact order of mixing is M = 6, for every prime p. For
f = 1 + u1·u2 + u1^2·u2 + u1^3·u2 + u1^4 + u2^2 + u1^4·u2^2 the algebraic Z^2-action attached to the module
Z[u1^{±1}, u2^{±1}]/⟨p, f⟩ is mixing of order 6 and not mixing of order 7. The polynomial f is absolutely
irreducible modulo every prime, so the condition on p in the problem holds for all primes. The proof combines
published results of Schmidt, Masser, and Derksen and Masser with a new elementary theorem: every non-zero
multiple of f over an algebraic closure of F_p has at least seven terms. The note is unrefereed.

## Statement checked
- **Primary source.** T. Ward, "Six problems in algebraic dynamics (updated December 2006)", Problem A,
  https://www.imath.kiev.ua/~skolyada/kevin.pdf (5 pages; read).
  - The polynomial in the source is exactly f. The source also writes out the defining relation of the subshift,
    with the support {(0,0), (1,1), (2,1), (3,1), (4,0), (0,2), (4,2)}; these are the same seven points.
  - M is defined there as the largest k for which k-fold mixing holds for all measurable sets. So "M = 6" means
    mixing of order 6 and not of order 7.
  - The source states 3 ≤ M < 7, by the methods of Einsiedler–Ward (2003) and Kitchens–Schmidt (1993), and asks
    for the exact value. Its 2006 update records Masser's theorem and says that computing the order of mixing of
    non-trivial examples remains a considerable problem.
- **Corpus record.** ulamai/UnsolvedMath, AMR-042-0001 (status `partially_solved`). Its statement agrees with the
  source.

## Readings
| Reading | Answer | Where |
|---|---|---|
| exact order of mixing M for a prime p for which f is irreducible modulo p | M = 6 | Theorems 1.1 and 1.2 |
| which primes p satisfy the hypothesis | all primes; f is even absolutely irreducible modulo p | Lemma 3.1 |
| least number of terms of a non-zero multiple of f over F_p (equal to M + 1) | 7, attained by f | Proposition 2.2, Theorem 1.2 |
| the same over an algebraic closure of F_p, and for Puiseux exponents | 7 | Theorem 1.2, Remark 2.3 |
| all non-mixing sets of cardinality 7 | p = 2: only the support of f, up to equivalence (by computer search); odd p: not determined | Section 8 |

## Results in the paper
- **Theorem 1.1.** For every prime p, f is absolutely irreducible modulo p and the action is 6-mixing but not
  7-mixing.
- **Theorem 1.2.** Every non-zero element of the ideal generated by f in the Laurent polynomial ring over an
  algebraic closure of F_p has at least seven terms.
- **Lemma 2.1.** In the function field K of the curve f = 0: if w^s is a constant times x^d·y^e, then s divides d
  and e, and w is a constant times a monomial. The constant field of K_0 = F_p(x)[y]/(f) is F_p, and the radical
  of the group generated by x, y in K_0 is F_p^*·⟨x, y⟩. The proof computes the divisors of x and y (p odd: simple
  zeros at the non-singular points (0, ±i) and (x_0, 0) with x_0^4 = −1; p = 2: x is ramified at the non-singular
  point (0, 1), and there are two unramified places over x = 1).
- **Proposition 2.2** (reduction). M + 1 is the least number of terms of a non-zero multiple of f over F_p. It
  uses three published results: Schmidt's criterion for non-mixing sets, Masser's theorem (Israel J. Math. 142
  (2004), Corollary on p. 190), and Lemma 5 of Derksen–Masser III (Ergodic Theory Dynam. Systems 38 (2018)), which
  needs n ≥ 2 and takes the radical inside K_0. Remark 2.3 gives a second route, through Kummer theory and an
  older form of the criterion, which does not use Lemmas 4 and 5 of Derksen–Masser III.
- **Proposition 4.3.** A non-zero multiple of f has at least six terms; with six terms its rows (by powers of
  u2) have 2, 2, 2 or 2, 1, 1, 2 terms.
- **Propositions 5.1 and 6.1.** Neither shape occurs. The proofs compare valuations and leading coefficients at
  the places over the roots of 1 + x^4, and use a Frobenius splitting and a computation in the basis {1, U},
  U = 1/(1 + y^2). The prime 2 has separate arguments.

## Computations (exact; scripts and outputs in reproducibility/)
The proofs do not depend on computation. The computations are checks, except that the statement on non-mixing
sets of cardinality 7 for p = 2 rests on a computer search.
- **Author** (`author/`).
  - `descent2.py`: an exhaustive search for p = 2 by descent along the binary digits of the exponents. There is
    no relation between r = 3, 4, 5, 6 pairwise distinct monomials in F_2(x)[y]/(f) (1, 11, 23 and 8,004 states
    expanded). That the set of reachable states is finite is established by the terminating runs, not in advance.
    Three variants without the symmetry reduction or the subsum pruning agree (543, 8,106 and 696,151 states).
  - `descent2_all7.py`: for r = 7 (38,031 states) the only relations are monomial multiples of f^(2^e).
  - `validate2.py`: on seven other curves over F_2 the search engine agrees with brute force in 17 comparisons.
  - `boxsearch.c`: no relation with at most six terms in the boxes (p; A, B) = (2; 48, 24), (3; 32, 16),
    (5; 26, 13), (7; 20, 10), (11; 16, 8), (13; 14, 7), smaller boxes for p = 17, 19, 23, 41, 43, and flat or tall
    boxes up to (2; 200, 6) and (2; 12, 100).
  - `structured_search.py`: no multiple of shape (2,2,2) with B ≤ 60 and exponents up to 6000, and none of shape
    (2,1,1,2) with B ≤ 30 and |m_2 − m_1| ≤ 200, for the 15 primes p ≤ 47. Control with b replaced by x + x^3.
  - `check_identities.py` (73 checks), `check_irreducible.py`, `check_local_analysis.py` (6 examples).
  - All of these programs were run again for this version (2026-10-03); the outputs agree with the saved ones.
- **Independent verification runs** (`independent_run/`; AI-assisted; code written without using the author's
  code).
  - A box search by a different method (images of the monomials in a finite quotient ring, exact verification of
    all candidates): no multiple with at most six terms in the boxes (2; 128, 64), (2; 256, 24), (2; 40, 120),
    (2; 400, 8), (2; 16, 200), (3; 80, 40), (3; 30, 80), (3; 250, 6), (3; 12, 120), (5; 64, 31), (5; 120, 5),
    (5; 10, 70), (7; 52, 25), (7; 90, 4), (7; 8, 50), (11; 30, 14), (13; 26, 12).
  - A structured search: shape (2,2,2) for all 0 < j < B ≤ 200 (p ≤ 13) and B ≤ 120 (17 ≤ p ≤ 47), with no
    bound on the exponents of the binomials; shape (2,1,1,2) for B ≤ 40, |m| ≤ 3000 (p ≤ 13) and B ≤ 30,
    |m| ≤ 1000 (17 ≤ p ≤ 47). Nothing found.
  - Controls: on curves with sparse relations the two box searches return the same lists; planted six-term
    multiples are found; the author's search engine agrees with the independent box search on three further
    curves over F_2.
  - The author's programs were also repeated in these runs, with identical outputs (all except the variant of the
    search for r = 6 without the symmetry reduction, which was repeated at the writing stage and in the second
    run).
- **Second independent verification run** (`independent_run_2/`; AI-assisted; code written without using the
  author's code or the code of the first run).
  - A third box search (images of the monomials in (F_p[Y]/(m))[x]/(f), i.e. in the basis 1, x, x^2, x^3; every
    hit tested by exact division): no multiple with at most six terms in the boxes (2; 88, 84), (3; 60, 56),
    (5; 42, 40), (7; 34, 32). These boxes are not contained in the boxes of the earlier searches.
  - Controls: in 20 small boxes (f and four other polynomials, p = 2, 3, 5, 7) the output is identical to a
    complete enumeration of all multiples in the box; the search finds f itself, and planted sparse multiples of
    control polynomials in larger boxes. With seven terms it finds only monomial multiples of the powers f^(p^e)
    in the boxes (2; 48, 24), (2; 24, 24), (3; 16, 12), (5; 20, 10), (7; 12, 8).
  - `identities.py`: 424 exact checks of the displayed identities of the paper (Lemma 3.1, relations (2), (3),
    (6), Lemmas 3.2, 3.3, 3.4, 3.6, the statements on Π_q, Ω_q, N_d, and the identities of Step 2 and Step 4 of
    Proposition 6.1); all passed.
  - All programs of `author/`, and the 17 box searches for Ward's polynomial and the structured searches of
    `independent_run/`, were run again from an extracted copy of the source archive. All outputs agree with the
    saved ones apart from running times (`independent_run_2/rerun_of_package.txt`). The control runs of
    `independent_run/` were not repeated.

## Independent adversarial audit
Two independent verification runs were made, both AI-assisted. Verdicts of the first run (2026-10-03):

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (source re-fetched; no misprint; M as defined in the source) |
| Reduction to sparse multiples | CONFIRMED, conditional only on the published results quoted |
| Proof of Theorem 1.2 | CONFIRMED (every step re-derived; no error found) |
| Computations | CONFIRMED (the author's runs repeated; independent searches found nothing) |
| Answer as posed and as intended | CONFIRMED (M = 6 for every prime p) |
| Novelty | CONFIRMED as far as can be checked (no priority claim) |
| Presentation | CONFIRMED_WITH_FIXES |

All required fixes of the first run were applied in the paper:
1. Proof of Proposition 6.1, end of Step 2: only ν = q_j ≥ 1 is stated there. That a divides the middle row is
   used for odd p at the start of Step 3, and for p = 2 only after 4 | q_j has been derived.
2. Proofs of Propositions 5.1 and 6.1 (Step 2): the explicit substitution ξ = x_0^{q_0}, x_0 = ξ^r with
   r·q_0 ≡ 1 (mod 8), replaces "taking q_0-th roots".
3. Section 2: Schmidt's criterion, Masser's theorem and Lemmas 4 and 5 of Derksen–Masser III are quoted from the
   published versions. The text says that the radical is taken inside K_0 and that Lemma 5 needs n ≥ 2. The
   Kummer-theory remark (Remark 2.3) was added.
4. Lemma 2.1 is stated as a self-contained lemma, with a full proof in Section 3.
5. Section 8 says that for p = 2 the control polynomial with b = x + x^3 is reducible. The control output was
   regenerated in full.
6. Section 8 says that the finiteness of the set of reachable states is established by the terminating runs.
7. The novelty statement credits the effective procedure of Derksen and Masser and their remark on shortest
   polynomials, and claims only the first explicit determination of M for this example, uniform in p.

Further changes made while writing, after the first run: the proof of absolute irreducibility (Lemma 3.1)
follows the shorter argument of the first run; the case analysis for the valuations in Step 1 of
Proposition 6.1 was shortened; short proofs were added to Lemma 3.3; examples showing that both hypotheses of
the reduction matter were added (Remark 2.4).

### Second independent verification run (2026-10-03)
This run checked the final text. It was made independently of the first run: the sources were fetched again,
all statements were re-derived from the text of the paper, and its programs were written before the packaged
programs were run.

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (source fetched again; the paper answers the question as posed) |
| Lemma 3.1, Lemma 3.3 and Step 1 of Proposition 6.1 (the arguments changed after the first run) | CONFIRMED |
| Lemma 2.1 and the proof of Theorem 1.2 (Sections 3 to 7) | CONFIRMED (line by line; no error found) |
| Reduction (Section 2) | CONFIRMED against the sources: (1.1), Lemma 4 and Lemma 5 of Derksen–Masser III, the Corollary on p. 190 of Masser's paper; both directions; conditional only on these published results |
| Remark 2.3 | CONFIRMED |
| Computations | CONFIRMED (own searches and identity checks; the programs of `author/` and the searches of `independent_run/` repeated with identical outputs) |
| The exhaustive search for p = 2 and the statement on non-mixing sets of cardinality 7 | CONFIRMED as stated |
| Novelty | No determination of M for this example found (no priority claim) |
| Presentation | CONFIRMED_WITH_FIXES |

Required fixes of the second run, all applied:
1. Verification paragraph: the two verification runs are described separately, and what the second run checked
   is recorded.
2. Remark 2.3: the theorem of Kitchens and Schmidt is quoted as the implication that is used; this is what the
   sources that were read (Masser, Section 5; Lind–Schmidt, Section 3) state.
3. Introduction: the effective determination is attributed to Part III of Derksen–Masser (with the main result
   of Part I), with the condition that the smallest order of non-mixing is at least 3 for the non-mixing sets.
4. Section 2: in the quotation of Lemma 4 of Derksen–Masser III the coefficients a_1, …, a_n are said to lie in
   K_0.
5. Abstract: the computer search for p = 2 confirms the main result for coefficients in F_2.
6. Section 8 (ii): the third box search, its boxes and its controls are recorded.
7. Scope paragraph: of Derksen–Masser I only the introduction was read, Part II was not seen; the two textbooks
   were not consulted again.

## Relation to the literature, novelty and scope
- **Sources read.** Ward's list; Einsiedler–Ward (arXiv:math/0204174; Theorem 3.1, Example 4.1, Remark 5.1(3));
  Masser 2004 in the published version (an archived copy of a repository file of the journal article);
  Derksen–Masser III in the published version (first-published-online form, from the first author's web page)
  and in the Basel preprint 2016-26; Lind–Schmidt (arXiv:1803.05862, Section 3); the introduction of
  Derksen–Masser I (arXiv:1010.4519). Bibliographic data and DOIs were checked with Crossref (again in the
  second run).
- **Not consulted.** Schmidt's book "Dynamical Systems of Algebraic Origin", Kitchens–Schmidt 1993,
  Derksen–Masser II and Ledrappier's note. What the paper says about the first two is taken from Masser 2004,
  Derksen–Masser III, Lind–Schmidt and Ward's list. The textbooks of Stichtenoth and Fulton are cited for
  standard facts and were not consulted again.
- **Searches (October 2026).** arXiv, Crossref, OpenAlex and zbMATH keyword searches; the OpenAlex lists of works
  citing Einsiedler–Ward, Kitchens–Schmidt (since 2004), Masser 2004 and Derksen–Masser I–III (fetched again in
  the second run: 102 distinct works); four web searches in total. No determination of the order of mixing of Ward's example was found.
- **What is known.** Derksen–Masser III give an effective procedure for the smallest order of non-mixing for any
  prime ideal and work out two examples, both with order of mixing 3. They remark that the problem is related to
  finding the shortest polynomial in an ideal, which is probably effectively solvable in positive characteristic.
  So for a fixed p the value of M is computable in principle. New here is the explicit value for Ward's example,
  for all primes at once, with an elementary proof.
- **Caveats.** The negative search is not a proof of priority. The non-mixing sets of cardinality 7 are
  determined only for p = 2, and only by computer.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/