# Verification report — OWR-1452-024 (Kraft–Russell generic equivalence problem, Oberwolfach Report 01/2007)

Verification date: 2026-09-30.

**Verdict.** The answer over F̄_p is negative.
- For the étale form asked in the report, this is classical: Russell's purely inseparable forms of A¹ refute
  it over every algebraically closed field of characteristic p (Proposition 6.1 of the paper, which is not
  counted among its results).
- The new content is that over F̄_p even the finite-degree form (Kraft–Russell 2014, Kaliman 2020) fails,
  and so does every weaker form. For every prime p there are two smooth affine families of threefolds over
  Y = A¹ ∖ {0, 1, −1} over F̄_p whose fibres over all closed points of Y are isomorphic as F̄_p-schemes, but
  which do not become isomorphic after any base change U → Y whose image contains the generic point
  (étale, of finite degree, or arbitrary). For p ≡ 1 (mod 4) the same holds for two smooth projective
  families of surfaces (blow-ups of E × E at two points, E: y² = x³ − x), over the surface E × E ∖ {0} or
  over a curve in it.
- The case of Q̄ asked in the same problem remains open. The note is unrefereed.

## Statement checked
- **Primary source.** Oberwolfach Report 01/2007, *Affine Algebraic Geometry*, Oberwolfach Rep. 4 (2007),
  no. 1, 5–82, doi:10.4171/OWR/2007/01; contribution of H. Kraft and P. Russell, pp. 69–70.
  - Page 69 was rendered from the report PDF and read (again in the second verification run, from the
    publisher's PDF, pp. 68–70). Proposition 1: k algebraically closed of infinite transcendence degree over
    the prime field; p: S → X, q: T → X morphisms of k-varieties whose (schematic) fibres over all x ∈ X are
    isomorphic; then there is a dominant étale φ: U → X with S ×_X U ≅ T ×_X U over U.
  - Problem 2 asks to show that the proposition holds over any algebraically closed field, or to give
    counterexamples over F̄_p or Q̄. The bars over F_p and Q are present in the report.
  - The points x are the closed points of X (the abstract of Kraft–Russell 2014 speaks of isomorphic
    "(closed) fibers"); if generic fibres were assumed isomorphic, the conclusion would follow by spreading
    out. The fibre isomorphisms are isomorphisms of k-schemes.
- **Later versions.**
  - Kraft–Russell, Transform. Groups 19 (2014) 779–792, doi:10.1007/s00031-014-9274-9. The publisher's
    version could not be accessed (anonymous requests to the publisher returned a JavaScript challenge).
    We read the journal-formatted accepted manuscript deposited in the University of Basel repository
    (edoc, https://hdl.handle.net/20.500.14716/83928) and arXiv:1204.3196v1.
    - Accepted manuscript: the Generic Equivalence Theorem is stated for affine morphisms with the
      conclusion "dominant morphism of finite degree"; Remark 2.2 says the authors do not know whether the
      theorem holds for all algebraically closed fields, e.g. for Q̄. Right after Theorem B of the
      introduction (a consequence of the Generic Equivalence Theorem) the authors note that in
      characteristic zero the dominant morphism can be chosen étale.
    - arXiv v1 (2012): the same theorem with the conclusion "dominant étale morphism", and the same
      remark (Remark 2). So the change from an étale to a finite-degree conclusion is due to Kraft and
      Russell themselves.
    - According to §2 of the accepted manuscript, a preliminary write-up of the theorem appeared in the
      lecture notes of the ICPA2006 school in Hanoi (*Polynomial Automorphisms and Related Topics*, eds.
      H. Bass, Nguyen Van Chau, S. Maubach, Publishing House for Science and Technology, Hanoi, 2007; cited
      there with the authors A. van den Essen, Ha Huy Vui, H. Kraft, P. Russell, D. Wright; Zbl 1128.14001).
      That volume was not seen.
  - Kaliman, Michigan Math. J. 69 (2020) 751–764, doi:10.1307/mmj/1585706558 (read as arXiv:1804.09747v3):
    Thm. 1.1 (restatement, finite degree), Thm. 2.5 (affineness not needed, any characteristic), Thm. 3.3
    (uncountable fields of characteristic 0, very general fibres), Rem. 3.12 (étale in characteristic 0
    by generic smoothness; unknown in positive characteristic), Conj. 4.2 (fields of finite transcendence
    degree over Q), Thms. 4.5, 4.7 (proper case).
- **Corpus record.** ulamai/UnsolvedMath, OWR-1452-024 (record 30000660, status `open`). The record asks
  the étale form. Its statement writes F_p and Q without the bars and does not say that the fibre hypothesis
  is over closed points; the source has F̄_p and Q̄, and closed points are meant.

## Readings
| Reading | Answer | Witness |
|---|---|---|
| Problem 2 over F̄_p, étale base change (Proposition 1 of the report; the form asked in the corpus record) | no | Proposition 6.1 (classical: Russell's purely inseparable forms of A¹, every p); Theorem 1.1 (affine, every p); Theorem 1.2 (projective, p ≡ 1 mod 4) |
| over F̄_p, dominant base change of finite degree (Kraft–Russell 2014, Kaliman 2020) | no (new as far as searched) | Theorems 1.1 and 1.2: no base change U → Y whose image contains the generic point works |
| over F̄_p, affine morphisms only | no | Theorem 1.1 (for the étale form also Proposition 6.1) |
| over F̄_p, proper (even smooth projective) morphisms | no, for p ≡ 1 (mod 4) | Theorem 1.2 |
| étale form over any algebraically closed field of characteristic p, any transcendence degree | no (classical) | Proposition 6.1: Russell's purely inseparable forms of A¹ |
| over Q̄ (and fields of characteristic 0, finite transcendence degree) | open | proper case proved by Kaliman (Thms. 4.5, 4.7); Proposition 7.1 excludes only the mechanism of Theorem 1.1 for families over punctured affine lines |

## Results in the paper
- **Lemma 2.1.** Over a principal ideal domain O, for μ ≠ 0, λ coprime to μ and r ∈ O², there is
  M ∈ SL₂(O) with M r ≡ λ r (mod μ); explicitly M = B D B⁻¹. The auxiliary generator e of dO + μO in the
  proof is not needed (μ′ = μ works as well); the paper keeps it because the programs of the finder and of
  the first verification run implement that form, and says so.
- **Lemma 3.1, Proposition 3.2.** X_φ (affine modification of G_m² × A¹ at {1, φ}) is a smooth irreducible
  affine threefold with units Ω*·u^Z·v^Z. X_φ ≅ X_ψ as Ω-schemes if and only if ψ ∈ GL₂(Z)·φ; an
  isomorphism of Ω-schemes covers either x ↦ M·x with M·φ = ψ, or the swapped case x ↦ ψ·(M·x) with
  M·φ = ψ⁻¹. The units argument uses F*(Ω*) = Ω*, i.e. that F is Ω-linear.
- **Theorem 1.1.** S: w z₁ = (u−1)(u−y), w z₂ = (y−1)(v−1) − y(u−1) over Y = A¹ ∖ {0, 1, −1}, and
  T = Fr*S (y ↦ y^p). Smooth affine morphisms of relative dimension 3, defined over F_p.
  - (i) At a closed point y ∈ F_q, write y = g^a, y + 1 = g^b; Lemma 2.1 with (Z, p, q − 1) gives
    M ∈ SL₂(Z) with M·c(y) = c(y)^p, so S_y ≅ T_y as k-schemes.
  - Since T = Fr*S and S is defined over F_p, each T_y is the Frobenius twist of S_y and is always
    isomorphic to S_y as an abstract scheme; the content of (i) is the k-linear isomorphism. The paper
    states this after Theorem 1.2.
  - (ii) At the generic point M·(t, t+1) = (t, t+1)^p forces M = pI by unique factorisation in k[t],
    and det(pI) = p² ≠ ±1.
  - Remark 4.1: for fixed M ∈ GL₂(Z) the set of closed y with M·c(y) = c(y)^{±p} (both cases) is finite,
    so the needed matrices are unbounded. Remark 4.2: the quasi-affine surfaces G_m² ∖ {1, c(y)} work too.
- **Theorem 1.2.** p ≡ 1 (mod 4), A = E × E, Y = A ∖ {0}, S = Bl_{({0}×Y)⊔Δ}(A × Y),
  T = Bl_{({0}×Y)⊔Γ_π}(A × Y). Smooth projective morphisms of relative dimension 2, fibres Bl_{0,a}(A) and
  Bl_{0,π(a)}(A); T = (π|_Y)*S, so again T_a is the Frobenius twist of S_a.
  - Lemma 5.1: E is ordinary; End(E_Ω) = Z[i], End(A_Ω) = M₂(Z[i]) and Aut_gp(A_Ω) = GL₂(Z[i]) for every
    algebraically closed Ω ⊇ F_p; π ∈ Z[i] has norm p.
  - Lemma 5.2: finitely generated Z[i]-submodules of E(F̄_p) are cyclic; E(F_{p^n}) ≅ Z[i]/(π^n − 1)
    (Lenstra 1996, Thm. 1(a); E(F̄_p) ≅ Z[i]_(π)/Z[i] by Lenstra's Thm. 3; a direct proof via Tate modules is
    included).
  - Lemma 5.3: Bl_{0,a}(B) ≅ Bl_{0,b}(B) as Ω-schemes for an abelian surface B iff b ∈ Aut_gp(B)·a.
  - Lemma 5.4: blow-ups of B × V along two disjoint sections are smooth projective over V, and their
    formation commutes with base change to points.
  - (i) E(F_q) is a Z[i]-submodule of E(F̄_p) because [i] is defined over F_p; Lemma 2.1 with
    (Z[i], π, π^n − 1) gives M ∈ SL₂(Z[i]) with M(a) = π(a), so S_a ≅ T_a as k-schemes.
  - (ii) At the generic point α(η) = π(η) with α ∈ GL₂(Z[i]) forces α = πI, which is not invertible.
  - Remark 5.5: a fixed α serves only finitely many closed points (α − πI is an isogeny, since
    det(α − πI) ≠ 0); the base can be shrunk to the curve (E ∖ {0}) × {0}; T is the pull-back of S along π|_Y.
- **Proposition 6.1** (classical in substance; not counted among the results). y^p = x + t x^p over A¹_t in
  characteristic p: all closed fibres ≅ A¹, trivial after t = s^p, not trivial after any étale base change
  hitting the generic point (leading coefficients would give t ∈ L^p). It already refutes the étale form of
  the Oberwolfach proposition over F̄_p, and in every positive characteristic.
  - That in characteristic 0 the finite-degree base change can be chosen étale is noted by Kraft and Russell
    (accepted manuscript, right after Theorem B) and by Kaliman (Rem. 3.12, generic smoothness).
  - Kaliman's Remark 3.12 is attached to his characteristic-zero Theorem 3.3 and asks whether the
    finite-degree morphism can be made étale in positive characteristic. In the natural reading (the
    finite-degree base changes of Kraft–Russell 2014 and Kaliman's Theorem 2.5) the answer is negative, and it
    follows immediately from these classical facts.
- **Proposition 7.1, Corollary 7.2.** Over a number field, morphisms c, c′ from a punctured affine line to
  G_m^r with c′(y₀) ∈ GL_r(Z)·c(y₀) at one suitable point y₀ satisfy c′ = γ·c globally (valuations at
  auxiliary primes); so families of the shape of Theorem 1.1 over Q̄ whose closed fibres are isomorphic as
  Q̄-schemes are isomorphic.

## Computations (exact; scripts and outputs in reproducibility/)
- **Finder** (`claimant/`, standard library only, about seven minutes).
  - Explicit M = B D B⁻¹ ∈ SL₂(Z) for all 21,926 admissible y in 32 fields F_q (p ≤ 13, q ≤ 4096), verified
    with schoolbook field arithmetic.
  - Exact Laurent-polynomial certificates of X_c(y) ≅ X_c(y)^p, in both directions, for all 1,348 admissible y
    in 16 fields with q ≤ 289; point-by-point check for q ≤ 16.
  - The construction of Theorem 1.2(i) for all 549,435 nonzero points of A(F_q), q ∈ {25, 125, 625, 169, 289};
    #E(F_q) = N(π^n − 1); E(F_q) cyclic over Z[i].
  - Identities of Proposition 6.1 for p ≤ 13.
- **First independent verification programs** (`verifier/`, separate finite-field and elliptic-curve code, no
  matrices B D B⁻¹, about one minute).
  - Lemma 2.1 on 20,000 random instances over Z and 20,000 over Z[i].
  - Orbits of SL₂(Z) (generators E12, E21) on (F_q*)² for 40 fields (p ≤ 53, q ≤ 1024): the number of orbits
    equals τ(q − 1) in each field, and c(y)^p lies in the orbit of c(y) for all 7,836 admissible y. The bound
    of Remark 4.1 holds for both signs for all 232 matrices with entries in [−3, 3] (q ≤ 256).
  - Orbits of SL₂(Z[i]) (four elementary generators) on E(F_q)² for 17 fields (q ≤ 841): the number of orbits
    equals the number of ideals of Z[i] containing π^n − 1, and π(a) lies in the orbit of a for all
    1,220,015 nonzero a (1,189,434 of them over the six fields with n ≥ 2, where π acts nontrivially).
  - Identities of Proposition 6.1 for p ≤ 23.
- **Second independent verification programs** (`independent_run_2/`, written before the other programs
  were read; own finite-field, Gaussian-integer and elliptic-curve arithmetic; about four minutes).
  - Lemma 2.1 (the construction of the proof and the simpler version μ′ = μ) on 10,849 instances over Z and
    15,258 over Z[i], random and adversarial: 0 failures.
  - Theorem 1.1(i) for all 47,698 admissible y in 63 fields (p ≤ 97, q ≤ 8192): M·c(y) = c(y)^p re-verified by
    schoolbook exponentiation, without log tables: 0 failures.
  - Orbits of SL₂(Z) = ⟨S, T⟩ and GL₂(Z) on (F_q*)² by breadth-first search in 29 fields (q ≤ 512): τ(q − 1)
    orbits; c(y)^p in the orbit of c(y) for all 2,580 y; a negative control is separated.
  - Remark 4.1: the bound holds for 8,752 (M, ±, field) triples; for q = 2^7 none of the 17,768 matrices of
    GL₂(Z) with entries of absolute value ≤ 30 works, and min |tr M| = 61, as the paper proves.
  - Smoothness: Jacobian rank 2 at all 16,944 F_q-points of the fibres of S and T for q ≤ 9.
  - Theorem 1.2: E ordinary for the 37 primes p ≡ 1 (mod 4) with 5 ≤ p < 400; Frobenius = π; E(F_q) ≅
    Z[i]/(π^n − 1) in 18 fields (q ≤ 4913); M(a) = π(a) checked with elliptic-curve arithmetic on 210,662
    points (all nonzero points for q ≤ 289, random samples plus the whole curve (E ∖ {0}) × {0} for larger q);
    orbits of SL₂(Z[i]) and GL₂(Z[i]) on E(F_q)² in 10 fields; Remark 5.5(1) for 288 matrices α.
  - Identities of Proposition 6.1 for p ≤ 31; Proposition 7.1 on an example over Q.
- **Reruns.** The finder's five programs were re-run on 2026-09-30 and reproduced the recorded outputs byte
  for byte. The second verification run extracted the previous release archive and re-ran all nine programs
  of `claimant/` and `verifier/` (Python 3.13.5): all nine outputs byte-identical, stderr empty, about
  6.3 minutes for the finder's programs and 1 minute for the others
  (`independent_run_2/rerun_release_summary.txt`). These programs are unchanged in this version. The
  programs of `independent_run_2/` were re-run from the new release archive: their outputs agree with the
  recorded ones except for the printed running times.
- The computations illustrate the closed-fibre statements for finitely many fields; the generic-point
  statements are proved in the paper and are not finite computations.

## Independent verification runs
All verification runs were AI-assisted (2026-09-30).

**First run.** It checked the proofs step by step, re-ran the finder's programs, and wrote the programs in
`verifier/`. Verdicts:

| Item | Verdict |
|---|---|
| Statement fidelity (report p. 69 read; closed points; bars on F̄_p, Q̄) | CONFIRMED |
| Proofs (Lemma 2.1, Lemma 3.1, Prop. 3.2 with the swapped case, Thms. 1.1, 1.2, Lemmas 5.1–5.4, Prop. 6.1, Prop. 7.1) | CONFIRMED |
| Computations | CONFIRMED |
| Answer as posed (F̄_p part negative; Q̄ part open) | CONFIRMED |
| Novelty (no earlier counterexample found; Prop. 6.1 classical) | CONFIRMED as far as can be checked (no priority claim) |
| Presentation | CONFIRMED_WITH_FIXES |

All six required fixes of the first run were applied:
1. The finiteness remark (Remark 4.1) includes the swapped case M·c(y) = c(y)^{−p}.
2. Proposition 6.1 is presented as classical in substance (Russell 1970; Kambayashi–Miyanishi 1978,
   Kambayashi–Wright 1985) and is not counted among the results.
3. The published Transformation Groups text could not be accessed; the paper says that the Generic
   Equivalence Theorem and Remark 2.2 are quoted from the accepted manuscript, and records the differing
   arXiv v1 wording (étale conclusion, Remark 2).
4. Lemma 5.1 states End(E_Ω) = Z[i] and Aut_gp(A_Ω) = GL₂(Z[i]) for every algebraically closed Ω ⊇ F_p,
   with Deuring's theorem (Silverman, Thm. V.3.1) and a proof; Lemma 5.2 cites Lenstra 1996, Thms. 1(a)
   and 3, for the Z[i]-module structure of E(F_{p^n}) and E(F̄_p) and adds a direct proof.
5. The scope of Proposition 7.1 (the former Remark D) is limited explicitly to morphisms from punctured
   affine lines to tori.
6. The paper says that the fibre hypothesis is over closed points, that the source has F̄_p and Q̄, and that
   the Q̄ non-proper case (Kraft–Russell 2014, Rem. 2.2; Kaliman 2020, Conj. 4.2) remains open.

A further read of the finished manuscript found no mathematical error. Its minor points were addressed:
the injectivity of Z[i] → End(E) is now justified (Silverman, Prop. III.4.2(a)); the bound in Remark 4.1 is
stated explicitly and the F_{2^7} example is proved in the text; the finiteness for a fixed α in Theorem 1.2 is
proved (Remark 5.5); the Hartshorne reference for blow-ups of varieties is Prop. II.7.16; the reference to
Kaliman's Remark 3.12 and to his Conjecture 4.2 is stated precisely; the Kraft–Russell numbering (accepted
manuscript versus arXiv v1) is explained; and notation clashes were removed.

**Second run.** It read pp. 68–70 of the report from the publisher's PDF, checked every proof line by line,
wrote the programs in `independent_run_2/` before reading any other program, re-ran all nine release
programs, checked the references and their numbering against the saved texts, and repeated the literature
search. Verdicts:

| Item | Verdict |
|---|---|
| Statement fidelity (Proposition 1 and Problem 2 on p. 69; closed points; k-isomorphic fibres) | CONFIRMED |
| Proofs (all, line by line) | CORRECT, no gap; precision points only |
| Computations (own programs; rerun of all nine release programs) | CONFIRMED (0 failures; 9/9 byte-identical) |
| Novelty | new as far as searched for the finite-degree and arbitrary-base-change forms over F̄_p; the étale form over F̄_p is refuted by Russell's classical forms (Prop. 6.1) |
| Presentation | minor fixes required (priority wording, credit, precision, disclosure) |
| Fatal problems | none |

All four required fixes of the second run were applied:
1. **Priority wording.** The paper no longer says that no earlier counterexample over F̄_p or Q̄ was found.
   It says that, apart from the classical inseparable phenomenon behind Proposition 6.1, which already
   refutes the étale form of the Oberwolfach proposition over F̄_p (indeed in every positive
   characteristic), no earlier counterexample over F̄_p to the finite-degree form of Kraft–Russell 2014 and
   Kaliman 2020 was found, and none over Q̄. The abstract, §1.2, the scope paragraph and this report say the
   same.
2. **Credit.** §1.1, Table 1 and Remark 6.2 cite the Kraft–Russell accepted manuscript (right after its
   Theorem B) next to Kaliman's Remark 3.12 for the characteristic-zero étale remark, say that the change
   from the étale conclusion (arXiv v1) to finite degree is Kraft and Russell's own, and say plainly that
   the negative answer to Kaliman's positive-characteristic question follows immediately from classical
   facts.
3. **Precision.** Theorems 1.1(i) and 1.2 state that the fibres are isomorphic as k-schemes, and a paragraph
   after Theorem 1.2 explains that each T_y is the Frobenius twist of S_y, hence always isomorphic to S_y
   as an abstract scheme, so that the content of (i) is the k-linear isomorphism. Proposition 3.2,
   Remark 4.2, Lemma 5.3 and Corollary 7.2 are stated for isomorphisms of schemes over the base field, and
   the proof of Proposition 3.2 says that it uses F*(Ω*) = Ω*. The proof of Theorem 1.2(i) says that E(F_q)
   is a Z[i]-submodule because [i] is defined over F_p, and a note after Lemma 2.1 says that the auxiliary
   element e is not needed.
4. **Disclosure.** The scope paragraph lists the cited works that were not seen: Russell 1970,
   Kambayashi–Miyanishi 1978, Kambayashi–Wright 1985 (whose hypothesis is quoted from Kraft–Russell 2014,
   Rem. 5.2), Deuring 1941, Kaliman–Zaidenberg 1999, the publisher's version of Kraft–Russell 2014, and the
   journal versions of Kaliman 2020, Poczobut 2025 and Bogomolov–Böhning–Graf von Bothmer 2016. §1.1 and the
   bibliography mention the preliminary write-up in the Hanoi ICPA2006 lecture notes (cited in
   Kraft–Russell 2014, §2), which was not seen. The number of general web searches is now three.

Optional suggestions of the second run that were applied: the abstract states the chronology (the 2007
statement with étale base change for arbitrary morphisms; the 2014 theorem for affine morphisms with finite
degree) and says "a pair of" families; Table 1 no longer cites the Oberwolfach report as a source for
"holds", its F̄_p row cites Proposition 1.3 for the étale column, and its caption credits the
characteristic-zero remark; the Kraft–Russell bibliography entry gives the repository address of the
accepted manuscript; the Verification paragraph describes the second run; the remarks on the auxiliary e
in Lemma 2.1 and on the Z[i]-submodule E(F_q) were added.

## Relation to the literature, novelty and scope
- **Searches (September 2026).** arXiv, Crossref, OpenAlex, zbMATH and Semantic Scholar, including the works
  citing Kraft–Russell 2014, Kaliman 2020 and Poczobut 2025, and three general web searches (one in each of
  the three rounds: finder, first and second verification run). Queries covered generic equivalence,
  isotriviality over countable fields and in positive characteristic, families with isomorphic closed
  fibres, fibrewise trivial families, the Fischer–Grauert theorem, forms of the affine line, and affine
  modifications of tori. In the second run OpenAlex refused anonymous requests (its free daily limit was
  exhausted), so the saved OpenAlex citing lists and Semantic Scholar were used instead.
- **Findings.** Apart from the classical inseparable phenomenon behind Proposition 6.1, which already
  refutes the étale form of the Oberwolfach proposition over F̄_p (indeed in every positive
  characteristic), nothing found gives an earlier counterexample over F̄_p to the finite-degree form of
  Kraft–Russell 2014 and Kaliman 2020, or to any weaker form, and nothing gives one over Q̄. The new content
  of the note is Theorems 1.1 and 1.2.
- **Credit.** The characteristic-zero reduction from finite degree to étale is in the Kraft–Russell
  accepted manuscript (right after Theorem B) and in Kaliman 2020, Remark 3.12; the finite-degree form of
  the theorem is Kraft and Russell's own. The negative answer to Kaliman's positive-characteristic question
  is an immediate consequence of Russell's classical forms of A¹.
- **Related work.**
  - Poczobut, Math. Z. 310 (2025), Art. 33 (arXiv:2108.10041): étale-local triviality of smooth projective
    fibrewise trivial families with reduced automorphism group scheme over fields of infinite transcendence
    degree; his §8 (arXiv version) has characteristic-p examples with non-reduced automorphism group
    schemes. Our families are not fibrewise trivial.
  - Bogomolov–Böhning–Graf von Bothmer, Eur. J. Math. 2 (2016) 45–54, Question 4.2 of the arXiv version
    (the birational analogue over all algebraically closed fields): not affected, since our families are
    birationally trivial.
  - Russell 1970; Kambayashi–Miyanishi 1978; Kambayashi–Wright 1985: purely inseparable forms of A¹ and
    flat families of affine lines (background for Proposition 6.1).
- **Caveats (sources not seen).** The publisher's version of Kraft–Russell 2014 (the accepted manuscript
  and arXiv v1 were read); the journal versions of Kaliman 2020, Poczobut 2025 and
  Bogomolov–Böhning–Graf von Bothmer 2016 (their arXiv versions were read); Russell 1970,
  Kambayashi–Miyanishi 1978, Kambayashi–Wright 1985, Deuring 1941 and Kaliman–Zaidenberg 1999; and the
  Hanoi ICPA2006 lecture notes with the preliminary write-up of the Generic Equivalence Theorem. These
  works are cited for background and attribution only. The hypothesis of Kambayashi–Wright is quoted from
  Kraft–Russell 2014, Remark 5.2, and Deuring's theorem is used in the form given in Silverman's book. This
  negative search is not a proof of priority.
- **Scope.** The note answers the F̄_p part of Problem 2 negatively, for the étale and the finite-degree
  forms, for affine and for smooth projective morphisms (the latter for p ≡ 1 mod 4); for the étale form
  alone the negative answer is classical. The Q̄ part remains open, and so does the finite-degree form over
  algebraically closed fields of characteristic p and finite positive transcendence degree.
- **Suggested corpus status.** `open` → solved (negative answer over F̄_p), with a note saying:
  - the corpus record asks the étale form, whose failure in characteristic p is classical (Russell's
    purely inseparable forms of A¹; Proposition 6.1);
  - the new content is the failure of the finite-degree and arbitrary-base-change forms over F̄_p
    (Theorems 1.1 and 1.2), for smooth affine and for smooth projective families;
  - the fibre hypothesis concerns closed points, and the source has F̄_p and Q̄;
  - the Q̄ case (non-proper morphisms; Kraft–Russell 2014, Rem. 2.2; Kaliman 2020, Conj. 4.2) remains open.

  A conservative alternative is `partially_solved`.

## Final wording change (2026-09-30, after the final readiness check)
- The scope paragraph of the paper now says that the affine counterexamples (Theorem 1.1) work for every
  p and the smooth projective ones (Theorem 1.2) for p ≡ 1 (mod 4). Before, it said "for affine as well as
  for smooth projective morphisms" without this qualifier.
- In this report the first verification run is now said to have checked the proofs step by step, and the
  F_{2^7} example of Remark 4.1 is said to be proved in the text.
- No mathematical statement, proof, program or recorded output changed. The final check rebuilt the paper
  from the new archive with tectonic (14 pages, no warnings, no overfull or underfull boxes), re-ran all
  twelve programs from a fresh extraction (the nine programs of `claimant/` and `verifier/` reproduced
  their recorded outputs byte for byte; the three programs of `independent_run_2/` agreed up to the printed
  running times), and regenerated source.zip, the zenodo/ copies and the checksums.

## 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/