# Verification report — KOU-21.76 (Kourovka Notebook Problem 21.76)

Verification date: 2026-09-28.

## Verdict
The note gives explicit, closed, non-completable irreducible elementary nets of every order n ≥ 3:
- over F(x,y) for every field F;
- over F_3(t) and F_2(t).

For p = 3 (and p = 2) it proves a characterisation: such nets exist over K iff K is not algebraic over F_p.

**Priority.** Problem 21.76 itself was, to our knowledge, first answered by Ya. N. Nuzhin, Sib. Math. J. 67 (2026)
840–845 (Corollary 1, online 30 July 2026). This note makes no priority claim for that answer.

Unrefereed.

## Statement checked
- Kourovka Notebook No. 21 (arXiv:1401.0300v46, 1 September 2026, still the current version on 28 September
  2026), Problem 21.76 (V. A. Koibaev).
- The problem asks whether irreducible closed elementary nets of order n ≥ 3 that are not completable exist over
  a field of odd characteristic.
- The definitions (net, elementary net, irreducible, closed, completable) are those of Problems 19.48 and 21.76.
- The corpus record (ulamai/UnsolvedMath, commit 2ea030b) lists KOU-21.76 as `partially_solved`, with "the
  existence question remains open". That is stale in view of Nuzhin (2026).

## Results and proof obligations checked
- **Lemma 2.1** (criterion σ_ij σ_ji σ_ij ⊆ σ_ij), **Lemma 2.2** (lifting a closed pair from R/J) and
  **Lemma 3.1** (free-product property of t_12(Fx), t_21(Fx); a special case of Koibaev, *Closed pairs*,
  2011, and of Nuzhin 2026, Lemma 8): checked line by line.
- **Theorem 1.1** (F(x,y)): checked.
- **Theorem 1.2(a)** (F_3(t)): checked.
  - The pair (F_3, F_3·ι) in SL_2(9) is one of the two exceptional cases of Levchuk's refinement of Dickson's
    theorem (Algebra and Logic 22 (1983) 306–316; statement taken from its zbMATH review, Zbl 0542.20027).
    This gives |H| = 120; the order was also checked by direct enumeration.
  - The closedness of the pair then follows from the Sylow 3-subgroup of order 3.
- **Theorem 1.2(b)** (F_2(t)): checked. H ≅ D_10, by a hand proof: the polynomial identity over F_4.
- **Theorem 1.2(c):** checked. The σ_ij are F_p[f]-modules, F_p[f] is a PID, and [F_p(t) : F_p(f)] = 2. So
  the hypotheses of the second paragraph of Kourovka 19.48 hold. Problem 21.76 uses 19.48 only for its
  definitions.
- **Proposition 5.1** (algebraic extensions of F_p: every irreducible elementary net of order ≥ 3 is
  completable): checked. It is a special case of Koibaev–Nuzhin, Sib. Math. J. 58 (2017) 109–112; a short
  direct proof is included.
- **Corollary 1.3:** checked. The transport to K ⊇ F_p(s) works because closedness and the completability
  criterion do not depend on the ambient field.
- **Remarks.**
  - 4.1: other lines over F_9 and F_4.
  - 4.2: an example over Q(i) via Z[i] → F_9. It consists of Z[3i]-modules, not Z[i]-modules, which is
    consistent with Koibaev 2020.
  - 5.2: for p ≥ 5, lifting from a field cannot give examples, by Levchuk's theorem.

## Computations (exact; scripts and outputs in reproducibility/)
Three independent sets of scripts (author, referee 1, referee 2) were used:
- Lemma 3.1 by exhaustive words.
- Theorem 1.1 and the lifted nets by exhaustive short words in generators of E(σ). Every transvection found has
  its entry in σ_ij:
  - 7.0M words over F_3(x,y);
  - 13.0M words over F_3(t);
  - 1.1M words over F_2(t);
  - 15.5M words over Z[i].
- The groups H over F_9 and F_4.
- Proposition 5.1 by brute force over small fields.
- Remark 5.2 over F_25, F_49, F_121, F_125 and eight small non-field rings. No closed pair violates a
  triple-product inclusion.
- Exceptional closed pairs occur only in characteristics 2 and 3.

## Independent adversarial audits
- **First audit** (earlier draft). It confirmed Theorem 1.1. It found the prior answer by Nuzhin 2026 and showed
  that the draft's finite-field theorem was a special case of Koibaev–Nuzhin 2017. It also found the F_3(t)
  example.
- **Second audit** (this version). No mathematical error; recommendation "publishable after fixes". All fixes
  were applied:
  - citations of Levchuk 1983, Koibaev 2011 (*Closed pairs*) and Nuzhin's Lemma 8;
  - the date of Notebook v46;
  - the characteristic-0 citation;
  - the scripts in the archive;
  - the definition of a closed pair;
  - the degree-2 argument;
  - the identification H ≅ SL_2(5);
  - the 19.48 wording and the context of Koibaev 2025 and Dryaeva–Koibaev–Nuzhin 2018;
  - Koibaev 2020 for Remark 4.2;
  - the sources of Lemma 2.1;
  - DOIs in the bibliography.

## Relation to the literature, novelty and scope
- **First answer.** Nuzhin 2026 answers Problem 21.76 first, with closures of subgroups over F(y,z) in every
  characteristic.
- **Plausibly new here:**
  - the one-variable examples in characteristic 3;
  - Corollary 1.3 for p = 3.
  - These combine Levchuk's exceptional pair with an elementary reduction modulo an ideal. Searches of zbMATH,
    Crossref, OpenAlex, mathnet.ru and arXiv in September 2026 found no earlier treatment.
- **Characteristic 2.** Examples were known earlier (Koibaev 2011). Its full text was not consulted, so the p = 2
  half of Corollary 1.3 may not be new.
- **Open.** For p ≥ 5 the case of transcendence degree one (F_p(t)) remains open.
- The negative search is not a proof of priority.

## 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/