# Verification report — KOU-19.93 (Kourovka Notebook 19.93 b), P. Spiga: lower p-central factors without a C_p wr C_p quotient)

Verification date: 2026-10-21.

**Verdict.** The note gives a **complete proof** (a disproof of part b)): part b) is false: for each of p = 2, 3, 5
no bound f(p,d) exists (for p = 5 with any d >= 3, for p = 3 with any d >= 5, for p = 2 with any d >= 7).
- **Theorem 1.** For (p, d) = (5, 3), (3, 5), (2, 7) and every K >= 0 there is a finite p-group Q with d(Q) = d, no
  epimorphic image isomorphic to C_p wr C_p, and a lower p-central factor of order > p^K. Q is a group Q_N of
  Section 5 of the note; the truncation level N and the index i of the large factor depend on K. Remark 7.2
  (direct product with C_p^m) gives every larger d.
- **Not new: the method.** The groups Q_N in truncated power-series algebras, the filtered Golod-Shafarevich
  inequality and the counting bound are those of M. Kinyon and J. Urban (preprint, Version 4, 19 September 2026,
  Section 3.33), who refuted part a) for every prime p >= 7 with d = 2 and x^p = y^p = (xy)^p = 1. Also theirs is the
  argument in F_p^2 that excludes C_p wr C_p for these relations (Lemma 3(i), (ii) and the case of a pair in (iii)).
  Their preprint declares AI assistance. The filtered inequality is Vinberg's (1965), in the form of Ershov's survey (Theorem 2.6,
  Proposition 2.7(a) of arXiv:1206.0490v1). The note proves both steps it uses (Lemmas 1 and 5) in full.
- **New.** The relators x_i^p, (x_i x_j)^p for (p, d) = (5, 3), (3, 5), to which the Kinyon-Urban argument applies
  pair by pair (Lemma 3); the twelve relators for (2, 7) with the bilinear-form criterion (Lemma 4) and the exact
  certificate C; the observation that the Kinyon-Urban polynomial 1 - 2t + 3t^p is positive on (0,1) for p = 3, 5
  and that their relations force an abelian group for p = 2.
- **Open.** (p, d) = (5, 2), (3, 2), (3, 3), (3, 4) and (2, 2), ..., (2, 6). A route to (5, 2) via the triangle
  group (5,5,5) is given as an unchecked sketch (Remark 7.5) and is not claimed.

The note is unrefereed and AI-assisted. The Notebook regards a problem whose AI solution has not been confirmed by
mathematicians as unsolved; part b) should therefore be regarded as open until the argument has been checked by
experts. The corpus status of the record (partially_solved) is unchanged by this note.

## Statement checked
- **Primary source.** E. I. Khukhro, V. D. Mazurov (eds.), Unsolved Problems in Group Theory. The Kourovka
  Notebook, No. 21, arXiv:1401.0300.
  - v48 (6 October 2026), page 152 (local copy and text extraction, line 5894); v49 (20 October 2026), printed
    page 153, read on arXiv (downloaded on 2026-10-21; the arXiv API lists v49 as the current version, updated
    2026-10-20). The text of 19.93 and its comment are the same in both.
  - Content (paraphrase): there is a function f: N x N -> N such that, if G is a finite p-group with d generators
    and no epimorphic image isomorphic to C_p wr C_p, then every factor of the p-lower central series of G has
    order at most f(p,d); a) for p >= 7; b) for p = 2, 3, 5. The proposer (P. Spiga, 19th issue, 2018) adds a
    comparison with Shalev's characterization of finitely generated p-adic analytic pro-p groups.
  - Marks: a) carries the star (confirmed solution), with the comment that the conjecture is refuted for p >= 7
    (AI solution confirmed by P. Spiga; M. Kinyon, J. Urban, preprint 2026, with the URL of the preprint). b)
    carries no mark.
- **Notebook web site.** The update file `2026/10/21upd-3.pdf` was read on 2026-10-21 (HTTP status 200); its header
  reads "November 2026 update" (sic). Problem 19.93 is on its PDF page 24 in the v49 form: a) starred, b) unmarked.
  The site and the preface of v49 state that problems with an AI solution not yet confirmed by mathematicians are
  marked with a sharp sign and should be regarded as unsolved.
- **Corpus record.** UnsolvedMath v1.8.0 (snapshot d371ee9), record KOU-19.93: status `partially_solved`; the
  statement agrees with the Notebook.

## Readings
- The hypothesis excludes epimorphic images of G itself, not sections and not quotients of subgroups; the
  Kinyon-Urban construction is read in the same way ("excludes wreath quotients of Q_N itself, exactly as asked").
- "With d generators": the groups have d(Q) = d, so both d(G) <= d and d(G) = d are covered.
- The conjecture asserts one function on N x N; for a refutation for a prime p one value of d suffices.
- "p-lower central series" = lower exponent-p central series P_1 = G, P_{i+1} = P_i^p [P_i, G]. If it is read as
  the Jennings-Zassenhaus series, Remark 7.1 gives the same conclusion.

## Results in the paper
- Lemma 1 (cumulative filtered Golod-Shafarevich inequality B_{N+1} >= 1 + d B_N - sum_k B_{N+1-delta_k}),
  self-contained proof; Lemma 2 (divergence of sum B_N tau^N when P(tau) < 0).
- Lemma 3 (W_p: pi = (s, a) is a surjective homomorphism; w^p = 1 iff s(v)a = 0; the axis argument: no generating
  d-tuple with y_i^p = (y_i y_j)^p = 1, including i > j by conjugation). Parts (i), (ii) and the case d = 2 of (iii)
  are the argument of Kinyon-Urban, Section 3.33; the note applies it to every pair.
- Lemma 4 (p = 2: D = C_2 wr C_2 dihedral; g^2 = z^{l1 l2}, [g,h] = z^{l1(g)l2(h)+l2(g)l1(h)}; psi(rho_k) =
  z^{alpha^T S_k beta}; psi onto iff alpha, beta independent, by the Burnside basis theorem) and certificate C.
- Lemma 5 (counting bound of Kinyon-Urban, proved in full); Proposition 6 (Q_N finite p-group, d(Q_N) = d,
  relators hold, no W_p quotient, B_N <= dim F_p[Q_N]/I^N); proof of Theorem 1 with the order of quantifiers.
- Exact values: 1 - 3t + 6t^5 at 1/2: -5/16; 1 - 5t + 15t^3 at 1/3: -1/9; 1 - 7t + 12t^2 at 7/24: -1/48.
- Remarks: 7.1 Jennings-Zassenhaus series; 7.2 every larger d; 7.3 open cases and a heuristic, marked as such;
  7.4 the pro-p limit; 7.5 sketch for (5, 2), not claimed; 7.6 the Kinyon-Urban polynomial (exact criterion
  2/(3p) < (p/(2(p-1)))^(p-1)); 7.7 Shalev's theorem (sections, not quotients: no conflict) and powerful groups.

## Computations (programs and outputs in reproducibility/)
- `scripts/d8_certificate.py` (numpy, 2 s): check A, ordered independent pairs orthogonal for S_1..S_m:
  16002, 7938, 3938, 1954, 1002, 530, 274, 122, 46, 22, 8, 2, 0; check B, all 8^7 maps to D: 2048256, 1016064,
  504064, 250112, 128256, 67840, 35072, 15616, 5888, 2816, 1024, 256, 0 epimorphisms (= 2^7 x check A).
- `scripts/wreath_check.py` (numpy, 7.5 s): Lemma 3(ii) for every element of W_2, W_3, W_5; the pair statement
  behind (iii) exhaustive over all pairs with u^p = v^p = 1 (2,025 pairs for p = 3; 31,640,625 for p = 5): 0
  violations.
- `scripts/gs_polys.py` (< 1 s), `scripts/truncation_dims.py` (7 s; B_N sanity values, not used in the proof).
- `check_note.py` (standard library, about 2 s; written with the note, independent code): 72 checks, 0 failures,
  including certificate C by both checks, 114 of the 336 single-entry changes of the table breaking the
  certificate, the degrees of the relation elements, the SHA-256 of `data/p2_relator_table.json`
  (ae6e2e599e6af8742255beaa3a75042860c6f17a562b49fb12d828b8679c58e4) and the equality of the table printed in the
  note with the data of `d8_certificate.py`.
- All four programs of `scripts/` were run again from a copy of the package on 2026-10-21; the outputs are
  byte-identical to the recorded ones.
- `independent_run_2/` (numpy, sympy, networkx, pypdf; about 25 s): the four programs of the third verification run
  (see below) with their recorded outputs.

## Independent verification runs
The results were first obtained, with proofs and the four programs of `scripts/`. Two independent verification
runs, both AI-assisted, followed (2026-10-21); both examined the first write-up of the result, not the text of the
note, and each re-derived the proofs and wrote its own programs. A third run examined the final text (below).

| Item | Run A | Run B |
|---|---|---|
| Statement, readings, record | CONFIRMED (v48 text; v49 date from the arXiv API) | CONFIRMED (v48, v49 and the site's update-3 file read) |
| Account of Kinyon-Urban (scope p >= 7, d = 2; index line "19.93 all"; AI declaration) | CONFIRMED against the preprint | CONFIRMED against the preprint |
| Lemma 1, Lemma 2 (against Ershov, Theorem 2.6, Proposition 2.7(a)) | CONFIRMED (re-proved; Ershov read) | CONFIRMED (re-proved; Ershov read) |
| Lemma 3 (p = 3, 5) | CONFIRMED (own code: exhaustive for p = 2, 3; 400,000 random pairs for p = 5) | CONFIRMED (own code, W_p as a permutation group, Phi by brute force: all 2,025 and 31,640,625 pairs; all 28,638,765 relator-satisfying 5-tuples in W_3, none generating) |
| Lemma 4 and certificate C (p = 2) | CONFIRMED (own code, D_8 as symmetries of a square; same prefix counts; 0 epimorphisms) | CONFIRMED (own code; bilinear count and brute force in three conventions; mutation tests: without rho_12 256 epimorphisms; 114 of 336 single-entry changes break the certificate) |
| Lemma 5, Proposition 6, assembly and quantifiers | CONFIRMED | CONFIRMED (Lemma 5 also tested on 8 small groups) |
| Remarks 7.1, 7.2, 7.4 | CONFIRMED | CONFIRMED |
| Remark 7.3 heuristic sentence | not proved; to be marked as heuristic | correct as stated for p-th-power relators |
| Exact values -5/16, -1/9, -1/48, -115/729 | CONFIRMED | CONFIRMED (exact root isolation also) |
| Re-run of the four programs | - | byte-identical outputs |
| Novelty | no prior treatment of b) found; not established beyond the searches | no prior treatment of b) found; not established beyond the searches |

No run found a gap in the proofs.

**Corrections required by run A** ("Required fixes" 1-5), all applied in the note:
1. Remark 7.3: the three-relators sentence is labelled a heuristic, not proved and not used.
2. Theorem 1 and its proof (Section 6) state that the truncation level N and the index i depend on K.
3. Section 1.1 says that v49 was read on arXiv (printed page 153) and gives the update-3 finding of the site.
4. The computational note on Lemma 3 (Section 9; this report) says that the pair check is exhaustive over all
   pairs with u^p = v^p = 1 (2,025 for p = 3; 31,640,625 for p = 5), and the Verification paragraph reports the
   enumeration of all 28,638,765 relator-satisfying 5-tuples for p = 3 by run B.
5. Section 1.3 states that the Kinyon-Urban counterexamples, like those of the note, have no C_p wr C_p quotient
   of the group itself, with a quotation of nine words.

**Corrections required by run B** ("Required fixes" 1-3 and suggestions 2-3), all applied:
1. The account of the Notebook's site: the update file 2026/10/21upd-3.pdf was read (HTTP status 200; header
   "November 2026 update" (sic); 19.93 a) starred, b) unmarked); the site's rule on unconfirmed AI solutions.
2. Full references, checked with Crossref and zbMATH: Lubotzky-Mann (J. Algebra 105 (1987) 484-505,
   doi:10.1016/0021-8693(87)90211-0), Vinberg (Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965) 209-214, Zbl 0171.29401;
   not read, cited as described by Ershov), Golod-Shafarevich (Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964) 261-272,
   Zbl 0136.02602; not read), Ershov (Internat. J. Algebra Comput. 22 (2012) 1230001,
   doi:10.1142/S0218196712300010; theorem numbers from arXiv:1206.0490v1), Shalev (J. Algebra 145 (1992) 204-208,
   doi:10.1016/0021-8693(92)90185-O, Zbl 0745.20029; only the zbMATH review read), Dixon-du Sautoy-Mann-Segal
   (CUP, 2nd ed. 1999, doi:10.1017/CBO9780511470882), Jennings (Trans. Amer. Math. Soc. 50 (1941) 175-185,
   doi:10.1090/S0002-9947-1941-0004626-6), Huppert (Endliche Gruppen I, Springer 1967,
   doi:10.1007/978-3-642-64981-3).
3. The docstring of the release copy of `d8_certificate.py` says "about 2 seconds" (it said "about 1 minute").
4. The p = 2 table is shipped in machine-readable form with its SHA-256, together with the brute-force check.
5. The route to (5, 2) suggested by run B is included as a sketch, plainly marked as unchecked and not claimed.

### Third verification run (final text of the note, 2026-10-21)
An AI-assisted run that examined the text of the note itself, whose proofs had been reworded since the first
write-up, as if nothing had been checked. Programs and outputs: `reproducibility/independent_run_2/`.

| Item | Result |
|---|---|
| Statement fidelity | CONFIRMED: v48 (local copy, p. 152) and v49 (downloaded, printed p. 153) read again, identical; a) starred, b) unmarked; preface rule on the sharp sign; the update file 2026/10/21upd-3.pdf read (HTTP status 200; header "November 2026 update"; 19.93 on PDF p. 24, a) starred, b) unmarked); corpus record (status partially_solved) agrees with the Notebook |
| Sources | Kinyon-Urban Version 4 (unchanged at the cited URL): introduction, index line, Section 3.33 read again; Ershov arXiv:1206.0490v1, Section 2.3 (Theorem 2.6, Proposition 2.7(a), attribution to Vinberg); Shalev's zbMATH review; 6 DOIs (Crossref) and 3 Zbl numbers (zbMATH) re-checked |
| Lemma 1 (both well-definedness claims, including N + 1 - delta_k <= 0; codimension of F_1), Lemma 2 (B_0 = 0) | CONFIRMED |
| Lemma 3 (including i > j, all d) | CONFIRMED; own code: model of the note and a permutation model without pi, all pairs (31,640,625 for p = 5), for p = 2, 3 all maximal cliques of the compatibility graph (none generates W_p) |
| Lemma 4, certificate C | CONFIRMED; Table 1 read from the PDF equals main.tex, the JSON file (SHA-256 as printed) and the DATA string; bilinear count, Gaussian elimination, brute force over all 8^7 maps in UT_3(F_2) and in the wreath product (prefix counts as printed, ratio 2^7), Lemma 4(c) on all maps; 114 of 336 single-entry changes; 2 surviving pairs without rho_12 |
| Lemma 5, Proposition 6 (I in F_2 also for p = 2; (d), (e)), assembly, quantifiers | CONFIRMED (Lemma 5 also tested on D_8, Q_8, C_4 x C_2, W_3, C_9 x| C_3) |
| Exact values and degrees | CONFIRMED: -5/16, -1/9, -1/48; lowest degrees exactly 5, 3, 2 |
| Remarks 7.1-7.7 | CONFIRMED; 7.5 plainly a sketch and not claimed, its statements not false (torsion of triangle groups, 25 x (-2/5) = -10, genus 6, nilpotency index 9 computed in F_5[C_5 x C_5]) |
| Re-run of the package from the source archive | recorded outputs reproduced (truncation_dims.out up to its printed running time) |
| Novelty | no answer to b) found (arXiv API, OpenAlex, Crossref, zbMATH, one web search); not certified |

No mathematical error and no gap was found. **Corrections required by the third run**, all applied:
1. Credit: Lemma 3(i), (ii) and the case of a pair in (iii) are the argument of Kinyon-Urban, Section 3.33; the
   note now says so at Lemma 3, in Section 1.3, in "Scope and priority" and in the abstract, and lists as new only
   the relator sets (R5), (R3) to which the argument applies pair by pair.
2. The sentence that the relations used for p >= 7 "do not work" for p = 2, 3, 5 now says that the
   Golod-Shafarevich argument with them fails (abstract, Section 1.3); Remark 7.5 sketches another route for p = 5.
3. The Verification paragraph and this report say that the first two runs examined the first write-up, and record
   this run; Section 9 and the README describe `independent_run_2/`.
4. Lemma 4(a): a clause justifies Z(D) = <z> (a centre of order 4 would make D/Z(D) cyclic).

## Relation to the literature, novelty and scope
- **Kinyon-Urban** (preprint cited in the Notebook; Version 4, 19 September 2026; the introduction, the index of
  results and Section 3.33 were read in full). The method of the note is theirs; they declare AI assistance (a
  48-hour experiment with GPT-6 Astra through Codex, later reviewed by an AI model). Their index lists 19.93 with
  the parts "all", while Section 3.33 proves the case of every fixed prime p >= 7. Section 3.33 also contains the
  argument behind Lemma 3 (the map onto F_p^2, the criterion s(v)a = 0, the pair argument), which the note credits
  to it. Nothing there treats p = 2, 3, 5.
- **Ershov's survey** (arXiv:1206.0490v1, Sections 1.1-1.2 and 2.3 read): Theorem 2.6 is the inequality of Lemma 1
  in generating-function form, attributed to Vinberg; Proposition 2.7(a) is the divergence statement.
- **Shalev (1992)**, by its zbMATH review: a finitely generated pro-p group is p-adic analytic iff for some n it has
  no closed section C_p wr C_{p^n}. This concerns sections; 19.93 excludes quotients C_p wr C_p. No conflict.
- **Searches (2026-10-21).** arXiv API (wreath products with the lower exponent-p central series, with
  Golod-Shafarevich groups and with p-adic analytic groups; the proposer's papers; all papers mentioning the
  Kourovka Notebook, about 120, with the abstracts of those from 2025-2026 screened; recent Golod-Shafarevich
  papers; triangle groups with p-group quotients), OpenAlex (keyword searches; the 20 works citing Shalev 1992),
  Crossref, the zbMATH API, a few web searches; repeated in part by the third verification run (arXiv API: the newest
  Kourovka papers up to 2026-10-18, wreath products with p-central series, Golod-Shafarevich papers, the proposer;
  OpenAlex; Crossref; zbMATH; one web search). Semantic Scholar refused automated access (HTTP status 429; not
  worked around). The repository file named in Kinyon-Urban's Section 3.33 was not located.
- **Caveats.** No printed answer to part b) was found in these sources; a search that finds nothing is not a proof
  of novelty, and no priority is claimed. Vinberg (1965) and Golod-Shafarevich (1964) were not read; the textbook
  results cited (Burnside basis theorem, powerful p-groups, the Jennings series) were not re-read for the note.

## 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/