# Verification report — OWR-14299292-013 (Stanojkovski's question on Vaughan-Lee's groups 8.5.14 and 8.5.15)

Verification date: 2026-10-14.

**Verdict.** The note answers the question as posed: it gives an isomorphism invariant, defined through orders of
subgroups and centralisers, that takes different values on the two groups, with a proof for every odd prime p and
every quadratic non-residue ω (so for every primitive ω, as in the source). Every statement it makes is proved. The
pair had been separated without automorphism groups before: Copetti's thesis (2004, Lemma 7.21), modelled on Brahana
(1951), distinguishes the two groups by a pencil of conics; the note proves that its invariant counts the same objects.
- **Settled.** For a group G of class 2 let 𝒜(G) be the set of abelian subgroups A ≥ G′ with |A : G′| = p² all of
  whose elements outside G′ have centraliser of index p² (breadth 2). Then |𝒜(G1)| = 1 (the subgroup ⟨b, c⟩G′) and
  |𝒜(G2)| = 3 (⟨b, c⟩G′, ⟨b, d⟩G′, ⟨c, e⟩G′), so G1 ≇ G2 (Theorem A). The commuting graph on the elements of breadth 2
  has 3p(p+1)²/2 edges for G1 and (3p³+6p²+p)/2 for G2, with explicit degree patterns (Theorem B). The group
  G1(ω) depends only on the square class of ω, and G1(s²) ≅ G2 by an explicit change of generators (Proposition C).
  The class number, the character degrees, the numbers of abelian subgroups containing G′ of each order and the lines
  of breadth-2 points coincide for G1 and G2 (Proposition 7.1). The breadth-2 locus is not projectively the same
  (Remark 7.2), and |𝒜(G)| − 1 is the number of double lines in the pencil of conics of Brahana and Copetti
  (Remark 7.3).
- **Credit.** The question is Stanojkovski's (Oberwolfach Report 27/2025, p. 1443); the pair is the one that the
  geometric invariants of O'Brien and Stanojkovski (J. Algebra 703 (2026) 266–278) did not separate in their
  computations for 3 ≤ p ≤ 37; the presentations, the automorphism group orders and the class numbers are Vaughan-Lee's
  (arXiv:1501.00678, §8.5). The classification of these groups is due to Brahana (1951, 1958), Copetti (2004) and
  Vaughan-Lee, and the first separation of this pair without automorphism groups is Copetti's Lemma 7.21, after Brahana
  (1951, pp. 548–549). Vaughan-Lee's numbers of descendants of order p⁹ (arXiv:1702.04898, p. 37) also differ, but are
  computed from the automorphism groups. Breadth, centralisers and the correspondence of Baer are classical.
  Proposition C is implicit in the completeness of Vaughan-Lee's list.
- **Limits.** "Simpler" is not a formal notion; the note argues why |𝒜(G)| qualifies and says that the proposers may
  judge otherwise; by Remark 7.3 it is a group-theoretic form of the classical invariant. Brahana's papers (Amer. J.
  Math. 73 (1951), paywalled; Illinois J. Math. 2 (1958), whose open copy refused the automated requests) were not
  read; what the note says about them is taken from Copetti and from O'Brien–Stanojkovski. No priority is claimed
  beyond the statements of the note.

The note is unrefereed.

## Statement checked
- **Primary source.** Computational Group Theory, Oberwolfach Reports 22 (2025), no. 2, pp. 1389–1450 (Report
  27/2025, workshop 1–6 June 2025; organisers Dietrich, Eick, Nebe, Roney-Dougal), doi:10.4171/owr/2025/27.
  - Read in the open PDF on the publisher's site, https://ems.press/content/serial-article-files/51860, fetched
    anonymously on 2026-10-14 (561,652 bytes, sha256 `818c47a49ce2fd05b13dd30e2bd9826f869fedf5e8b1c3a9d6d77bf0c1e356b0`,
    identical to the copy fetched on 2026-09-29 for the corpus work, and fetched again by the third verification run
    with the same hash). Pages read as rendered images: p. 1443 (PDF page 55; the problem) and pp. 1399–1400 (PDF
    pages 11–12; abstract of Stanojkovski's lecture, joint with O'Brien).
  - The problem: p an odd prime, ω a primitive element of F_p; in the variety of groups of exponent p and class 2,
    G1 = ⟨a,b,c,d,e | [c,b], [d,b], [e,c], [e,d], [d,c] = [b,a], [e,b] = [c,a], [e,a] = [d,a]^ω⟩ and
    G2 = ⟨a,b,c,d,e | [d,a], [e,a], [c,b], [d,b], [e,c], [d,c] = [b,a], [e,b] = [c,a]⟩, labelled 8.5.14 and 8.5.15
    in Vaughan-Lee's list, whose automorphism group orders differ. The question: whether there is a "simpler
    invariant" for non-isomorphism testing of these groups. Magma code at the O'Brien–Stanojkovski repository.
- **Vaughan-Lee, arXiv:1501.00678v1.** Read: pp. 1–5 (introduction; Theorem 3: 70 groups of order at most p⁸; §2
  conventions: presentations in the variety, ω primitive throughout "for consistency", any non-residue for a group of
  order p⁶) and pp. 28–30 (§8.5, groups 8.5.11 to 8.5.17). The presentations of 8.5.14 and 8.5.15 agree with the
  source; both groups have p⁵ + 2p⁴ − p³ − p² conjugacy classes; |Aut| = 2(p−1)(p²−1)p¹⁷ and 2(p−1)³p¹⁷; the matrices
  for Aut(8.5.14) contain the blocks [[α, β], [ωβ, α]] in the rows and columns of b and c. The entries give no
  character degrees.
- **O'Brien–Stanojkovski, arXiv:2411.19555v3** (25 March 2026; J. Algebra 703 (2026) 266–278,
  doi:10.1016/j.jalgebra.2025.08.029): read completely (9 pages). §2.1 Baer correspondence; Theorem 2.3 (the
  invariants); §4: Brahana [4, 5], Copetti [9], Vaughan-Lee [29]; §4.1: 22 classes of 5-generator groups of order p⁸
  with |G′| = p³ (Copetti, Vaughan-Lee; case (j) of Copetti's Theorem 7.22 = group 8.5.j); for p ∈ {3, …, 37} the
  invariants identify twenty classes and "do not distinguish between (14) and (15)"; Table 2 rows (14), (15):
  n_p(I4(B)) = p, deg I4(B) = 3, n_p(I3(B•)) = 2p³ − p, degprim = (1, 3) for both.
- **Copetti, Finite-dimensional Lie algebras of nilpotency class 2, Master's thesis, ANU, 2004** (read by the third
  verification run in the copy of the ANU repository, handle 1885/150852): Chapter 1 (Brahana's 1951 and 1958 papers
  treat the 5-generator case; Chapter 7 determines the 22 algebras of dimension 12 "and 8") and Chapter 7 completely.
  Theorem 7.22 lists the orbits; orbit 14 = ⟨[a1,a2]+[a3,a4], [a1,a3]+[a2,a5], [a1, ωa4+a5]⟩ and orbit 15 =
  ⟨[a1,a2]+[a3,a4], [a1,a3]+[a2,a5], [a4,a5]⟩, which are the spaces of commutator forms of G1(1/ω) ≅ G1 and of G2.
  Lemma 7.21 (printed pp. 82–84): the orbits 14 and 15 are different, because the pencil of conics Q_x ∩ ρ,
  x ∈ ⟨u,v⟩ ([u,v] the only commutator of ρ, Q_x : x1B5 − x2B4 + x3B3 − x4B2 + x5B1 = 0 with the Plücker quadrics B_i)
  contains double lines for orbit 15 and none for orbit 14; the proof "is modelled on Brahana's proof which can be found
  in [6], p 548-549. It is also outlined in [7], p 650".
- **Repository** github.com/eamonnaobrien/Invariants-class2-pgroups: the README and the file `reps-5-3.m` (matrices
  B14, B15; `w := PrimitiveElement(F); // needs to be a nonsquare`), fetched again by the third verification run;
  `invariants.m` read by the first two verification runs.
- **Corpus record.** ulamai/UnsolvedMath, OWR-14299292-013 (dataset version 1.6.0, snapshot commit c6e7f41…;
  upstream status `open`). Its statement is the source's question. Another snapshot of the collection (revision
  372682f) restates it with both presentations: "Find an invariant simpler than automorphism-group order that
  proves G1 and G2 are nonisomorphic". Both are faithful to the source.

## Readings
| Reading | Answer | Where |
|---|---|---|
| An invariant, simpler than \|Aut\|, that separates G1 and G2 for every odd p (ω primitive, as in the source) | \|𝒜(G1)\| = 1, \|𝒜(G2)\| = 3 | Theorem A |
| The same for every quadratic non-residue ω | the same | Theorem A |
| ω a quadratic residue (not the source's hypothesis) | G1(ω) ≅ G2; nothing to separate | Proposition C(b), Remark 4.2 |
| A second invariant of the same kind | edges of the commuting graph on the elements of breadth 2: they differ by p | Theorem B |
| An invariant already in the literature | the pencil of conics of Copetti's Lemma 7.21 (after Brahana 1951): double lines only for 8.5.15; \|𝒜(G)\| − 1 is their number | Remark 7.3, §1.1, §1.3 |
| An invariant inside the framework of O'Brien–Stanojkovski (determinantal invariants of B and B•) | their numerical invariants do not separate the pair (their computations, p ≤ 37; the note proves that the Pfaffian ideals and I4(B) coincide); the rank locus of B• as a configuration does: exactly two hyperplanes meet the breadth-2 locus in p² + 2p + 1 points for G2, none for G1 | Section 7, Remark 7.2 |
| Cheaper counts: breadth distribution, class number, character degrees, numbers of abelian A ≥ G′ of each order (Qiao's polynomial), commuting r-tuples, lines of breadth-2 points | all coincide for the two groups | Proposition 7.1 and the text after it |
| "Simpler" | informal; the note gives three reasons (orders of subgroups and centralisers only; computable from breadths and a count of lines; uniform proof) and leaves the judgment to the proposers | Section 1.3 |

## Results in the paper
- **Lemma 2.1.** G_i is the Baer group H(B_i) on F_p⁵ × F_p³ of an explicit alternating map B_i; |G_i| = p⁸ (upper
  bound from the relations, lower bound from H(B_i)); C_G(x) = ker(ad_x) × W, breadth = rank ad_x; Lemma 2.2: 𝒜(G) ↔
  good lines (isotropic lines of breadth-2 points) of P⁴(F_p).
- **Section 3.** The net of forms f_y and its Gram matrix M(y); the five 4×4 Pfaffians and the kernel vector v(y);
  exactly one form of rank 2 (y1 = y2 = 0) with 3-dimensional radical R; the Pfaffian ideal
  (y1², y2², y1y2, y1y3, y2y3) for both groups (Lemma 3.2). The breadth-2 locus S = P(R) ⊔ Q, |S| = 2p² + 2p + 1,
  Z(G) = G′ (Lemma 3.3). The scroll Z = {rank N(X) ≤ 1}: S = P(R) ∪ Z, P(R) ∩ Z = D = P⟨e2, e3⟩; the lines all of
  whose points lie in Z are D, ℓ∞ and the rulings ℓ_t = P⟨P_t, Q_t⟩ (Lemma 3.4).
- **Theorem A.** Case 1 (lines in P(R)): only D = ⟨b, c⟩G′. Case 2: a good line not in P(R) has at least 3 points on
  Z, so it lies on Z and is ℓ∞ or a ruling; B(P_t, Q_t) = t(ω − t²)w1 + (t² − ω)w2 for G1 (zero iff t² = ω: never,
  ω a non-residue) and 2t²w1 − 2tw2 for G2 (zero iff t = 0), and ℓ∞ is isotropic only for G2. Remarks 4.1–4.3:
  where the hypotheses enter; for ω = s² the extra members ⟨b + c/s, a + sd + e/s⟩G′ and ⟨b − c/s, a − sd − e/s⟩G′;
  the conjugate rulings ℓ_{±√ω} over F_{p²}.
- **Theorem B.** Degrees from the identities B(x, v(y)) = (y2 L_x(y), −y1 L_x(y), 0) for x ∈ R and
  B(v(z), v(y)) = Λ(y, z)·(cross product), Λ = ωy1z1 − y2z2 (G1), y1z2 + y2z1 (G2).
- **Proposition C.** (a) a, b^{1/s}, c, d^{1/s}, e^s satisfy the relations of G1(ωs²); (b) for ω = s²,
  a′ = (d^ω e^{−1})², b′ = bc^{1/s}, c′ = bc^{−1/s}, d′ = a^s d^{s²} e, e′ = a^{−s} d^{s²} e satisfy the relations of
  G2, determinant 16s². Remark 6.1: relation to Vaughan-Lee's list and automorphism orders; Remark 6.2: the
  repository encodes 8.5.14 as G1(1/ω), the same group.
- **Proposition 7.1** and Section 7: counts that coincide, now with the character degrees (p⁵ of degree 1, p⁴ − p³ of
  degree p, p⁴ − p² of degree p², from Z(G) = G′ and the ranks of the forms of the net) and the numbers of commuting
  r-tuples; Qiao's polynomial; the determinantal invariants of O'Brien–Stanojkovski (n_p(I4) = p; every 4×4 minor of
  M(y) is ±Pf_iPf_j, so I4(B) = (y1,y2)²(y1,y2,y3)² for both groups and deg I4 = 3 for every p, the multiplicity of the
  point y1 = y2 = 0; n_p(I3(B•)) = 2p³ − p = 1 + (p − 1)(2p² + 2p + 1); degrees (1, 3) consistent with the plane and the
  cubic scroll). Remark 7.2: in P(V/D) the rulings form the conic X1² = X4X5 and P(R) is the point r = (0 : −ω : 1)
  (G1), resp. (1 : 0 : 0) (G2); a ruling is isotropic iff it is the only ruling in its hyperplane through P(R), i.e. a
  point of contact of a tangent through r; |𝒜(G)| − 1 = the number of rational tangents; the breadth-2 loci are not
  projectively equivalent (two hyperplanes meet S in p² + 2p + 1 points for G2, none for G1). Remark 7.3: Copetti's
  conic of x ∈ R⊥ is {y : x(v(y)) = 0}, the union of the lines through c that correspond to the rulings in the
  hyperplane x = 0; its double lines are the tangent hyperplanes of Remark 7.2; |𝒜(G)| = 1 + (number of double lines).

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **Program of the note** (`check_note.py`, sympy, 7 seconds; 197 checks, ALL CHECKS PASSED: True). From the
  presentations as text: relation vectors, rank 7, the commutator tables. Symbolic identities: M(y), Pfaffians, v(y),
  M(y)v(y) = 0, R, N(v(y)) of rank 1, B(P_t, Q_t), B on ℓ∞, B|_R, B(x_R, v(y)), B(v(z), v(y)), the counting identities,
  Proposition C (seven relations each, determinant 16s²), the two extra members for ω = s² with the corrected sign,
  B14 = G1(1/ω), B15 = G2. Enumeration for p = 3, 5, 7, G2 and G1(ω) for every ω ∈ F_p^× (15 groups): breadths, the
  decomposition S = P(R) ∪ Z, all 2-planes (1210, 20306, 140050), isotropic planes p³ + 2p² + p + 1, no isotropic
  3-space, p² + 2p + 2 lines inside S, |𝒜| with the predicted members (1 for the 6 non-residues, 3 for G2 and the 6
  residues), degrees and edges of Γ, class numbers; the Hilbert function 1, 3, 6, 10, 3, 3, 3, 3, 3 of F_p[y]/I4(B).
  Deliberately wrong predictions (the sign of e/s; a coefficient of B(P_t, Q_t)) and a wrong presentation were
  reported as failures.
- **Programs of the first write-up** (`original/`): symbolic checks of the identities and of Proposition C; the
  per-point enumeration in 17 runs (p ∈ {3, 5, 7, 11, 13}, with non-primitive non-residues 6 mod 7, 10 mod 11,
  5 mod 13 and the residues 2, 4 mod 7, 3 mod 11, 3 mod 13); the search for the generators of Proposition C(b).
- **Verification run A** (`verification_run_A/`): own symbolic re-derivation (tables derived from the presentations);
  enumeration of all 2-planes by reduced row echelon forms, breadths from 3×3 minors, radicals by elimination, for the
  source's presentations at every odd p ≤ 23 and for the repository's B14, B15 at p ≤ 13, with residues and
  non-primitive non-residues (35 runs); the group of order 3⁸ itself by multiplication (centraliser orders
  {6561: 27, 729: 1350, 243: 5184}, class number 369, 1210 subgroups A ≥ G′ with |A : G′| = 9, 49 abelian,
  |𝒜| = 1, 3, 3 for G1(2), G2, G1(1)); |Aut| at p = 3 (3¹⁵ · 288 for G1(2), 3¹⁵ · 144 for G2 and G1(1), as
  Vaughan-Lee's formulas give); the Hilbert function of I4(B) over Q.
- **Verification run B** (`verification_run_B/`): own symbolic re-derivation; enumeration of all 2-planes and isotropic
  3-spaces for p ≤ 13 and every ω ∈ F_p^× (40 runs), and of the lines on Z for p ≤ 11; a per-point algorithm with
  deliberately wrong inputs at p = 5, 7, 17 (all detected; at p = 17: (|S|, |𝒜|, |E|) = (613, 1, 8262) for G1(3),
  (613, 3, 8245) for G2 and G1(1)); the groups themselves by explicit multiplication at p = 3 (G1(2), G2, G1(1)) and
  p = 5 (G1(2): order 390625, class number 4225, 181 abelian A ≥ G′ of order p⁵, |𝒜| = 1; 14 minutes); a re-run of all
  programs of the first write-up from a copy (all outputs identical).
- **Third verification run** (`independent_run_2/`, see below): Brahana's model with the Plücker relations, p ≤ 13 for
  every ω and p = 17, 19 for selected ω (59 groups, every check OK); 75 symbolic identities; the groups of order 3⁸ in a
  polycyclic model; wrong-input tests; the re-run of the whole package.
- **Re-runs of the package.** At the writing (2026-10-14, from a copy): `check_note.py`, all programs of `original/`
  and the quick programs of both runs A and B, outputs identical (`reproducibility/RERUN_LOG.txt`). By the third
  run, from an extracted copy of `source.zip`: every program with a recorded output, the long ones included,
  114 comparisons, all byte-identical (`independent_run_2/r2_package_rerun.txt`).

## Independent verification runs
The results were first obtained with proofs and the programs in `original/`. Three independent verification runs,
all AI-assisted, followed (2026-10-14); each wrote its own programs from the definitions. Runs A and B worked
separately on the results before the note was written, and neither saw the other's report; the third run examined
the final text of the note and the package (below).

| Item | Run A | Run B |
|---|---|---|
| Statement, presentations, conventions against the source and Vaughan-Lee | CONFIRMED (commutator convention irrelevant in class 2; the repository's B14 is G1(1/ω)) | CONFIRMED |
| Lemma 2.1 (order p⁸, Baer model), Lemma 2.2 | CONFIRMED (re-derived; order argument without free groups; group check at p = 3) | CONFIRMED (group checks at p = 3, 5) |
| Lemmas 3.2, 3.3 (net, breadth-2 locus) | CONFIRMED (re-derived) | CONFIRMED (re-derived) |
| Lemma 3.4 (scroll, lines on Z) | CONFIRMED (re-derived) | CONFIRMED (re-derived; lines on Z enumerated for p ≤ 11) |
| Theorem A | PROVED (re-derived; enumerations p ≤ 23, repository matrices p ≤ 13) | PROVED (re-derived; enumerations p ≤ 13, all ω; p = 17) |
| Theorem B | PROVED | PROVED |
| Proposition C | PROVED (\|Aut\| at p = 3 consistent) | PROVED |
| Sources and literature | mostly accurate; three wrong sentences in the first write-up (fixes 1, 2, 8 below); Brahana and Copetti missing | open in print; prior work: none answering the question; Qiao's polynomial as the closest count |
| Overall | CONFIRMED with minor fixes | CONFIRMED (one sign misprint) |

Neither run found a mathematical error in a theorem, proposition or lemma.

### Third run: on the final text (`reproducibility/independent_run_2/`)
The third run, also AI-assisted, examined the final text of the note and the package line by line, as if nothing had
been checked, with programs of its own (Python with numpy and sympy); verdict **CONFIRMED WITH REQUIRED FIXES**, no
mathematical error, one major credit correction.
- **Sources.** It read again p. 1443 and pp. 1399–1400 of the Oberwolfach report, pp. 1–5 and §8.5 of Vaughan-Lee's
  list, §§1, 2, 4 and Table 2 of O'Brien–Stanojkovski, §1 of Qiao and the file `reps-5-3.m`, and it read for the first
  time Copetti's thesis (Chapters 1 and 7; the earlier attempts had failed) and Vaughan-Lee arXiv:1702.04898 (p. 37).
  Brahana's papers could not be read (1951: paywalled; 1958: the open copy on Project Euclid refused the automated
  requests).
- **Proofs.** Every lemma, theorem, proposition and remark was re-derived, including the parts added at the writing
  (a3 = 0 and Proposition 7.1(d)). No error was found. One sentence was false: the end of Section 7 said that |𝒜(G)|
  is not a function of the rank loci; in fact the position of P(R) relative to the scroll determines |𝒜(G)| and is
  different for the two groups (now Remark 7.2).
- **Literature.** Copetti's Lemma 7.21 (after Brahana 1951, pp. 548–549) separates the orbits 14 and 15, that is
  G1 and G2, by a pencil of conics; the run proved that its double lines are exactly the hyperplanes through P(R)
  containing one ruling, i.e. the isotropic rulings, so |𝒜(G)| = 1 + (number of double lines) (now Remark 7.3).
  Vaughan-Lee 2017 (p. 37) gives different numbers of descendants of order p⁹ for the two groups ((p+37)/2 and
  (p+81)/2, p ≥ 5), computed from the automorphism groups. Brahana's 1958 paper was missing.
- **Own computations.** `r2_enum.py` (relation space K ⊂ Λ²V from the presentation text; breadth = dim(x∧V + K) − 7;
  isotropic lines as the decomposable vectors of K, by the Plücker relations on all points of P(K)): for G2, B14, B15
  and G1(ω) for every ω ∈ F_p^×, p = 3, 5, 7, 11, 13, and five groups each at p = 17 and 19, all statements of
  Theorems A and B, Proposition 7.1 (with the character degrees), Remarks 4.2, 5.1, 7.2 and 7.3 (double lines computed
  from Copetti's definition with the 4×4 Pfaffians) — 1069 checks, all OK. `r2_symbolic.py`: 75 identities (among them
  the tables derived from the presentation text, every 4×4 minor of M(y) = ±Pf_iPf_j, Copetti's quadric = x·v(y),
  Proposition C, the polar lines and pencils of Remarks 7.2 and 7.3), all OK; mutation tests detected. `r2_group.py`:
  the groups of order 3⁸ in a polycyclic model by multiplication only (associativity, exponent, relations, class
  number 369, 1210 subgroups A ≥ G′ with |A : G′| = 9, 49 abelian, |𝒜| = 3, 1, 3 for G2, G1(2), G1(1)).
  `r2_wrong_input.py`: three wrong inputs, all detected.
- **Package.** Re-run from an extracted copy of `source.zip`: 114 comparisons, all byte-identical, the long runs
  included.

**Corrections required by the runs**, all applied in the note:
1. (A1) n_p(I3(B•)) = 2p³ − p = 1 + (p − 1)(2p² + 2p + 1), the affine breadth-≤ 2 locus including 0: Section 7.
2. (A2) deg I4(B) = 3 is the multiplicity of the point y1 = y2 = 0 in y-space, not the degree of the scroll; the
   scroll accounts for the 3 in degprim = (1, 3): Section 7 (Hilbert function recomputed by `check_note.py`).
3. (A3, B1) The second extra member for ω = s² is ⟨b − c/s, a − sd − e/s⟩G′: Remark 4.2 (and tested in
   `check_note.py`).
4. (A4) Brahana (1951) and Copetti (2004, Theorem 7.22) as the classification sources: Sections 1.1, 1.3, "Scope and
   priority" (now read in the case of Copetti; see 12).
5. (A5) The coincidence of all Theorem 2.3 invariants is O'Brien–Stanojkovski's computational finding for p ≤ 37;
   the note proves the coincidence of the Pfaffian ideals, of I4(B) and of the point counts and component degrees of
   the rank loci: Sections 1.1, 7, "Verification".
6. (A6) The restatement "Find an invariant simpler than automorphism-group order …" is located (snapshot 372682f of
   the collection); version 1.6.0 carries the source sentence: Section 1.1.
7. (A7) Vaughan-Lee fixes ω primitive throughout; the blocks [[α, β], [ωβ, α]] have determinant α² − ωβ², the norm
   form: Section 1.1 (Conventions), Remark 6.1.
8. (A8) The 2016 J. Algebra paper (vol. 468, 440–447) is by Seungjai Lee: reference [Lee16].
9. (A9, cosmetic) The sentences on "only Vaughan-Lee's orders" and on the newest arXiv item of Stanojkovski are not
   repeated in the note.
10. (A10, B2) The repository's normalisation ω → 1/ω of B14, and the invariant computed for B14, B15: Remark 6.2,
    Section 8.
11. (B2) Qiao's totally-isotropic polynomial (arXiv:2408.09963, §1.3.3) coincides on the pair; 𝒜(G) refines its
    coefficient a2 by breadth: Section 7.
12. (third run) Credit Copetti's Lemma 7.21 (after Brahana 1951, pp. 548–549, and Brahana 1958, p. 650, according to
    Copetti) for the first separation of the pair without automorphism groups, and relate it to |𝒜(G)|: abstract,
    Sections 1.1, 1.2, 1.3, Remark 7.3, "Scope and priority", "What was read".
13. (third run) Replace the false closing sentence of Section 7 ("|𝒜(G)| is not a function of these rank loci") by
    Remark 7.2: the tangents through r, |𝒜(G)| − 1 = number of rational tangents, the breadth-2 loci not projectively
    equivalent, the hyperplane counts.
14. (third run) Add Brahana 1958 (Illinois J. Math. 2 (1958) 641–717, doi:10.1215/ijm/1255448335) to the
    classification sources: Section 1, references.
15. (third run) Vaughan-Lee 2017 does treat the pair (descendant numbers, p. 37): Section 1.3, "Scope and priority".
16. (third run) Add the character degrees to Proposition 7.1 (part (e), with proof), and the commuting r-tuples; mention
    them in Section 1.3.
17. (third run) Run B enumerated the lines on Z for p ≤ 11 (not 13): "Verification".
18. (third run) Record the third run: Section 8, this report, `reproducibility/independent_run_2/`, the package README.
In addition, I4(B) is now shown to be the square of the Pfaffian ideal for both groups (deg I4(B) = 3 for every p),
Section 7 is titled "Other invariants of the pair", and Remark 4.3 points to Remark 7.2.

## Relation to the literature, novelty and scope
- **Searches (14 October 2026; first write-up, the three runs, and at the writing).** arXiv API (authors Stanojkovski
  and O'Brien; abstracts with "exponent p" and "class 2"/"class two", "Vaughan-Lee", "Brahana", "Pfaffian",
  "alternating bilinear", "commutator matrix", "abelian subgroups", "breadth", "non-isomorphism"), Crossref, zbMATH,
  Semantic Scholar, OpenAlex citing works (none for O'Brien–Stanojkovski; three for Vaughan-Lee 2015: S. Lee 2016,
  Vaughan-Lee arXiv:1702.04898, Wan–Ye–Zhang arXiv:2003.12802; six for Brahana 1951 and five for Brahana 1958, among
  them Vaughan-Lee's paper on groups of order p⁸ and exponent p, IJGT 2015, and Freeman 1974 on the lines of PG(4, q)),
  the repository, Copetti's thesis, seven web searches.
  - The pair was separated without automorphism groups by Copetti (Lemma 7.21) after Brahana (1951); the proposers'
    March 2026 version mentions only Vaughan-Lee's automorphism group orders for this pair.
  - Vaughan-Lee 2017 (p. 37): descendant numbers (p+37)/2 and (p+81)/2 (p ≥ 5) — automorphism-group based.
  - Qiao (arXiv:2408.09963): counts of abelian subgroups containing G′ by order — the same for both groups;
    Wan–Ye–Zhang classify such groups through orbits of subspaces of alternating matrices, for five generators and
    |G′| = p³ only at p = 3 (abstract).
- **What was read.** See "Statement checked"; of Qiao, §1 (pp. 1–5) of arXiv v2; of Vaughan-Lee 2017, pp. 1–2 and 37;
  of the other works found, abstracts. Not read: Brahana 1951 (paywalled) and Brahana 1958 (the open copy on Project
  Euclid refused the automated requests of the third run); what the note says about them is taken from Copetti and
  O'Brien–Stanojkovski.
- **Novelty.** To our knowledge the invariant |𝒜(G)| with Theorem A and its uniform proof, Theorem B and Proposition C
  are not in print in this form; Proposition C is implicit in the completeness of Vaughan-Lee's list; by Remark 7.3,
  |𝒜(G)| is a group-theoretic form of the invariant of Brahana and Copetti, which separated the pair first. A negative
  search is not a proof of novelty.
- **Scope.** "Simpler" is informal; the note does not claim that the proposers will regard |𝒜(G)| as simpler.

## 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/