# Verification report — OWR-14299906-003 (Freslon–Gerontogiannis–Skalski: is QAut(A) a quantum subgroup of the quantum isometry group of O_A?)

Verification dates: 2026-09-30 (first AI-assisted verification run, on an earlier draft) and 2026-10-01 (second
AI-assisted verification run, on the final text).

**Verdict.** The answer is negative in general, and it depends on the reading.
- *Canonical reading.* For every primitive A with QAut(A) ≠ Aut(A), the homomorphism u_ij ↦ p_ij, C(G^1_A) → C(QAut(A)), of
  FGS Proposition 5.11 does not factor through C(G^ℓ_A) for any ℓ ≥ L_A, in particular not through C(G^∞_A).
  Consequently, the natural action S_i ↦ Σ_j p_ij ⊗ S_j of QAut(A) on O_A is not D-isometric (Theorem 1.1).
- *Abstract reading.* For the Cuntz algebras O_N (A = J_N) and for A = J_N − I_N, N ≥ 4, QAut(A) = S_N^+ is not a quantum
  subgroup of G^∞_A, not even of G^2_A, under any surjective quantum group homomorphism (Corollary 1.3). The same holds
  for a primitive 5×5 matrix A_hub at every level ℓ ≥ 3, although S_4^+ = QAut(A_hub) is a quantum subgroup of G^2_{A_hub}
  (Example 4.2). On the other hand, for two primitive 4×4 matrices A_D and A_E, QAut(A) is non-classical and is a quantum
  subgroup of G^∞_A through a non-canonical embedding (Proposition 1.4).
- Whether a quantum group such as S_M^+ (M ≥ 4) can be a quantum subgroup of G^∞_A through an embedding without fixed
  vertices is open (Problem 4.3). The note is unrefereed.

## Statement checked
- **Primary source.** A. Skalski (joint work with A. Freslon and D. Gerontogiannis), "Quantum isometry groups of
  log-Laplacians on Cuntz–Krieger algebras", Oberwolfach Reports 23 (2026), no. 1, pp. 228–231, Report 4/2026,
  doi:10.4171/OWR/2026/4. Question 1 is on p. 230.
  - It recalls that G^1_A contains QAut(A) as a quantum subgroup and asks whether the same holds for G^∞_A when
    QAut(A) ≠ Aut(A).
  - The abstract assumes throughout that A is primitive (p. 229).
- **Same question in the paper.** A. Freslon, D. M. Gerontogiannis, A. Skalski, arXiv:2601.02835 (v1 6 Jan 2026, v3 14 Jul
  2026; to appear in Comm. Math. Phys. according to the arXiv comment), doi:10.48550/arXiv.2601.02835.
  - Section 5.3, just before Proposition 5.11, says it is unclear whether G^∞_A contains QAut(A). The passage is the same
    in v1, v2 and v3.
  - Definitions used in the note (numbering of v3): Definition 3.2 (actions), 3.4 (D-isometric), 3.6 (G^ℓ_A),
    Proposition 4.2 (the action φ_A), Theorem 4.5 (finality of G^∞_A), Definition 5.9 (QAut(A)), Proposition 5.11
    (QAut(A) ⊂ G^1_A and Aut(A) ⊂ G^∞_A). A quantum subgroup is given by a surjective unital *-homomorphism intertwining
    the coproducts (Section 5.3); surjective Hopf *-homomorphisms of the polynomial algebras are also used there, and the
    note's results hold in both settings.
  - Standing assumption: A is primitive. All matrices in the note (J_N, J_N − I_N, A_D, A_E, A_hub, A_7) are primitive.
- **Corpus record.** ulamai/UnsolvedMath (version 1.6.0), OWR-14299906-003, status `open`. Its statement text is taken
  from the Skalski abstract.

## Readings
| Reading | Answer | Witness |
|---|---|---|
| (Q_can) does u_ij ↦ p_ij factor through C(G^∞_A)? | no, for every primitive A with QAut(A) ≠ Aut(A) | Theorem 3.1: the canonical relations of levels ≤ L_A force all p_ij to commute |
| the natural action of QAut(A) on O_A is D-isometric? | no, for every such A | Theorem 1.1, via the finality theorem FGS 4.5 |
| (Q_abs) is there any surjective quantum group homomorphism C(G^∞_A) → C(QAut(A))? | no for A = J_N, J_N − I_N (N ≥ 4) and A_hub | Corollary 1.3, Example 4.2 |
| (Q_abs) for general A | depends on A: yes for A_D and A_E | Proposition 1.4 |
| (Q_abs) with QAut(A) ⊇ S_M^+, M ≥ 4, in general | open | Problem 4.3 (excluded when the embedding fixes a vertex, Theorem 1.2(b)) |

## Results in the paper
- **Theorem 3.1.** Let A be irreducible and p a magic unitary with Ap = pA. If all products p_{α1β1}···p_{αnβn} over
  admissible words of length n ≤ L_A are partial isometries, then the p_ij commute. Proof: for a ≠ c, insert row sums
  along a shortest path from a to c; this writes p_ab p_cd as a sum of partial isometries with orthogonal ranges and
  sources, and a product of two projections is a partial isometry only if they commute. The bound L_A is attained
  (Remark 3.2).
- **Theorem 1.1** (canonical reading) follows from Theorem 3.1 and FGS Theorem 4.5.
- **Property (G).** A compact quantum group has (G) if its polynomial algebra is non-commutative and generated by the
  coefficients of every non-trivial finite-dimensional unitary representation. S_M^+ (M ≥ 4) has (G) by Banica's fusion
  rules (Math. Ann. 314 (1999), Theorem 4.1); cited, not re-proved.
- **Theorem 1.2.** No quantum group with (G) is a quantum subgroup of G^ℓ_A (a) for ℓ ≥ 2 and A ∈ {J_N, J_N − I_N};
  (b) for ℓ ≥ L_A + 1 through a homomorphism that fixes a vertex; (c) for ℓ ≥ L_A through a homomorphism with
  π(u_ij) ∈ ℂ·π(p_ij).
  - (a) uses that all q_ab commute with all p_cd in C(G^2_A) (for J_N − I_N with a row-sum step).
  - (b) uses a path U → F → ⋯ → F → U through fixed vertices and two uniformity lemmas from AP′ = Q′A, which make the
    full blocks commute.
  - (c) uses Theorem 3.1.
- **Corollary 1.3.** N ≥ 4, A ∈ {J_N, J_N − I_N}: QAut(A) = S_N^+ ≠ S_N, and no S_M^+ (M ≥ 4) is a quantum subgroup of
  G^ℓ_A, ℓ ≥ 2.
- **Example 4.2.** A_hub: S_4^+ = QAut(A_hub) is a quantum subgroup of G^2 but not of G^3; so arguments using level 2 only
  cannot work for general A.
- **Proposition 1.4** (Propositions 5.1, 5.3, Lemma 5.2). QAut(A_D) = dual of ℤ2 ∗ ℤ2 embeds by u ↦ diag(s, t, 1, 1).
  QAut(A_E) = QAut(C_4) = H_2^+ embeds by placing the 2×2 cubic matrix (a b; c d) on the vertices {1, 3} and 1 at 2 and 4;
  all four coproduct formulas are proved. H_2^+ has the non-trivial group-like unitary 2a² − 1, so it lacks (G).

## Computations (scripts and outputs in reproducibility/)
- **Lead** (`lead/check_note.py`, standard library, exact rational arithmetic, about 20 s): 36 checks of every finite claim
  of the note, including primitivity, L_A and |Aut(A)| of all matrices; exact failure levels of the canonical relations
  (J_4, J_4 − I_4: level 2; A_hub: level 3; A_7: level 5, lower levels hold); the explicit model of Lemma 5.2(e) with all
  relations and coproducts evaluated in its tensor square; the maps of Propositions 5.1 and 5.3 on all words of length
  ≤ 6. All pass.
- **Finder** (`finder/`, numpy/scipy, numerical): Lemma 2.2 on random partial isometries; Theorem 3.1 level by level on five
  graphs; the identity (1) and the orthogonality sums; Propositions 5.1 and 5.3 in random models; facts used only in an
  earlier approach. Reruns reproduced all five outputs byte for byte.
- **First independent verification run** (`independent/`, numpy, written separately): v1 (Example 4.2), v2 (Remark 3.2),
  v3 (Propositions 5.1 and 5.3 in fresh random models, all words of length ≤ 7, all four coproducts, the group-like
  unitary). Reruns reproduced the outputs.
- **Second independent verification run** (`independent_run_2/`, own README; standard library except one numpy script;
  written independently of all other code):
  - `r2_exact_checks.py`: 81 exact checks of the finite claims of the final text, all pass. They use a new rational
    non-commutative 4×4 magic unitary and two exact models of Lemma 5.2: the note's model and a new model on
    Q^3 ⊕ Q^2. They cover all four coproduct formulas and Δ(g) = g ⊗ g in the tensor square, the maps of
    Propositions 5.1 and 5.3 on all admissible word pairs of length ≤ 7 (and on all word pairs, admissible or not,
    of length ≤ 7 for Proposition 5.1 and ≤ 5 for Proposition 5.3),
    Σ_k w_ik ⊗ w_kj = Δ(w_ij), exact failure levels of the canonical relations (J_4, J_4 − I_4: 2; A_hub: 3;
    A_7: 5; A_D, A_E: 2), and identity (1) with the telescoping sums.
  - `r2_exhaustive_small_graphs.py`: Theorem 3.1 on 908 (matrix, model) pairs. These are all primitive matrices of size
    4 or 5 with two disjoint transpositions in Aut(A) (up to relabelling), and all primitive matrices of size ≤ 6 with
    four twin vertices. The first failure is always at a level ≤ L_A, with equality in 207 cases. In the four-twin
    family it never comes later than the word length k + 2 used in the proof of Theorem 1.2(b).
  - `r2_numeric_models.py`: Lemma 2.2(a) on random partial isometries, and a complex Pauli-type model of S_4^+
    (failure levels 2, 2, 3, 5 for J_4, J_4 − I_4, A_hub, A_7).
  - All programs of `lead/`, `finder/` and `independent/` were rerun from an extracted copy of the source archive; all
    nine outputs are byte-identical to the recorded ones.

## Independent AI-assisted verification runs
### First run (2026-09-30, on an earlier draft)
Verdicts of the first AI-assisted independent verification run:

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (FGS's primitivity assumption had to be stated) |
| Proofs | CONFIRMED_WITH_FIXES: the canonical-inclusion theorem, the J_N and J_N − I_N results and the two examples were confirmed; the draft's general no-go theorem for quantum groups with (G) was invalid as proved (its last step used commutation of full blocks, but only one row against one row was proved; A_hub is an instance where the draft's argument would run at level 2 although S_4^+ is a quantum subgroup of G^2) |
| Computations | CONFIRMED (reruns byte-identical; new checks v1–v3) |
| Answer as posed | CONFIRMED (negative in general, in both readings for O_N; the abstract reading depends on A) |
| Novelty | CONFIRMED as far as can be checked (no priority claim) |
| Presentation | fixes required |

All required fixes were applied in the note:
1. The draft's general no-go theorem was replaced by the proved Theorem 1.2 (cases (a) J_N and J_N − I_N at level 2,
   (b) fixed vertex at level ≥ L_A + 1, (c) canonical type at level ≥ L_A), with the two uniformity lemmas and the
   path through the fixed vertices written out.
2. Corollary 1.3 (O_N and J_N − I_N) is proved directly from Lemma 2.3(b), including the row-sum step for J_N − I_N.
3. The draft's corollary on matrices with ≥ 4 twin vertices is not claimed; it appears only as the open Problem 4.3,
   together with the open fixed-point-free case.
4. FGS's standing assumption that A is primitive is stated, and all matrices used are primitive.
5. Proposition 5.3 now checks all four coproduct formulas (Lemma 5.2(d)), and Lemma 5.2(e) records the non-trivial
   group-like unitary 2e − 1 = 2a² − 1 of H_2^+, so H_2^+ lacks property (G).

In addition, Example 4.2 (the audit's counter-instance) is included to show that level 2 does not suffice, Banica's
fusion rules are cited precisely (Theorem 4.1 of Math. Ann. 314 (1999)), and Remark 3.2 records that the bound L_A in
Theorem 3.1 is attained.

### Second run (2026-10-01, on the final text)
The second AI-assisted independent verification run was done before any further edit. It checked the text line by line
against the primary sources, which were fetched anonymously:
- the Oberwolfach Report 4/2026 PDF (pp. 229–230 rendered and read);
- the LaTeX sources of arXiv:2601.02835 v1, v2 and v3. The passage of Section 5.3 with Proposition 5.11 is byte-identical
  in all three versions, and so is the theorem numbering;
- Banica, Math. Ann. 314 (1999), Theorem 4.1 (arXiv:math/9811060);
- Banica–Bichon–Collins, Section 6 (cubic unitaries; arXiv:math/0701859).

| Item | Verdict |
|---|---|
| Statement fidelity (OWR p. 230; FGS v1–v3 Section 5.3) | CONFIRMED |
| Theorems 1.1 and 3.1, Remark 3.2 | CONFIRMED |
| Theorem 1.2 (a), (b), (c), Lemma 4.1, Corollary 1.3 | CONFIRMED (every step checked; (b) is consistent with Example 4.2, whose level-3 argument is the case k = 1 of the proof of (b), while level 2 lies below the range L_A + 1 = 4 of (b)) |
| Example 4.2; Problem 4.3 (fixed-point-free case stated as open) | CONFIRMED |
| Propositions 5.1 and 5.3, Lemma 5.2 (d), (e) | CONFIRMED |
| Cited results (FGS v3 numbering: Definitions 3.2, 3.4, 3.6, 5.9, Remark 3.8, Lemma 3.10, Propositions 3.12, 4.2, 5.10–5.12, Theorem 4.5, Section 2; Banica Theorem 4.1; BBC07 Section 6) | CONFIRMED |
| Computations (new code; byte-identical reruns) | CONFIRMED |
| Novelty and credit | CONFIRMED (no priority claim) |
| Presentation | fixes required (below) |

It found no mathematical error. Its required fixes were all applied:
1. The Verification paragraph and the reproducibility README now say that all results are proved in the text; the
   earlier wording was removed.
2. The Verification paragraph names a single first run on the earlier draft and describes this second run.
3. The abstract and the introduction state precisely what remains open. This is whether a quantum group such as S_M^+
   can be a quantum subgroup of G^∞_A, for general A, through an embedding that fixes no vertex.
4. The Scope paragraph dates the literature search 1 October 2026 and names the fixed-point-free case (Problem 4.3) as
   open.
5. The closing remark of Section 5 now says that the embeddings fix vertices, so by Theorem 1.2(b) the two quantum
   groups cannot have property (G). Indeed they have non-trivial group-like unitaries (Lemma 2.5(a)).
6. The fixed vertex in Theorem 1.2(b) is renamed z, which avoids a clash of notation with its proof.
7. This report records the run.
8. The code and outputs of the run are included in `reproducibility/independent_run_2/`.

Minor changes were also made:
- "Skalski's abstract in Oberwolfach Report 4/2026";
- the orbit relation of Lemma 2.4(a) is credited to Lupini–Mančinska–Roberson (J. Funct. Anal. 279 (2020), 108592);
- one sentence contrasts Theorem 1.1 with Joardar–Sharma (for the spectral triple of Farsi et al., the quantum
  automorphism group of a strongly connected graph acts isometrically);
- "not isometric for this spectral triple" in the abstract;
- a separate symbol for the antipode;
- Problem 4.3 notes that the answer is negative for J_N and J_N − I_N.

## Relation to the literature, novelty and scope
- **Searches (30 September and 1 October 2026, anonymous, logged).** arXiv API: authors Freslon, Gerontogiannis,
  Skalski, Joardar, Mandal, Banica, Schmidt, Weber, and keyword searches on quantum isometries and quantum automorphisms of Cuntz–Krieger and graph
  C*-algebras; Crossref (DOIs of all references verified); OpenAlex (FGS has cited_by_count 0); zbMATH; one web search.
  - The newer related works (Gerontogiannis–Goffeng, arXiv:2605.31390; Ismert, arXiv:2608.23520; Karmakar–Mandal,
    Studia Math. 2026, on quantum automorphism groups of direct sums of Cuntz algebras) do not treat the question.
    The second run repeated the arXiv, Crossref, OpenAlex and zbMATH searches and one web search on 1 October 2026,
    and found nothing new.
  - Related work: Schmidt–Weber (CMB 2018) and Joardar–Mandal (IDAQP 2018) on quantum symmetries of graph C*-algebras;
    Joardar–Sharma (J. Noncommut. Geom., 2025) on isometric actions for a different spectral triple (there the quantum
    automorphism group of a strongly connected graph acts isometrically). None concerns G^∞_A.
- **Novelty.** Modest. The question was posed as open by Freslon, Gerontogiannis and Skalski (arXiv:2601.02835, OWR 4/2026);
  the arguments are short and use standard tools (partial isometry criteria, orbits of magic unitaries as in
  Lupini–Mančinska–Roberson, Banica's fusion rules). This negative search is not a proof of priority.
- **Scope.** Settled: the canonical reading for all primitive A; the abstract reading for J_N, J_N − I_N (N ≥ 4) and A_hub;
  existence of positive cases (A_D, A_E). Open: a characterization of the A for which QAut(A) is a quantum subgroup of
  G^∞_A, in particular whether quantum groups with property (G) can embed without fixed vertices (Problem 4.3).

## 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/