# Verification report — OWR-17294-017 (Lebed's question on the bijectivity of the quantum symmetriser)

Verification date: 2026-10-01 (revised after the independent AI-assisted verification runs of 2026-09-30 and 2026-10-01).

**Verdict.** The answer is negative. Farinati and García Galofre's Question 44, whether the image
A ⊗ B ⊗ A of their comparison map (B the Nichols algebra of −σ, A = kM(X, σ)) is a bimodule resolution of A, fails
for the permutation rack X = Z/4, x ◁ y = x + 1 (bijective, left and right non-degenerate, neither involutive nor
idempotent). Proved in the text by explicit arguments (the matrices were also checked by computer):
- an explicit 2-cycle z of weight 3 that is not a boundary; its real and imaginary parts z_re, z_im have integer
  coefficients and are independent in H_2 (over every field of characteristic ≠ 2);
- with coefficients k_0 (X acting by 0), for every quotient or subcomplex of the braided complex, the map induced by the
  quantum symmetriser misses part of HH_3(A; k_0), and dually in cohomology;
- a criterion: A ⊗ B ⊗ A is a resolution iff A is Koszul and B_n = K_n for all n (K the Koszul dual coalgebra);
- every finite permutation rack whose permutation has a cycle of length divisible by 4 (resp. 3) has non-zero homology
  in weight 3 (resp. 4) (characteristic 0).

Not new in substance: for finite bijective left non-degenerate non-involutive solutions with finite-dimensional B, in
particular the dihedral quandle R_3, the negative answer follows from the remark after Question 44 and Jespers–Kubat–Van
Antwerpen (TAMS 2019; Theorem 4.6 in the numbering of arXiv:1812.02026v2). By exact computation: the dimensions for Z/4,
H_2 = Q^2 in weight 3 and H_3 = Q^3 in weight 4; the cyclic racks Z/m fail for all 3 ≤ m ≤ 8; 13 of the 29 isomorphism
classes of bijective solutions on three points fail (ten of them are also covered by the finite-dimensional case, the
rack Z/3 by the family theorem). Classes that pass have only passed finite tests. The note is unrefereed.

## Statement checked
- **Primary source.** V. Lebed, "Yang–Baxter cohomology: Diagonal deformations and knot invariants", pp. 3237–3238 of
  Oberwolfach Report 51/2019, Mini-Workshop: Algebraic Tools for Solving the Yang–Baxter Equation (organised by
  E. Jespers, V. Lebed, W. Rump, L. Vendramin), Oberwolfach Rep. 16 (2019), no. 4, 3207–3242, doi:10.4171/OWR/2019/51
  (published 2020). The report was read in the published PDF.
  - Problem 2 is on p. 3237: "Is the quantum symmetriser QS bijective for general solutions?"
  - Just before it, QS is introduced as an explicit map relating H*(X, r) to HH*(kM(X, r)); just after it, the answer is
    said to be known to be positive for involutive and idempotent solutions [FGG16, Leb17].
- **What "bijective" means** (read in the arXiv versions; numbering as there).
  - Farinati–García Galofre, JPAA 220 (2016), arXiv:1508.07970v2. Theorem 32 and Corollaries 42–43: the comparison map f
    from their d.g. bialgebra to the bar resolution is given by the quantum symmetriser of −σ, with image A ⊗ B ⊗ A.
    Question 44 (p. 22): is Im f = A ⊗ B ⊗ A a resolution of A as an A-bimodule? The remark after it (p. 23): positive for
    involutive solutions in characteristic 0, for the flip and σ = Id in any characteristic; a positive answer with
    dim B < ∞ forces finite global dimension. Proposition 24: the involutive case.
  - Lebed, IJAC 27 (2017), arXiv:1607.08081v1. Section 6: it is natural to ask whether the (reduced) quantum symmetriser
    can be turned into a quasi-isomorphism; Theorem 6: quasi-isomorphisms from the critical complex for idempotent
    braidings, with arbitrary linear bimodule coefficients (Definition 3.1 allows zero actions).
  - Lebed, J. Algebra 564 (2020), arXiv:1612.05768v1, Section 6 (p. 17): unknown in general; the involutive case is known
    for the constant character in characteristic 0.
  - Lebed, JKTR 27 (2018), arXiv:1801.08315v2: describing the kernel and image of QS is an open question.
  - D. Yang, J. Algebra 451 (2016), Question 7.5 (arXiv:1506.03117v1): a related question.
- **Corpus record.** ulamai/UnsolvedMath, OWR-17294-017: "Is the quantum symmetriser QS bijective for general
  set-theoretic Yang--Baxter solutions?" (upstream status `open`). The statement is faithful to the source.

## Readings
| Reading | Refuted? | Witness |
|---|---|---|
| QS bijective at chain level (literal) | trivially, for every finite solution (QS_2 = 1 − σ kills the orbit sums) | not used |
| Lebed's critical complex applied to bijective σ (literal) | fails already for the flip, which is involutive | not used |
| FGG Question 44: A ⊗ B ⊗ A → A exact | yes | Z/4: z_re, z_im (Thm 1.3(a)); Thm 1.4; R_3 (Cor 1.5); census |
| QS a quasi-isomorphism on some quotient (or subcomplex) of the braided complex, coefficients k_0 | yes, for every quotient and subcomplex | Z/4: weight-3 part of HH_3(A; k_0) is K_3 (dim 49), image B_3 (dim 47) (Thm 1.3(b)); R_3: off-diagonal Tor (§5) |
| constant character k_1, the quotient by ker QS | yes, for this quotient only | Z/4: H_3(B, ∂) = Q, H_3(M; Q) = 0 (Remark 3.3, exact computation) |
| constant character k_1, other quotients | not studied | – |
| FGG's quadratic quotient by the orbit sums (§3.6) | yes | H_2 of the quotient ≅ H_2(A ⊗ B ⊗ A) ≠ 0 for Z/4 (Remark 3.4) |

## Results in the paper
Names used in the finder's notes → numbering in the paper (the same mapping is in `reproducibility/README.md`).
- **Theorem A → Theorem 1.3 and §3.** Lemma 3.1 (relations in the eigenbasis u_j), the matrix (3) of QS_3 on
  W = span(u1u1u0, u1u0u1, u0u1u1), the linear form ℓ = (i, 1+i, 1) vanishing on its columns with ℓ(u0u1u1) = 1, the
  quantum Serre element κ = (ad_c u1)^2(u0), the cycle z (5) and z_re, z_im (6); the proof works over every field of
  characteristic ≠ 2 (by extension of scalars to k(i), now spelled out in the text). Remark 3.2 (A is PBW, hence Koszul), Remark 3.3 (k_1), Remark 3.4 (quadratic quotient).
- **Lemma 1 → Lemma 2.2** (Euler characteristic); **Proposition 2 → Proposition 2.3** (criterion, with a
  self-contained proof); Lemma 2.1 (B_n ⊆ K_n); Lemma 2.4 (coefficients k_0).
- **Theorem C → Theorem 1.4 and §4.** Reduction to two eigenvectors; the formula (8) for QS_3 on W for any eigenvalue
  λ, with determinant (1−λ)^4(1+λ^2); for λ = ω an explicit 4×4 matrix, a linear form ℓ' = (−1, 2ω, 2+2ω, 1) and the
  cycle z'. Remark 4.1 (Serre elements, dimensions, controls, and where the classes of Z/5 and Z/7 lie).
- **Theorem B → Corollary 1.5 and §5**, credited to FGG16 and JKVA19 (see also CJKVA, Cor. 7.11), with two further
  published routes (Roos' non-Koszulity of FK_3 as recorded by Walton–Zhang; the Hilbert series of FK_3^! from
  Walton–Zhang, Thm 3.10(1), giving 1 − t^6).
- **Census → §6**; Table 2 (cyclic racks Z/m, m ≤ 8).

## Computations (exact unless stated; scripts and outputs in reproducibility/)
- **Author** (`lead/check_note.py`, standard library only, exact arithmetic over Q and cyclotomic fields implemented in
  the script, about 15 s): 108 checks, ALL CHECKS PASSED.
  - Z/4 in the X-basis over Q: dim A_n (n ≤ 7), dim B_n = 1, 4, 14, 47, 152 and dim K_n = 1, 4, 14, 49, 171 (n ≤ 4),
    B_n ⊆ K_n, homology of the complex in weights ≤ 4 (weight 3: H_2 = 2; weight 4: H_3 = 3; nothing else), z_re and
    z_im are cycles and independent modulo boundaries (ranks 47 → 49), the identities in A_2, the k_1 computation.
  - Z/4 in the eigenbasis over Q(i): (2), Lemma 3.1, the matrix (3) (computed from the X-basis and from the diagonal
    model), ℓ, κ, (4), the normal words, Table 1 up to n = 5, the cycles z and z̄ (cycle, not a boundary),
    z = z_re + i z_im, and the homology per multidegree in weights 3 and 4.
  - Two-letter models: (8) as integer polynomials in λ, det = (1−λ)^4(1+λ^2); for λ = ω over Q(ω) the 3×3
    determinant −9, the 4×4 matrix exactly as printed, ℓ', the Serre elements, z' (cycle, not a boundary); the
    multidegree complexes (3, 6, 2) and (4, 8, 3) with one-dimensional homology; controls λ = −1, ζ_6, ζ_12 exact.
  - R_3: dim A_n = 1, 3, 5, 6, 6, 6, 6, 6, 6 (n ≤ 8), dim B_n = dim K_n = dim T(V)/(orbit sums)_n = 1, 3, 4, 3, 1, 0,
    H_A(t)H_B(−t) = 1 − t^6 + O(t^9), homology exact in weights ≤ 5 and H_3 = Q in weight 6.
  - Table 2: exact dims of B_n and K_n (n ≤ 4) for Z/m, 2 ≤ m ≤ 8, over Q(e^{2πi/m}); dim A_n both from monoid classes
    and from the eigenbasis.
- **Finder** (`claimant/`, Python with numpy). X-basis computations with exact fraction-free elimination and ranks
  modulo two large primes. The chain map property of f on all basis elements of weight ≤ 3 for six solutions; the
  homology of Im f and of the quadratic quotient with exact ranks (Z/4, Z/3, R_3); the certificate of Section 3 (checks B1–B3);
  the complexes for k_0 and k_1; the Koszul bimodule complexes; Tor via the bar complex (mod p); the census of all
  bijective solutions on 2 and 3 points; Theorem C examples. `run_all.sh` was rerun for the release (88 s): all outputs
  reproduced apart from timing lines.
- **Independent verification run 1** (`verifier/`, AI-assisted, separate code in exact arithmetic over Q(i),
  Q(ζ_3), Q(ζ_5), Q(ζ_7)). Z/4 in the eigenbasis (n ≤ 5) and in the X-basis (exact ranks of QS_3, QS_4); the 3×3 block;
  homology per multidegree; k_1; two-letter blocks; R_3 (to weight 7, and the quadratic dual); Z/5 and Z/7 exactly
  (Euler coefficients −11 t^4 and 2 t^3); Z/m for m ≤ 8 modulo a prime; an own enumeration of the census (73 solutions
  in 29 classes on 3 points, 8 involutive; the same 13 failing classes with the same coefficients). All scripts were
  rerun for the release and reproduced their recorded outputs.
- **Independent verification run 2** (`independent_run_2/`, AI-assisted, 2026-10-01; own code written from the text of
  the paper, standard library + numpy, about 2 minutes). X-basis integer matrices with ranks over Q certified by the
  Hadamard bound (if the largest rank r found modulo a set of primes has a prime product exceeding the product of the
  r+1 largest column norms, then r is the rank over Q), and an own exact implementation of Q(ζ_m); B_n in the eigenbasis
  also by restriction of scalars to Q.
  - Z/4 over Q: solution properties, dim A_n (n ≤ 8), B_n (n ≤ 4), K_n (n ≤ 5), the homology in weights ≤ 4 (only
    H_2 = 2 in weight 3 and H_3 = 3 in weight 4), the identities in A_2, z_re and z_im (cycles, independent modulo
    boundaries), Remark 3.3; over Q(i): Table 1 to n = 5, Lemma 3.1, the actions of c ⊗ id and id ⊗ c, (3), ℓ, κ, the
    normal words, and the homology per multidegree in weights 3 and 4.
  - Positive characteristic: z_re and z_im are independent modulo boundaries over F_p for every odd p ≤ 31 (over F_3,
    dim B_3 = 46) and are boundaries over F_2, as the hypothesis char k ≠ 2 suggests.
  - Two-letter models: (8) and its determinant as polynomials in λ; for λ = ω the 3×3 determinant −9, the 4×4 matrix
    entry by entry, ℓ', the rank 3, the kernel (ad_c u_1)^3(u_0) = v_1 + (2+2ω)v_2 + 2ωv_3 − v_4, z' (cycle, middle
    factors in B_3, not a boundary); the complexes (3, 6, 2) and (4, 8, 3) with one-dimensional homology; these parts
    are exact for λ = −1 and primitive 5th, 6th, 7th, 8th and 12th roots of unity.
  - Table 2 exactly for every 2 ≤ m ≤ 8 (all entries, dim A_n also from monoid classes, H_A H_K(−t) = 1 + O(t^5)); the
    homology of Z/5 in weight 4 (eleven one-dimensional H_3 classes, e.g. in 3α_1+α_2) and of Z/7 in weight 3 (H_2 in
    α_1+α_2+α_4 and α_3+α_5+α_6).
  - R_3: dim A_n = the Walton–Zhang series (n ≤ 8), dim B_n = dim E_n = dim K_n = 1, 3, 4, 3, 1, 0, 0, the FK_3 relations,
    1 − t^6, homology exact in weights ≤ 5 and H_3 = 1 in weight 6; B is 12-dimensional over F_2, F_3, F_5, F_7 as well.
  - Census, by a third own enumeration: 5 solutions in 5 classes on two points (3 involutive; none fails up to weight
    8, with dim B_n exact for n ≤ 8); 73 solutions in 29 classes on three points (8 involutive), the same 13 failing
    classes (four with 1 − t^4 and dim B = 16, three with 1 − 3t^4, six with 1 − t^6 and dim B = 12), all left and right
    non-degenerate; dim B_6 certified exactly for the 8 non-involutive classes that pass.
  - Theorem 1.4 on actual racks (X-basis): cycle types 4+1 (H_2 = 4 in weight 3), 3+1 and 3+2 (H_3 = 9 and 15 in
    weight 4); for comparison 2+2 (not covered) has H_3 = 3 in weight 4.
  - The release package was rerun from an extracted copy of source.zip: `lead/check_note.py` (ALL CHECKS PASSED),
    `claimant/run_all.sh` (all 12 outputs identical apart from timing lines) and all `verifier/` scripts (identical).

## Independent verification
### Run 1 (2026-09-30, AI-assisted)
| Item | Verdict |
|---|---|
| Source fidelity (OWR 51/2019 pp. 3237–3238; FGG16 v2; Lebed IJAC v1, plactic v1, JKTR v2) | CONFIRMED; two location slips (fix 2) |
| Proofs (Lemma 1, Proposition 2, Theorems A–C) | CONFIRMED (re-derived in the AI-assisted verification run) |
| Computations | CONFIRMED with own code (see above); the finder's scripts rerun with identical outputs |
| Answer as posed | CONFIRMED (negative), including the strongest reading with coefficients k_0 |
| Novelty | no published answer found; missed prior art for R_3 (fix 1) |
| Classification | paper candidate (short note), after minor fixes |

Required fixes of run 1 and how they were applied:
1. *Credit the three-line route for R_3* (the remark after Question 44 with JKVA19, Thm 4.6, and CJKVA Cor. 7.11).
   **Applied.** Corollary 1.5 is stated as a consequence of [FGG16] and [JKVA19], with [CJKVA26, Cor. 7.11]; the
   paragraph before it in the introduction and the proof in §5 describe the route; the R_3 statement is a corollary of
   published results; "Scope and priority" says that this case is not new in substance and names the two further
   published routes (Roos' result as recorded by Walton–Zhang, and Walton–Zhang, Thm 3.10(1)). The novelty wording is
   limited to "we found no earlier treatment of the example Z/4, of Theorem 1.3(b) or of Theorem 1.4" and "the
   criterion is elementary and presumably known to experts".
2. *Source locations.* **Applied.** [Leb19] is cited with pp. 3237–3238, Problem 2 on p. 3237; the plactic remark is
   cited as [Leb20, §6], and "Scope and priority" states that numbering refers to the arXiv versions (for the plactic
   paper, v1).
3. *Evidence versus proof.* **Applied.** §6 says that a class that does not fail has only passed a finite test; results
   obtained only modulo primes are described as evidence (Verification item 2, `reproducibility/README.md`); Z/5 and Z/7
   are cited as exact (Table 2, now computed exactly for all 2 ≤ m ≤ 8 by the author's program and, for m = 5, 7, by the
   verification run).
4. *Scope the k_1 claim.* **Applied.** Remark 3.3 concerns only the quotient by ker QS ("We did not study other
   quotients for these coefficients"), repeated in the concluding remarks; the "every quotient" statement is Theorem
   1.3(b) for k_0, and for R_3 it rests on the off-diagonal Tor (§5).
5. *Finite X.* **Applied.** The introduction says "for finite X the chain map QS_2 = 1 − σ kills the sums over σ-orbits".
6. *(Optional) Gateva-Ivanova 2018, Prop. 6.5.* **Applied**: not cited. The verification run observed that for R_3,
   where dim A_2 = 5 = 2|X| − 1, the direct computation gives dim A^!_3 = 3, while it read Prop. 6.5(2)(i) of
   arXiv:1808.03938v4 as asserting A^!_3 = 0 under this condition; we did not investigate this further. We also do not
   use the statement relating Koszulity and involutivity for square-free solutions quoted in JKVA19, Remark 4.7 from an
   earlier version of that preprint; its version 4 reports that statement as incorrect (its Example 3.13). The H_A
   cross-check with Dietzel–Feingesicht–Lebed, which the run could not confirm, was replaced by Walton–Zhang, Thm 3.10(1).

Additional changes by the author in the first revision:
- Theorem 1.3 is stated over every field of characteristic ≠ 2 (the proof in the text only uses that 2 is invertible); the
  computational statements are over Q.
- The non-boundary arguments now use explicit linear forms: ℓ for Z/4 and, for Theorem 1.4 with λ = ω, the printed 4×4
  matrix, the form ℓ' and the cycle z'. The finder's proof of Theorem C(b) used an Euler characteristic with ranks from a
  modular lower bound and a Serre-element upper bound; the new argument does not need modular ranks.
- The formula (8) for the 3×3 block for every eigenvalue λ; the proofs of Proposition 2.3 and Remark 3.4 made
  self-contained.
- The author's own program `lead/check_note.py`, which checks every printed matrix, form and cycle exactly.
- Bibliographic data of all references verified on Crossref and with the arXiv export API (this revision).

### Run 2 (2026-10-01, AI-assisted)
The second independent verification run concentrated on what was added after run 1: Theorem 1.3 over every field of
characteristic ≠ 2, the explicit forms and cycles, (8) and its determinant, the self-contained proof of Proposition 2.3,
Remark 3.4, Table 2, the citation of Jespers–Kubat–Van Antwerpen, the coefficient statement for k_0 and the census. It
re-read the sources (the OWR report, rendered on p. 3237; FGG16 v2 including §§2–3.6, Remark 26, Theorem 32, Corollaries
42–43, Question 44 and the remark after it; Lebed IJAC v1, plactic v1, JKTR v2; JKVA v1 and v2 and the abstracts of the
journal version and of the corrigendum; CJKVA v2; Walton–Zhang; Andruskiewitsch–Schneider, Prop. 2.11 and Lemma 3.7).

| Item | Verdict |
|---|---|
| Source fidelity | CONFIRMED |
| Proofs, including the parts added after run 1 | CONFIRMED; no mathematical error; the descent from k(i) to k was only implicit (fix 1) |
| Computations | CONFIRMED with own code (see above); the release package reproduces |
| Answer as posed | CONFIRMED (negative), including every quotient of the braided complex with coefficients k_0 |
| Novelty and credit | no published answer found; citation and credit precision (fixes 2, 6, 8) |
| Presentation | minor wording and notation (fixes 3, 4, 5, 7, 9, 10, 11) |

Required fixes of run 2 and how they were applied (all **applied**):
1. *Descent from L = k(i) to k.* §3 now says that the objects over L arise by extension of scalars (C_L = C ⊗_k L,
   H_*(C_L) = H_*(C) ⊗_k L); the proof of Theorem 1.3(a) deduces the independence of [z_re], [z_im] in H_2(C) from that of
   [z], [z̄] in H_2(C_L); (b) notes that B_3 ≠ K_3 over k; the proof of Theorem 1.4 notes that classes over L suffice.
2. *Unambiguous citation of JKVA19.* The theorem is cited as [JKVA19, arXiv v2, Thm. 4.6]; the bibliography entry and
   "Scope and priority" say that the published version numbers the corresponding characterisation of involutive
   solutions as Theorem 4.5 (as cited in CJKVA26, §7), that the published version was not seen, and that version 2
   contains the global-dimension condition; [CJKVA26, Cor. 7.11] is given as well.
3. *Remark 4.1.* "The failures of Z/5 and Z/7 involve more eigenvectors" was replaced by an accurate statement (the
   two-letter parts are exact for primitive 5th, 7th and 8th roots of unity as well; the classes of Z/5 and Z/7 lie in
   multidegrees such as 3α_1+α_2 and α_1+α_2+α_4).
4. *Abstract.* The permutation-rack statement now says "in characteristic zero", as Theorem 1.4 does.
5. *Notation in the proof of Theorem 1.4.* The generic word in θ and ρ is now y = y_1y_2y_3y_4 (v_1, …, v_4 are the basis
   words).
6. *Census credit.* §6 now says that all 13 failing classes are left and right non-degenerate, so that Corollary 1.5
   already covers the ten with dim B < ∞, and Theorem 1.4 covers the rack Z/3; it records that dim B_6 of the eight
   passing classes was certified exactly.
7. *Remark 3.4.* The factorisation through D/⟨ω⟩ is justified by the multiplicativity of f [FGG16, Cor. 42].
8. *Involutive case of Question 44.* Cited as [FGG16, Prop. 24 and the remark after Question 44].
9. *Verification records.* Item 3 of the Verification paragraph describes both runs; this report and
   `reproducibility/README.md` describe run 2, whose code and outputs are in `reproducibility/independent_run_2/`; the
   wording "by hand", "hand proof" and "hand certificate" was removed.
10. *Bibliography.* The volume (43) of the MSRI Publications series was added to [AS02].
11. *Theorem 1.3(b).* The generic complex is now called Y, to avoid confusion with FGG's d.g. bialgebra 𝒟.

## Relation to the literature, novelty and scope
- **Read** (arXiv versions unless stated): the OWR report (published PDF); FGG16 (v2); Lebed IJAC (v1), plactic paper
  (v1), JKTR survey (v2); D. Yang (v1); Jespers–Kubat–Van Antwerpen (v2, Theorem 4.6 and Remark 4.7);
  Colazzo–Jespers–Kubat–Van Antwerpen (v2, Corollary 7.11); Walton–Zhang (v1, Introduction, Def. 0.1, Thm 3.10);
  Andruskiewitsch–Schneider, "Pointed Hopf algebras" (§2.3, Lemma 3.7). In the verification run also: Covez–Farinati–
  Lebed–Manchon, Lebed–Szymik, Dietzel–Feingesicht–Lebed, Lebed (2024), Farinati (2024), Gateva-Ivanova (2018, v4),
  Gateva-Ivanova–Majid (2024), Bardakov–Elhamdadi–Singh (2025), Shi (2023); none answers the question. In verification
  run 2 also: Jespers–Kubat–Van Antwerpen v1 (where the characterisation is Theorem 4.5, without the global-dimension
  condition), the AMS abstracts of the journal version and of the corrigendum (which corrects the journal's Theorem 4.4),
  Kubat's abstract in the same Oberwolfach report (pp. 3214–3215, citing the global-dimension equivalence as Theorem 4.6
  of the arXiv version), and the abstracts of Saito–Zappala (2025), Inoue–Itaba (2026) and Dietzel–Feingesicht–Lebed
  (2025); none answers the question.
- **Not consulted directly** (bibliographic data verified on Crossref): Roos (1999), Fomin–Kirillov (1999),
  Milinski–Schneider (2000), Polishchuk–Positselski (2005), Priddy (1970), and the published journal versions. The facts
  about FK_3 used in the paper are recorded in Walton–Zhang and were checked by our computations; the facts about Koszul
  algebras are standard. The paper cites the global-dimension theorem of JKVA19 as Theorem 4.6 of arXiv:1812.02026v2;
  Colazzo–Jespers–Kubat–Van Antwerpen cite it as "[arXiv version, Theorem 4.6] (see also [journal version,
  Theorem 4.5])", so the journal numbers the corresponding characterisation 4.5; whether the journal statement already
  contains the global-dimension condition was not checked (the journal version was not seen).
- **Searches (anonymous, logged).**
  - Finder (2026-09-30): arXiv API (quantum symmetrizer with Hochschild or Yang–Baxter; authors Farinati, García
    Galofre, Lebed; structure monoid with Koszul or Nichols; braided homology; Fomin–Kirillov with Koszul or quandle;
    permutation racks), Crossref, OpenAlex (works citing FGG16; two queries rate-limited), zbMATH (works citing FGG16 and
    Lebed17), one web search.
  - Verification run (2026-09-30): arXiv API author lists and 15 keyword queries; Semantic Scholar citations of FGG16
    (15), Lebed17 (24), the plactic paper (11) and the OWR report (3); one web search.
  - Author (this revision; 2026-10-01 local time, 2026-09-30 21:14–21:28 UTC): Crossref metadata for every DOI of the
    bibliography and arXiv API metadata for every cited preprint; arXiv recency queries (authors Lebed and Farinati,
    "quantum symmetrizer"); no web search.
  - Verification run 2 (2026-10-01 local time, 2026-09-30 22:14–22:44 UTC; log in the audit folder): the sources above;
    Crossref metadata for all 15 DOIs of the bibliography (all confirmed) and two keyword searches; 12 arXiv API queries
    (quantum symmetrizer; authors Lebed, Farinati, García Galofre; Nichols + Yang–Baxter + Hochschild; structure monoid
    + Koszul; braided (co)homology; structure algebra + global dimension; permutation racks; Fomin–Kirillov + Koszul;
    and others); OpenAlex (lookups; the queries for citing works were rate-limited); Semantic Scholar citations of FGG16
    (15), Lebed17 (24) and the plactic paper (11); the zbMATH Open API; one web search.
  - No resolution of Question 44 or of Problem 2 beyond the involutive and idempotent cases was found, and no statement
    of the implication for finite-dimensional B. This negative search is not a proof of priority.
- **Scope.** The negative answer is proved in the text for Z/4 and for the permutation racks of Theorem 1.4, and follows
  from published results for finite-dimensional B (Corollary 1.5). Computational, with exact arithmetic: Table 1 beyond
  the multidegrees treated in the text, the full homology in weights ≤ 4, Remark 3.3, Table 2, the R_3 homology, and the census.
  Open: which solutions satisfy Question 44 (by Proposition 2.3, those with A Koszul and B = K); the group-algebra
  variant (FGG16, Remark 33); the involutive case in positive characteristic; other quotients for the constant
  character.

## Suggested corpus update (not performed)
Change OWR-17294-017 from `open` to `solved` (negative answer) and add a note citing this preprint:

> Negative answer. For the permutation rack X = Z/4, x ◁ y = x+1, the complex A ⊗ B ⊗ A (the image of the
> Farinati–García Galofre comparison map, B the Nichols algebra of −σ) has an explicit non-trivial 2-cycle in weight 3,
> so it is not a resolution (their Question 44); with coefficients on which X acts by 0, the weight-3 part of HH_3 has
> dimension 49 (over Q) while the quantum symmetriser reaches only 47 dimensions, for every quotient of the braided
> complex. In characteristic zero, A ⊗ B ⊗ A is also not a resolution for any finite permutation rack with a cycle of
> length divisible by 3 or 4. For non-involutive non-degenerate
> solutions with finite-dimensional Nichols algebra (e.g. the dihedral quandle R_3) the failure follows from a remark
> of Farinati–García Galofre and Jespers–Kubat–Van Antwerpen (TAMS 2019). Positive results remain for involutive
> (characteristic 0) and idempotent solutions.

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