# Verification report — OWR-8415347-019 (Freslon, with Franz and Skalski: residual nilpotence of the augmentation ideal; the final problem on O_N^+)

Verification date: 2026-10-10 (version 1.0). Version 1.1 of the same day is described in the section "Version 1.1" below.

**Verdict.** The note answers the **final problem** of the source for every N ≥ 2 and determines the nilpotent
residual of O(O_N^+) for every N ≥ 2; every statement it makes is proved. The scope is as follows.
- **Settled.**
  - N = 2, 3: no group without elements of order two (in particular no torsion-free group, hence no residually
    torsion-free nilpotent group) has a dual, however embedded in O_N^+, which together with SO_N topologically
    generates O_N^+ (Theorem 1.1).
  - N ≥ 4: O_N^+ = ⟨SO_N, dual of F_2⟩, with the free group F_2 embedded by two rotation blocks (Theorems 1.2 and
    1.3). F_2 is residually torsion-free nilpotent (Magnus). Hence the answer to the final problem is yes exactly
    for N ≥ 4 (Corollary 1.4).
  - The nilpotent residual I^∞ of O(O_N^+) is zero for N ≥ 4 (O_N^+ is strongly connected) and is the kernel of
    O(O_N^+) → O(SO_N) for N = 2, 3 (Corollary 1.5). This answers the "main question" of the source.
- **Known before.** The case N = 2 of the negative answer and of the residual (Franz, Freslon and Skalski 2023,
  Lemma 4.7 with Proposition 4.5). Proposition 3.1 on group duals in O_N^+ follows from known results
  (Woronowicz; Banica, Bhowmick and De Commer; Banica and Patri). The statements of Lemmas 4.8 and 4.9 are cases
  of theorems of Chirvasitu (2020: Theorem 3.4 for SU_2, Theorem 3.1 for N = 2); the proofs in the note are new.
- **Relies on a published theorem.** For N ≥ 5 the proof is an induction resting on Theorem 4.1(1) of Brannan,
  Collins and Vergnioux (2017); its proof is recalled in Appendix A of the note. For N = 4 the proof needs only
  standard facts (representations of sl_2, basic facts on CQG algebras).
- **Open.** The general question of the source (for which Hopf algebras the nilpotent residual vanishes) is a
  programme and is not addressed. The Gaussian part of O_N^+ is not determined for N ≥ 4. Which torsion-free
  quotients of F_2 can replace F_2 is not known (Question 7.1; for the Heisenberg group there is only a test).

The note is unrefereed. Two independent verification runs examined the first written version of the results,
and a third one examined the final text of the note; all three were AI-assisted.

## Statement checked
- **Primary source.** A. Freslon (joint work with U. Franz and A. Skalski), "Residual nilpotence of the
  augmentation ideal", problem contribution in: Quantum Groups – Algebra, Analysis and Category Theory,
  Oberwolfach Report No. 44/2021, Oberwolfach Rep. 18 (2021), no. 3, 2397–2458, doi:10.4171/OWR/2021/44; the
  problem is on pp. 2452–2453.
  - Read in the publisher's PDF (two independent text extractions of the two pages agree). The third
    verification run fetched the file again (576,713 bytes, sha256
    `d848e0dc511e4b5f25aa161784583cb9fd59eae679dd208545718bb56c42079e`, identical to the copy used before) and read
    the two pages in a third extraction.
  - Content: for a Hopf algebra A with augmentation ideal I, the nilpotent residual I^∞ is the intersection of
    the powers I^n. Three questions: (1) for which A is I^∞ = 0; (2) what is I^∞ for O(O_N^+); (3) is there a
    discrete group Γ, residually torsion-free nilpotent, such that O_N^+ is topologically generated by SO_N and
    the dual of Γ.
  - Facts recorded there: classical compact groups (I^∞ = 0 iff connected), group algebras (iff Γ residually
    torsion-free nilpotent), non-trivial idempotents (then I^∞ ≠ 0), and U_N^+ (I^∞ = 0, by topological generation
    by U_N and the dual of F_N, with a reference to Chirvasitu 2020).
  - N is not quantified, no embedding of the dual of Γ is prescribed, and "topologically generated" refers to
    Chirvasitu's paper.
- **Corpus record.** ulamai/UnsolvedMath, OWR-8415347-019 (dataset version 1.6.0; upstream status
  `partially_solved`). The statement of the record is a damaged extraction of the printed page (the hat on Γ and
  the symbol O_N^+ are lost, the sentences are out of order). The source decides; the record's assessment says
  that the final existence question was unresolved.
- **Conventions.** Compact quantum groups through CQG algebras; quantum subgroups = surjective Hopf
  *-homomorphisms (Chirvasitu 2020, Franz–Freslon–Skalski 2023); topological generation = Chirvasitu's
  Definition 0.1 (no proper quotient Hopf *-algebra through which the given quotients factor), which is the
  notion used in the source and in Franz–Freslon–Skalski, Proposition 4.5.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Final problem "for a given N" | no for N = 2, 3; yes for N ≥ 4 | Corollary 1.4 |
| Final problem "for all N at once" | no (it fails for N = 2, 3); one group, F_2, serves all N ≥ 4 | Corollary 1.4 |
| The dual of Γ may be embedded in any way | the negative result covers every surjective Hopf *-homomorphism O(O_N^+) → C[Γ]; the positive result gives one explicit embedding | Theorem 1.1, equation (1) |
| A Hopf *-homomorphism O(O_N^+) → C[Γ] which is not surjective | its image is the group algebra of a finitely generated subgroup, which is again residually torsion-free nilpotent; the answer does not change | proof of Proposition 3.1 |
| Quantum subgroups and group duals defined through C*-algebras (universal, reduced, intermediate) | the same notion | Remark 2.4 |
| Another copy of SO_N in O_N^+ | there is none: every surjective Hopf *-homomorphism O(O_N^+) → O(SO_N) differs from the standard one by conjugation with an element of O_N and has the same kernel (this rests on a standard fact on the automorphisms of SO_N, quoted without proof) | Remark 2.5 |
| Topological generation in the sense of Brannan–Collins–Vergnioux (intertwiners, fixed vectors) | equivalent for O_N^+; both implications are proved | Propositions 2.2, 2.3 |
| "Torsion-free" instead of "residually torsion-free nilpotent" | the negative result holds for all groups without elements of order two | Theorem 1.1 |
| Main question (I^∞ of O(O_N^+)) | zero for N ≥ 4; kernel of the map to O(SO_N) for N = 2, 3 | Corollary 1.5 |
| General question (all Hopf algebras) | not treated | Section 7.3 |

## Results in the paper
- **Propositions 2.2, 2.3.** For O_N^+: equality of fixed vectors on all tensor powers of u implies topological
  generation (the direction used), and conversely.
- **Proposition 3.1.** A quantum subgroup of O_N^+ which is the dual of a group without elements of order two is
  given, in a real orthonormal basis, by k ≤ N/2 rotation blocks R(g_i) and ones; the g_i generate the group.
- **Theorem 1.1.** N = 2, 3: such a dual is a classical subgroup of SO_N; the generated quantum subgroup is SO_N.
- **Theorem 1.2.** N = 4: Fix_{SO_4}(u^{⊗m}) ∩ Fix_{dual F_2}(u^{⊗m}) = Fix_{O_4^+}(u^{⊗m}) for all m. Proof:
  the model C^4 = V ⊗ V' with so_4 = sl_2 ⊕ sl_2 (Lemma 4.2); invariants supported on one pairing (Lemma 4.5);
  the word of a pair of sign vectors in terms of block weights (Lemma 4.6); trivial words have trivial prefixes
  (Lemma 4.7); the vanishing lemma, by a lowest-weight argument (Lemma 4.8); the slice lemma (Lemma 4.9);
  Proposition 4.3. Only the elementary inclusion TL_4(m) ⊂ Fix_{O_4^+} is used; Banica's description of the fixed
  vectors of O_4^+ is a by-product (Corollary 4.4).
- **Theorem 1.3.** N ≥ 4: the same equality and O_N^+ = ⟨SO_N, dual F_2⟩, also for F_k with 2 ≤ k ≤ N/2. Proof by
  induction from N = 4 with Theorem 5.1 (Brannan–Collins–Vergnioux), whose proof is recalled in Appendix A.
- **Lemma 5.2 (Magnus).** The Magnus map of C[F_2] is injective; the powers of the augmentation ideal of C[F_2]
  intersect in zero; F_2 is residually torsion-free nilpotent. Classical; short proofs included.
- **Corollary 1.4.** The answer to the final problem.
- **Lemma 6.1.** For a bialgebra in characteristic zero: if [K, K] ⊂ K_3, then A/K_∞ is commutative.
- **Corollary 1.5.** The nilpotent residual of O(O_N^+). **Corollary 6.3.** Gauss(O_N^+) = SO_N for N = 2, 3.
- **Remarks 4.11, 4.12.** Relation to the theorems of Brannan–Collins–Vergnioux and of Chirvasitu; a remark on a
  step in the proofs of two statements in the arXiv versions of two papers of Chirvasitu, with an explicit
  element of the kernel of the map considered there (for three tori) and exact computations.
- **Question 7.1.** Whether the dual of the Heisenberg group H_3 can replace the dual of F_2 (tested for m ≤ 12
  only; no claim).

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation. The programs test the statements for small tensor powers and parameters.
- **Programs with which the results were first obtained** (`original/`, numpy; about four minutes): the equality
  of Theorem 1.2 directly in C^4 for m ≤ 8 (exact) and that of Theorem 1.3 for N = 5 (m ≤ 8) and N = 6 (even
  m ≤ 8) (tests); Proposition 4.3 for m ≤ 10; Lemma 4.9 for all 2^m sign vectors, m ≤ 12; Lemma 4.5 for all
  noncrossing pairings, m ≤ 12; Lemma 4.6 for all pairs, m ≤ 9; Lemma 4.7 on 637,420 words; Lemma 4.8 on 42,884
  configurations, with controls; the Heisenberg test for m ≤ 10.
- **Verification run A** (`verification_run_A/`): Theorem 1.2 directly in C^4 by an exact and a floating-point
  method (m ≤ 8) and by invariant theory (m ≤ 10); Lemma 4.9 for all sign vectors, m ≤ 12; Proposition 4.3 with
  all equations, m ≤ 10; Lemmas 4.6, 4.7, 4.8 on 87,380 pairs, 3,368,420 words and 299,202 configurations;
  Theorem 1.3 for N = 5, 6, 7 and m ≤ 8 (for N = 5, 6 by two unrelated methods), with a control (F_2 replaced
  by Z^2); Theorem 5.1 for N = 4, 5, 6 (and N = 3, outside the theorem) and k ≤ 8.
- **Verification run B** (`verification_run_B/`): Theorems 1.2/1.3 for N = 4 (m ≤ 12), N = 5 (m ≤ 10), N = 6
  (m ≤ 8, also for F_3), N = 7 (m ≤ 7), for N ≤ 6 and smaller m also by a second method; one rotation block for
  N = 2, 3 (the intersection is all of Fix_{SO_N}); Lemmas 4.7 and 4.8 on 11,068 trivial words and 171,607
  configurations; the fixed-vector form of U_2^+ = ⟨U_2, dual F_2⟩ for all colour words of length ≤ 12; the
  quotients O(O_N^+)/K_n (for N = 3: O(O_3^+)/K_n = O(SO_3)/K_n for all n ≤ 10, exact); the two exact
  computations of Remark 4.12 (142 of 144) and floating-point experiments on larger representations; the
  Heisenberg test for m ≤ 12 (exact for these m).
- **Program written with the note** (`writing_stage/check_magnus.py`, standard library, about 40 seconds):
  Lemma 5.2(a) for the 485 elements of F_2 of word length ≤ 5 (their images under the Magnus map are linearly
  independent; exact rank over Q of the truncated series).
- **Verification run 3** (`independent_run_2/`, numpy, about five minutes; programs written before the other
  parts of the package were opened):
  - Theorem 1.2 directly in C^4, from the equations of the Lie algebra of SO_4 on the basis tensors with trivial
    word: the intersection is exactly TL_4(m) for m ≤ 8 (dimensions 1, 0, 1, 0, 2, 0, 5, 0, 14). Theorem 1.3 in
    the same way: exact for N = 5 (m ≤ 7), N = 6 (m ≤ 6), N = 7 (m ≤ 5). Control with Z^2 in place of F_2:
    dimensions 4 and 25 for m = 4, 6. The test after Question 7.1 (Heisenberg group): exact for m ≤ 8.
  - Lemma 4.9 for all 8,191 sign vectors of length ≤ 12; Lemma 4.6(c) for all 22,369,621 pairs (s, t) of these
    lengths; Lemma 4.5 for all noncrossing pairings of at most 12 points.
  - Lemma 4.7 on 3,802 trivial words found among 2,289,966 enumerated words and on 300,000 random trivial words
    (up to 16 syllable pairs); Lemma 4.8 on 373,539 configurations (8,016 with a weight of trivial word); no
    violation; controls with a hypothesis dropped (23 words, 30 configurations violate the conclusion).
  - Remark 4.12(ii), with a criterion derived again: for the tori (z, x, z) and the representation
    V_1 ⊗ z ⊗ V_2 ⊗ z^{-1} ⊗ V_1 the products of the isotypic projections span a space of dimension 142 of 144
    (over Q; 1331 products), and 144 for (z, x, z, x); the same numbers with the group of order two. A second
    program computes the map from its definition on the 144 coefficients: rank 142, and the two elements
    ψ_+ z ψ_0 z^{-1} ψ_- − ψ_- z ψ_0 z^{-1} ψ_+ (the element of the remark) and its off-diagonal analogue are mapped
    to zero. For 20 representations of dimension up to 125 the span is everything with four, five and six tori
    alternating between T_z and T_x (ranks modulo a prime); with three tori the deficits 2, 21, 48, 148 recorded
    by run B in floating point are found again modulo a prime.
  - Appendix A: Lemma A.1 (n ≤ 4, m ≤ 8); for N = 4, 5, 6 and k ≤ 8, 7, 6: span{T^1_p} = span{A_p}, linear
    independence of the vectors x_p of Step 3 (and of the vectors y_{p,j} for k ≤ 7, 6, 6), and dimension
    |NC_2(k)| of the invariant vectors.
  - Lemma 5.2(a) for the same 485 elements (the truncation must reach degree 10); the expansion used in
    Remark 6.2; the properties of the group of triples of Section 7.1 (on a box); the coefficients of
    Example 4.10.
- **Re-runs.** On 2026-10-10, when the note was written, the programs listed in `reproducibility/run_quick.sh`
  (53 outputs, including seven slower ones) were run again, one process at a time. All outputs are identical to
  the recorded ones up to fields that record running times. Details: `reproducibility/README.md`,
  `reproducibility/RERUN_LOG.txt`. Verification run 3 repeated this from an extracted copy of the archive (53
  outputs reproduced, 0 different), and then ran the quick part once more with the thirteen steps of its own
  folder added (59 outputs reproduced, 0 different); the logs are in
  `reproducibility/independent_run_2/rerun_of_the_package/`.

## Independent verification runs
The results were first obtained with complete proofs. Three independent verification runs, all AI-assisted,
followed on 2026-10-10; each wrote its own programs. Runs A and B examined the first written version of the
results and did not use each other's report: run A the statement, Proposition 3.1 and Theorems 1.1–1.3 with the
cited results; run B Lemma 6.1, Corollary 1.5, the consistency with known results, the computations, the
remark on two proofs in the literature, and the relation to the literature. Run 3 examined the final text of
the note.

| Item | Run A | Run B | Run 3 (final text) |
|---|---|---|---|
| Statement and conventions against the source | CONFIRMED (source read; two non-blocking clarifications) | not its part | CONFIRMED (publisher's file read again; scope sentences and title checked) |
| Equivalence of the two notions of topological generation for O_N^+ | proved (Propositions 2.2, 2.3) | the direction needed for Corollary 1.5(a) proved again | CONFIRMED (Lemma 2.1, Propositions 2.2, 2.3, Remarks 2.4, 2.5 re-derived) |
| Proposition 3.1, Theorem 1.1 | CONFIRMED (re-derived; holds for groups without elements of order two) | computation for N = 3 consistent | CONFIRMED |
| Theorem 1.2: Lemmas 4.1, 4.2, 4.5, 4.6, 4.7, 4.8, 4.9, Proposition 4.3 | CONFIRMED, each lemma re-derived | tested by its own programs | CONFIRMED on the final text (each proof re-derived; tested by its own programs) |
| Theorem 1.3 | CONFIRMED (three steps; the cited theorem re-derived) | fixed-vector equality established by computation for the ranges listed above | CONFIRMED |
| Brannan–Collins–Vergnioux, Theorem 4.1(1): statement and hypotheses | CONFIRMED in the arXiv version; proof re-derived | not its part | Appendix A compared line by line with Section 4.1 of the arXiv version: CONFIRMED; no circularity |
| Lemma 5.2 (Magnus map) | injectivity on the group proved | quoted | CONFIRMED (the proof for the group algebra, written with the note, re-derived) |
| Lemma 6.1, Corollary 1.5, Remark 6.2, Corollary 6.3 | not its part | CONFIRMED (re-proved; N = 3 confirmed by an exact computation up to K_10) | CONFIRMED |
| Consistency with known results | no contradiction found | CONFIRMED (ten checks) | Section 7.1 CONFIRMED (the group of triples verified) |
| Relation to Chirvasitu's theorems; Remark 4.12 | not its part | CONFIRMED WITH CORRECTIONS (the remark was completed and made precise) | CONFIRMED WITH CORRECTIONS (description of the step compared with both arXiv texts; computations repeated; an explicit element of the kernel added) |
| Numbered citations, bibliography | cited statements checked | statements of Franz–Freslon–Skalski checked | every numbered citation compared with the arXiv sources; all 21 literature entries verified again |
| Novelty and credits | not its part | CONFIRMED WITH CORRECTIONS (credits added) | CONFIRMED (searches repeated; two attributions made exact) |

No run found a gap or a wrong statement in the results and their proofs. Run B found that the account of
the literature in the first written version was incomplete in one point (a preprint of Chirvasitu,
arXiv:1904.13190, had been listed among the works citing his paper as not relevant, whereas it states a general
theorem behind his Theorem 3.4), that the identification of Lemma 4.8 with a case of that theorem was missing,
and that credits were missing; see the corrections below.

**Clarifications recommended by run A**, all made in the note:
1. which direction of the fixed-vector criterion is used where, with direct proofs (Section 2.3);
2. quantum subgroups defined through O(G) or through C*-algebras (Remark 2.4);
3. other copies of SO_N (Remark 2.5);
4. the factorisation through O(SO_N) in the proof of Theorem 1.1;
5. Theorem 1.2 does not depend on Banica's theorem (Section 4.2, Corollary 4.4);
6. Proposition 3.1 and Theorem 1.1 are stated for groups without elements of order two.

**Corrections required by run B**, all made in the note:
1. Lemma 4.8 is the fixed-vector form of the case G = SU_2 of Chirvasitu's Theorem 3.4, and Lemma 4.9 that of the
   case N = 2 of his Theorem 3.1; the statements are credited to him, the proofs are new; his second preprint
   (arXiv:1904.13190) is cited; the observation on the proofs is confined to Remark 4.12 and limited to what is
   certain (Remarks 4.11, 4.12).
2. Proposition 3.1 is presented as a specialisation of results of Woronowicz, of Banica, Bhowmick and De Commer
   and of Banica and Patri.
3. The embedding of the dual of F_2 in O_4^+ is credited to Franz, Freslon and Skalski (Section 1.2).
4. Theorem 4.2 of Brannan–Collins–Vergnioux is cited, and the structure of Theorem 1.2 through U_2^+ is stated
   (Remark 4.11).
5. Lemma 6.1: the characteristic of the field, the justification of the grading, the pointer to Milnor–Moore and
   Quillen.
6. Corollary 1.5(b): the invariance of K_∞ under the involution; Remark 6.2 on N ≥ 4.
7. Corollary 1.5(a): the direction of the criterion; the Magnus map instead of a reference (Lemma 5.2).
8. Section 7.1: an example of a strongly connected quantum group with disconnected classical version; correct
   wording of the example on quantum subgroups.
9. Section 7.2: the question on the Heisenberg group is attributed to Franz, Freslon and Skalski and stated as a
   question, with the tested range.
10. Corollary 6.3 on the Gaussian parts for N = 2, 3.
11. The description of what is new (Section 1.3) follows the findings of run B.
12. The sources not consulted are named ("Scope and priority").

**Added when the note was written, after runs A and B; examined by run 3:**
- the proof of Lemma 5.2(a) for the group algebra (linear independence of the images of the group elements under
  the Magnus map), which replaces a reference; it is tested by `writing_stage/check_magnus.py`. Run 3: re-derived
  line by line; correct.
- the presentation of the proof of Brannan–Collins–Vergnioux in Appendix A as an induction starting from
  Corollary 4.4 (run A had re-derived the proof with Banica's theorem as an input). Run 3: Lemma A.1,
  Proposition A.2 and Corollary A.3 agree with Lemma 4.5, Proposition 4.6 and the proof of Theorem 4.1 of the
  arXiv version; the linear independence of noncrossing pairing vectors is used once, in dimension N − 2 ≥ 2;
  Corollary 4.4 comes from Section 4 alone and Theorem 5.1 is used only for N ≥ 5, so the argument is not circular.
- the description of the group in Section 7.1 by triples of integers instead of a presentation. Run 3: group
  law, torsion-freeness, commutators (0, 0, 2n) and abelianisation Z^2 × Z_2 verified.
- Figure 1. Run 3: blocks, letters, block weights, exponents and the two compatible pairings verified.
- Remark 2.5 in its sharper form. Run 3: the quoted fact (every automorphism of the Lie group SO_N is the
  conjugation by an element of O_N) is the correct statement for every N ≥ 2, also for N = 2, 4 and 8; it
  remains quoted without proof.

**Corrections required by run 3**, all made in the note or in the package:
1. Remark 4.11(3): Banica's embedding is an embedding of O(U_2^+) into O(T) * O(SU_2).
2. Remark 4.12(i): the hypothesis G ≠ {e}.
3. Remark 4.12(ii), (iii): the explicit element of the kernel with its verification; the exact statement on four,
   five and six tori; the representation in the case of the group of order two; the two arXiv versions named.
4. Section 1.2: the copy of the dual of F_2 is implicit in the proof of Proposition 6.5 of Franz–Freslon–Skalski
   (which finds the dual of the Heisenberg group through O_2^+ * O_2^+).
5. After Proposition 3.1: Theorem 1.5 of Banica–Patri is attributed there to Woronowicz.
6. Appendix A: both changes with respect to the source are named (induction instead of Banica's theorem;
   Lemma A.1 instead of the quoted linear independence).
7. Remark 2.5: the quoted fact holds for every N ≥ 2, also for N = 8.
8. Section 7.1: the subalgebra of O(U_N^+) is the one generated by the products u_ij^* u_kl.
9. Section 8: the programs of run 3 and its re-run of the package.
10. The Verification paragraph: final state.
11. "Scope and priority": what run 3 read; its searches.
12. The package: the folder `independent_run_2/`; in the README the number of the equation of the arXiv PDF
    corresponding to the "map (8)" of the program names (it is equation (9) there), the wording of the notes, and
    the re-runs; this report.

## Relation to the literature, novelty and scope
- **Searches (10 October 2026).** When the results were first obtained: eight queries to the arXiv listing
  interface, the lists of works citing Franz–Freslon–Skalski 2023 and Chirvasitu 2020 in OpenAlex, three zbMATH
  queries, two web searches. Run A: two web searches. Run B: the arXiv listing interface was not available on
  that day (rate limit), and OpenAlex (keyword queries, citing works, recent works), zbMATH (five queries),
  Crossref and three web searches were used instead. Run 3: 21 queries to the arXiv listing interface (topics
  and authors), OpenAlex (works citing Franz–Freslon–Skalski 2023 and 2025, Chirvasitu 2020 and
  Brannan–Collins–Vergnioux; nine keyword queries), six zbMATH queries, two Crossref queries and one web search.
  - No answer to the final problem, no determination of the nilpotent residual of O(O_N^+) for N ≥ 3, and no
    statement that O_N^+ is topologically generated by SO_N and the dual of a torsion-free group was found.
  - Preprints not yet indexed by these services may have been missed.
- **What was read.** The two pages of the source; of Franz–Freslon–Skalski 2023 (arXiv v3) the introduction and
  Sections 3–6; of Franz–Freslon–Skalski 2025 (arXiv v2) the introduction and Section 4; Chirvasitu 2020 (arXiv
  v1, the only arXiv version) and Chirvasitu's preprint arXiv:1904.13190v1 (the only version) completely; of
  Brannan–Collins–Vergnioux (arXiv v2) Sections 2.3, 3.3, 4.1; of Banica–Patri (arXiv v3) Sections 1 and 6. All
  theorem numbers in the note are those of these arXiv versions. The bibliographic data of all 21 literature
  references were checked with Crossref, zbMATH or the arXiv listing, and once more by run 3.
- **Not consulted.** The published versions of Brannan–Collins–Vergnioux (the publisher's site refused
  automated access), of Chirvasitu 2020, and of the two papers of Franz, Freslon and Skalski. The other
  references were not read; what the note says about them is standard or taken from the papers read.
- **Caveats.**
  - Remark 4.12 concerns the arXiv versions of two texts; the published version of Chirvasitu 2020 was not
    compared. The remark does not put the theorems in doubt, and Lemmas 4.8 and 4.9 prove the cases needed. Its
    statement on three tori is proved in the remark by an explicit element and does not depend on a program.
  - Results of the literature which depend on U_N^+ = ⟨U_N, dual F_N⟩ are not used in the proofs.
  - The statement on the automorphisms of SO_N in Remark 2.5 is quoted as a standard fact, without proof.
  - This negative search is not a proof of priority, and no priority is claimed.
- **Scope.** The note answers the final problem of the source for every N ≥ 2 and determines the nilpotent
  residual of O(O_N^+) for every N ≥ 2. The general question of the source stays open, and the Gaussian part of
  O_N^+ is not determined for N ≥ 4.

## Version 1.1 (10 October 2026)

Version 1.1 changes the following and nothing else:
- one sentence in Remark 4.12(iii): "A. Chirvasitu has informed us (personal communication, October 2026) that he is
  aware of this error in the argument." The wording is the one he asked for;
- the paragraph "Version history" at the end of the note and the date line of the title page.

The theorems, the proofs, the programs and their recorded outputs are unchanged, so the verification described above
applies to version 1.1 as well. The PDF was rebuilt from the changed source with the same tool; the two changed pages
(16 and 23 of 26) were compared with version 1.0.

## 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/