# Verification report — AMR-103-0120 (Guadagnini–Pilo conjecture: quantum invariants and the fundamental group)

Verification date: 2026-10-04. Paper: "A Conjecture of Guadagnini and Pilo on Quantum Invariants and the
Fundamental Group: Counterexamples and a Classification by Gauge Group" (27 pages).

**Verdict.** The conjecture is false as stated. For G = SU(4), SU(5) and Spin(7) there are lens spaces with the
same fundamental group whose quantum invariants are nonzero and have different absolute values; this is proved by
exact computation from the Reshetikhin–Turaev surgery formula. Using the published lens space formula of Hansen
and Takata (Theorem 5.1 of their paper, not reproved), the paper also shows: among the simply connected compact
simple groups, the conjecture holds exactly for SU(2), SU(3), Sp(2n), G_2, Spin(8) and F_4, and fails for SU(N)
with N ≥ 4, Spin(m) with m = 7 or m ≥ 9, E_6, E_7 and E_8. The projective versions of the invariant are not
decided. The note is unrefereed.

## Statement checked
- **Primary source.** E. Guadagnini, L. Pilo, "Three-manifold invariants and their relation with the fundamental
  group", Comm. Math. Phys. 192 (1998) 47–65, doi:10.1007/s002200050290; the arXiv version hep-th/9612090 was
  read (the journal version was not available).
  - The invariant is defined through Chern–Simons theory with a simple compact gauge group. The conjecture: for
    non-vanishing invariant, its absolute value depends only on the fundamental group.
  - Section 4 proves this for lens spaces and SU(2). Section 5 reports numerical agreement for SU(3), for some
    lens spaces with p ≤ 20 and 3 ≤ k ≤ 50.
- **Problem list.** T. Ohtsuki (ed.), "Problems on invariants of knots and 3-manifolds", Geom. Topol. Monogr. 4
  (2002) 377–572, Conjecture 7.5 (Section 7.1; arXiv:math/0406190 read): for non-vanishing τ^G_r(M), the absolute
  value |τ^G_r(M)| depends only on π_1(M). There r = k + h^∨, and the neighbouring problems speak of simply
  connected compact simple Lie groups G and of the projective versions τ^{PG}_r.
- **Corpus record.** ulamai/UnsolvedMath, AMR-103-0120 (version 1.6.0, status `open`). Its statement agrees with
  Conjecture 7.5.
- **Which invariant.** G simply connected compact simple; the Reshetikhin–Turaev invariant of the modular category
  of Bakalov–Kirillov (Theorem 3.3.20) at q = exp(πi/(mκ)), κ = r = k + h^∨, whose simple objects correspond to
  all integrable highest weights of level k; normalization τ(S^3) = 1. For algebras that are not simply laced,
  the parameter called r by Hansen and Takata is mκ. For SU(3) the matrices printed by Guadagnini and Pilo were
  compared with this invariant (Remark 3.4); the values printed in their tables are reproduced.

## Readings
| Reading | Answer | Where |
|---|---|---|
| the conjecture as stated, for an arbitrary gauge group | false | Theorem 1.1: SU(4), r = 12, L(8,1) and L(8,3); SU(5), r = 10, L(5,1) and L(5,2); Spin(7), r = 15, L(10,1) and L(10,3) |
| the same for lens spaces only (the case treated in the source) | false, by the same examples | Theorem 1.1 |
| for which simply connected compact simple G it holds | exactly SU(2), SU(3), Sp(2n), G_2, Spin(8), F_4 (uses the formula of Hansen and Takata) | Theorem 1.2, Corollary 1.3 |
| SU(2), all closed oriented 3-manifolds (the source proves the lens-space case) | true | Theorem 6.3 with n = 1, Proposition 7.1 |
| SU(3) (numerical evidence in the source) | true (uses the formula of Hansen and Takata) | Theorem 6.1 |
| restricted to homotopy equivalent manifolds | still false: SU(7), r = 14, L(7,1) and L(7,2) | Remark 8.2 (Theorem 5.1; Table 1: floating point and a finite-field certificate) |
| projective versions (SO(3), PSU(N), …) | not decided | Remark 8.4 |

## Results in the paper
- **Theorem 1.1 / Theorem 3.1** (independent of Hansen–Takata). Exact identities in a cyclotomic ring for
  R′(a) = (S′T′^{a_1}S′ ⋯ T′^{a_n}S′)_{ρρ}:
  - SU(4), r = 12: |τ(L(8,1))| = 16 + 8√3, |τ(L(8,3))| = 56 + 32√3;
  - SU(5), r = 10: |τ(L(5,1))|² = 3475 + 1550√5, |τ(L(5,2))| = 45 + 20√5;
  - Spin(7), r = 15: |τ(L(10,1))| = 10√10·cos(π/12)/Π, |τ(L(10,3))| = 10√10·sin(π/12)/Π, ratio 2 − √3.
- **Proposition 2.1.** The modulus of the surgery formula for a chain of unknots with continued fraction ±p/q is
  |τ(L(p,q))|; it does not depend on the sign and orientation conventions. Remark 2.2 (slam-dunk moves) and
  Remark 3.2 (linking forms) identify the chains with the lens spaces.
- **Section 4** (lattice Gauss sums; everything about τ uses Hansen–Takata, Theorem 5.1, and is tagged "uses
  Theorem 2.3"): Lemma 4.1 (the Gauss sum G_q(y) is zero or has modulus p^{l/2}|R|^{1/2}, with support y_0 + Λ_0);
  Proposition 4.2 (only a coset W_q of the stabilizer H contributes; the phases Ψ_q(w) do not depend on choices);
  Corollary 4.3 (the case p | κ); Lemma 4.4 (orientation); Proposition 4.5 (hypothesis (K)); Proposition 4.6
  (non-degenerate case, complete proof); Corollary 4.7 (transfer by a Galois automorphism).
- **Section 5** (failures): Theorem 5.1 (SU(N), N = ap, p | κ); Proposition 5.2 (SU(5), every κ ≥ 10 divisible by
  5); Corollary 5.3 (effective: every multiple κ ≥ Np^5 of p); Proposition 5.4 (SU(4), p = 8, κ ≡ 4 mod 8, ratio
  cot(π/κ)); Theorem 5.5 (every N ≥ 4; N = 6, 8, 12 computer-assisted); Proposition 5.6 (Spin(7), p = 10, κ ≡ 5 mod
  10, ratio tan(5π/(4κ))); Propositions 5.7, 5.8 (Spin(2n+1), n ≥ 4; Spin(2n), n ≥ 5); Proposition 5.9 (E_6, E_7,
  E_8, computer-assisted, with the sums for E_6 and E_7 printed).
- **Section 6** (the statement holds): Theorem 6.1 (SU(3), G_2, Spin(8), F_4), with the computer-assisted Lemma
  6.2 (Laurent-polynomial identities; the tables for A_2 and G_2 are printed in the paper, those for D_4 and F_4
  are in the package); Theorem 6.3 (Sp(2n), every n ≥ 1; a determinant that splits into Vandermonde blocks).
- **Section 7.** Proposition 7.1: for every G and r, the conjecture is equivalent to its lens-space case
  (Turaev, formula (II.2.3.a) and Theorem II.5.4; Kirillov Jr., Theorem 4.2; Aschenbrenner–Friedl–Wilton,
  Theorem 2.3 of the arXiv version, which rests on geometrization).
- **Remark 8.1.** Two closed formulas of Theorem 2 of Guadagnini and Pilo (arXiv version) are inaccurate (p = 4
  with r ≡ 3 mod 4; gcd(r, p) = 2 with p/2 odd, p ≥ 6, r ≥ 4). The correct values do not depend on q, so their
  conclusion is not affected.

## Computations (scripts and outputs in reproducibility/)
- **Exact (integer arithmetic).**
  - Theorem 3.1: `author/scripts/verify_counterexamples.py`, `closed_forms_exact.py`; independently
    `independent_run_neg/vexact_main.py`, `vexact2.py` (root systems from Euclidean coordinates; 8 to 20 minutes
    per group) and `independent_run_2/r2_exact_thm11.py`, `r2_exact_modular.py` (explicit coordinates; 64-bit
    integers as long as an a priori bound allows it; seconds to one minute per group). All three check that
    formulas (4) and (6) agree exactly in the six cases; the two runs check S′² = ±c²C, (S′T′)³ = scalar·C, the
    value 1 on S^3, and further chains, among them (7,−1), (2,2,2,2,2,2,2) for L(8,1) and (3,4,1), (3,2,−1),
    (4,1,4) for L(8,3).
  - Theorem 5.5 for N = 6, 8, 12: `exceptional_N.py`; independently `vexact2.py` and `r2_exact_misc.py`.
  - Proposition 5.9: `theorem1_exact.py`, `exceptional_terms.py`; independently `r2_orbits_E.py` (enumeration of
    the orbits Wρ; the printed sums for E_6 and E_7 are reproduced term by term; for E_8 the orbit of ρ modulo 5L
    has 48384 points, which gives |H| = 14400).
  - Lemma 6.2: `laurent_tables.py` (all sets and polynomials; the output gives the number of elements of each
    set, every P_1, and every P_q or its relation to P_1), `criterion_tables.py`; independently
    `independent_run_pos/tablesC.py` and `independent_run_2/r2_laurent.py` (agreement with the output of
    `laurent_tables.py` in all 126 cases (Λ, q); Table 2 reproduced).
  - Rows F of Table 1 (finite-field certificates, Lemma 3.3): `sun_modular_det.py` (SU(6), SU(7), SU(8)),
    `e8_level5.py`; self-test `certificate_selftest.py`; independently `r2_sun_det.py` and `r2_coset_surgery.py`
    (two primes each; for E_8 with an element of order 4κ).
  - The pair L(64,9), L(64,25) for sl_4 mentioned by Hansen and Takata: both absolute values are 2√2 at κ = 6
    (`lens64_check.py`, `r2_exact_thm11.py lens64`; exact).
- **Floating point.** Table 1 from the surgery formula: `vsun.py`, `vsurg.py`, `vcoset.py` (first verification
  run), `r2_sun_det.py`, `r2_table1.py`, `r2_coset_surgery.py` (third run; all twelve rows), the author's programs
  for the rows F, `minimal_check.py` and `independent_check_short/lead_sun.py` for SU(4), SU(5), SU(7). Remarks
  3.4 and 8.1: `gp_su3_check.py`, `su2_gp_check.py`; `r2_gp_su3.py`, `r2_su2_gp.py`.
- **Tests** (no statement depends on them).
  - Formula (6) against the surgery formula: 15549 + 32850 + 586073 + 91772 triples, fifteen types of rank ≤ 5
    including B_2, B_3, B_4, C_3, C_4, C_5, G_2, F_4; largest relative deviation below 2·10^-10.
  - The closed formulas as printed (`printed_formulas_check.py`, `r2_closed_forms.py`, `r2_sp2n.py`) and all
    intermediate statements against brute force.
  - Scans: the author's (formula of Hansen and Takata) and those of the second verification run (surgery
    formula; about 2.2·10^8 triples in floating point and 3·10^7 modulo two primes) found no pair with two
    different nonzero moduli for the groups of Theorem 1.2(a). Floating-point thresholds are unreliable when
    S_ρρ is very small (F_4 at κ ≥ 21: redone in modular arithmetic).
- **Reruns for this version (2026-10-04).** All of the author's programs, all programs of the first verification
  run, and all programs of the second run except its two long scans were run again from an extracted copy of the
  source archive; the outputs agree with the saved ones apart from running times.

## Independent adversarial audit
Three independent verification runs were made, all AI-assisted (2026-10-04). The first two checked the author's
working notes, one part each; the paper was written afterwards and follows their reports. The third run checked
the complete text of the paper. No run found an error in a statement. The table summarizes their findings.

| Item | First run (counterexamples, Sections 3–5 of the notes) | Second run (Sections 6–7 of the notes) | Third run (complete paper) |
|---|---|---|---|
| Statement fidelity | CONFIRMED (sources read: Guadagnini–Pilo, Ohtsuki's list, Hansen–Takata) | CONFIRMED (transcription of Hansen–Takata, Theorem 5.1 checked against the source) | CONFIRMED (TeX sources of Ohtsuki's list, Guadagnini–Pilo, Hansen–Takata, Kirillov Jr., Aschenbrenner–Friedl–Wilton fetched again and compared; tables of Guadagnini–Pilo for SU(3) reproduced) |
| Proofs | CONFIRMED; one sketch (Proposition 4.6) completed; effective bound and Spin(7) family supplied | CONFIRMED; one sketch (Proposition 4.6) completed; references of Proposition 7.1 checked and made precise | CONFIRMED; all proofs re-derived, including the parts written after the first two runs |
| Computations | CONFIRMED (Theorem 3.1 reproduced exactly with own code; Table 1 reproduced from the surgery formula) | CONFIRMED (tables of Lemma 6.2 reproduced with own code; no counterexample found for the groups of Theorem 1.2(a)) | CONFIRMED (Theorem 3.1, Theorem 5.5 for N = 6, 8, 12, Proposition 5.9, Lemma 6.2, Table 2 reproduced exactly with own code; Table 1 reproduced, rows F also in finite fields; package rerun) |
| Answer as posed | CONFIRMED (false for general G) | CONFIRMED (true for the six families, assuming Hansen–Takata) | CONFIRMED |
| Novelty | CONFIRMED as far as can be checked (no priority claim) | CONFIRMED as far as can be checked (no priority claim) | CONFIRMED_WITH_FIXES (a preprint of Kuriya on a perturbative version and a paper of Sokolov on SU(2) had to be mentioned) |
| Presentation | CONFIRMED_WITH_FIXES | CONFIRMED_WITH_FIXES | CONFIRMED_WITH_FIXES |

Required fixes of the first two runs, all applied in the paper:
1. The invariant is specified in Theorem 1.1 and Section 2.2 (simply connected G, all integrable weights of level
   k, r = k + h^∨; relation to the parameter mκ of Hansen and Takata); the projective versions are stated as
   not decided (Remark 8.4).
2. What is independent of Hansen–Takata (Theorem 1.1, the floating-point values of Table 1) is separated from
   what uses it (Theorem 1.2 and Corollary 1.3 say so in their statements; the statements on τ in Sections 4
   to 6 carry the tag "uses Theorem 2.3"; Section 4 states it at the beginning).
3. The SU(N) theorem has an explicit level (Corollary 5.3: κ ≥ Np^5).
4. Spin(7): exact ratio 2 − √3 (Theorem 1.1) and the family of Proposition 5.6 with proof.
5. Proposition 4.6 has a complete proof.
6. Proposition 4.2 states and proves that Ψ_q(w) does not depend on the choices.
7. Step 3 of the theorem on SU(3), G_2, Spin(8), F_4 is the computer-assisted Lemma 6.2, in the form of
   Laurent-polynomial identities; tables for A_2 and G_2 in the text (Table 2), complete tables for D_4 and F_4 in
   the package (`laurent_tables.py` and its output).
8. Theorem 6.3: units q are represented by positive integers; the integer v is chosen for the pair (q_1, q_2);
   the use of negative indices in Step 4 is pointed out.
9. The chains are identified with the lens spaces by an argument (Proposition 2.1, Remarks 2.2 and 3.2), and the
   orientation convention is fixed once (Section 2.3).
10. Scans are labelled as tests, with the caveat on floating-point thresholds; the determinant scan for Sp(2n),
    n ≤ 12, is no longer quoted in the paper.
11. The sentence about earlier tests for SU(3) and SU(4) was removed.
12. The finite-field certificates are described precisely (Lemma 3.3).
13. Proposition 7.1 quotes the precise statements used (Turaev (II.2.3.a) and Theorem II.5.4, Bakalov–Kirillov
    Remark 4.1.14, Kirillov Jr. Theorem 4.2, Aschenbrenner–Friedl–Wilton Theorem 2.3).
14. Credits: Guadagnini–Pilo, Ohtsuki's list, Jeffrey, Yamada, Li–Li, Takata, Kohno–Takata, Zhang–Carey, Zhang,
    Hansen–Takata (Theorem 5.1, Proposition 5.2), Kac–Peterson, Reshetikhin–Turaev, Kirby–Melvin, Turaev,
    Bakalov–Kirillov, Kirillov Jr., Wenzl, Aschenbrenner–Friedl–Wilton; the paragraph "Scope and priority" says
    which of them were not read.

Required fixes of the third run, all applied in the paper:
1. A preprint of T. Kuriya, "The LMO invariant and Guadagnini–Pilo's conjecture for lens spaces" (cited in
   arXiv:0803.1732; not obtained; reported to treat a version of the conjecture for perturbative invariants), is
   mentioned in the Introduction and in "Scope and priority"; the statements on the literature search were made
   precise.
2. M. V. Sokolov, Math. Notes 61 (1997), is cited next to Yamada (1995); both are known from zbMATH summaries.
3. The abstract states the dependence on the formula of Hansen and Takata where the classification is first
   mentioned.
4. The tag "uses Theorem 2.3" was added to Propositions 4.2, 4.5, 4.6, 5.2, 5.4, 5.6, 5.7, 5.8 and Corollaries
   4.3, 4.7, 5.3.
5. Lemma 3.3: for E_8 the finite-field computation uses an element of order 4κ = 2M; the text says so.
6. Table 2: the caption speaks of the nonzero vectors in L.
7. Remark 8.1(b): the two expressions agree for r = 2 and differ for r ≥ 4.
8. The sentence on Remark 5.3(a) of Hansen–Takata says that their statement is about the invariants themselves
   and that only the absolute values are considered here.
9. Proof of Proposition 5.9: wording; the order of H is confirmed by counting the orbit of ρ modulo 5L.
10. The Verification paragraph and Section 9 record the third run; the sentence on parts not checked by an
    independent run was removed.
11. Small points: the convention 1/0 = ∞ for continued fractions; Remark 8.2 notes that the separation of L(7,1)
    and L(7,2) does not depend on Hansen–Takata; the Sources paragraph names the zbMATH summaries.
12. Package: folder `independent_run_2` with a README; README, this report and the metadata updated.

## Relation to the literature, novelty and scope
- **Searches (October 2026).** arXiv, Crossref, OpenAlex and zbMATH; the works citing Guadagnini–Pilo (five in
  OpenAlex: Ionicioiu–Williams 1998, Williams 2000, Ohtsuki's list, Hikami 2006, Guadagnini–Mancarella 2010) and
  Hansen–Takata (45 and 19 records); web searches. All requests were anonymous.
  - No counterexample to the conjecture and no proof of the lens-space statement for a group other than SU(2)
    was found. The theses of Pilo (1997) and Oriti (2003) describe the general case as open.
  - A preprint of T. Kuriya with the title "The LMO invariant and Guadagnini–Pilo's conjecture for lens spaces"
    is cited in his arXiv paper 0803.1732. It could not be obtained and is in none of the four databases.
    According to a web-search summary it treats a version of the conjecture for the perturbative invariants
    τ^{PG} of lens spaces. Its precise statements are not known to us; Theorem 1.1 is not affected.
  - Yamada (1995) and Sokolov (1997) treat the absolute values (Turaev–Viro invariants) of lens spaces for
    SU(2); they were not read (zbMATH summaries).
  - Hansen and Takata announce (Remark 5.3 of their paper) a paper with detailed calculations of the Gauss sums;
    it was not found. The same remark says that the sl_4 invariant with κ = 6 distinguishes L(64,9) and L(64,25);
    the absolute values of these two invariants are equal (2√2, exact computation), so this pair is not a
    counterexample.
- **Caveats.** Not read: Jeffrey 1992, Yamada 1995, Sokolov 1997, Li–Li 1996, Takata 1996, Kohno–Takata 1993,
  Zhang–Carey 1996 (known from titles, abstracts, summaries and citations); the journal version of
  Guadagnini–Pilo; Kuriya's preprint. Read in arXiv or online versions: Hansen–Takata (Sections 2, 4, 5),
  Bakalov–Kirillov (Section 3.3, Remark 4.1.14), Kirillov Jr. (Section 4), Aschenbrenner–Friedl–Wilton
  (Section 2.1), Turaev's book (Sections II.2, II.5). Once the formula of Hansen and Takata is available, the
  mechanism is elementary and may be known to specialists. No specialist was consulted. This negative search is
  not a proof of priority, and none is claimed.
- **Scope.** Theorem 1.1 is unconditional (exact computation). The classification (Theorem 1.2, Corollary 1.3)
  depends on Theorem 5.1 of Hansen and Takata, which was not reproved; it was compared with the surgery formula
  exactly in eight cases and numerically in more than 700000 cases. Open: the projective versions of the
  invariant; the complete list of pairs (p, r) at which the statement fails for the groups of Theorem 1.2(b).

## 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/