# Verification report — OWR-3385-008 (Below–Krummeck–Richter-Gebert conjecture, Oberwolfach Report 2/2009, Problem 6)

Verification date: 2026-09-30.

**Verdict.** The conjecture is true. For every n ≥ 3, every field K and all points A_1, …, A_n, A'_1, …, A'_n in K²
with [A'_iA'_j] ≠ 0 for i ≠ j, the alternating sum S_n = Σ_{π_1 = 1} sgn(π) α(π(A) | π(A')) of cyclic determinant
quotients is 0. Equivalently, the bracket polynomial obtained by clearing denominators vanishes identically over Z,
hence over every commutative ring. The proof is complete, self-contained and valid in every characteristic; no step
depends on a computation. For n = 1 the quotient is 0/0 and for n = 2 the sum is ([A_1A_2]/[A'_1A'_2])², so the
statement concerns n ≥ 3, as in the sources. The note is unrefereed. This report was revised on the same day after a
second independent verification run and after a final readiness check (see "Independent verification").

## Statement checked
- **Problem list.** Oberwolfach Report No. 2/2009, *Discrete Differential Geometry* (organisers A. I. Bobenko,
  R. Kenyon, J. M. Sullivan, G. M. Ziegler), Oberwolfach Rep. 6 (2009), no. 1, pp. 75–144, doi:10.4171/OWR/2009/02.
  *Open problems in discrete differential geometry*, collected by G. Rote, pp. 106–110 (Problems 1–14); Problem 6
  (J. Richter-Gebert) is on p. 108.
  - The report PDF was fetched anonymously from EMS Press (70 pages, sha256 978d8638…3f13021) and page 108 was read
    as a rendered image. Rote's separate copy of the problem list has the same text.
  - The problem: 2n points in K² over a commutative field K; the sum over π = (1, π_2, …, π_n) ∈ S_n of
    σ(π) α(π(A_1, …, A_n) | π(A'_1, …, A'_n)) vanishes, where α is the cyclic quotient of 2 × 2 determinants
    ([A_1A_2]⋯[A_nA_1] over [A'_1A'_2]⋯[A'_nA'_1]) and σ(π) is the sign.
  - Status given there: trivial by symmetry for n = 3, 4, 7, 8, 11, 12, …; proved for n = 5 and 6; apparently true for
    n = 9 and 10.
- **Original source.** A. Below, V. Krummeck, J. Richter-Gebert, *Complex matroids phirotopes and their realizations in
  rank 2*, in: Discrete and Computational Geometry (Goodman–Pollack Festschrift), Algorithms Combin. 25, Springer 2003,
  pp. 203–233, doi:10.1007/978-3-642-55566-4_9. Read in full in the author version linked from Richter-Gebert's
  publication list at TU München (22 pages, sha256 93fb121f…87413bf1). Section 6, Conjecture 6.1 is the statement
  above. Details used in the paper:
  - π acts by π(A_1, …, A_n) = (A_{π(1)}, …, A_{π(n)}) (the printed definition has a small typo).
  - The authors clear the common denominator and obtain a multihomogeneous bracket polynomial.
  - The restriction π_1 = 1 is deliberate: without it the sum vanishes trivially for even n. Each cycle then appears
    twice, which makes the cases n ≡ 0, 3 (mod 4) trivial.
  - n = 5 is, up to relabelling and a factor 2, the five-point formula (8) of their Lemma 3.4 (realizability of
    uniform non-chirotopal rank-2 phirotopes on five points): they fix the point E instead of 1, and (8) runs over 12 of
    the 24 orderings, using the reversal symmetry. Their Theorem 5.2 proves n = 5 for arbitrary A, A' in C².
  - n = 6 was checked with Mathematica, and n ∈ {9, 10, 13, 14} numerically. They write that a general proof seemed out
    of reach.
- **Corpus record.** ulamai/UnsolvedMath, record OWR-3385-008 (record_id 30001137), status `open`.
  - In version 1.6.0 (commit c6e7f41), the version cited in the paper, the title field is the first sentence of the
    statement: "Let K be a commutative field and let A_1,…,A_n,A′_1,…,A′_n be points of K^2." The statement fields
    (`original`, `upstream` and `for_research`) are the OWR text. They define α and ask whether the alternating sum
    over π = (1, π_2, …, π_n) is zero "for every n". `statement.clean` is null.
  - In a later copy of the dataset (commit 372682f, saved 2026-09-29; the record was updated on 2026-09-16), the record
    is titled "Alternating Cyclic-Quotient Identities over Commutative Fields" and has a cleaned statement of the same
    question (statement status `corrected_verified`). The status is still `open`.
  - The answer to the question as posed is: yes for every n ≥ 3; n = 2 is false and n = 1 is undefined (Remark 1.5(a)).
  - The first version of this report gave the title of commit 372682f as the title of the record. This bullet replaces
    it.

## Readings
| Reading | Proved? | Where |
|---|---|---|
| sum over π with π_1 = 1, π(A) = (A_{π(1)}, …, A_{π(n)}), all [A'_iA'_j] ≠ 0 (the source) | yes, every n ≥ 3, every field | Theorem 1.2 |
| π(A) read as (A_{π⁻¹(1)}, …, A_{π⁻¹(n)}) | yes (same sum, since π ↦ π⁻¹ preserves π_1 = 1 and the sign) | Conventions, §1 |
| polynomial form: common denominator cleared (the authors' "bracket polynomial"), no condition on the points | yes, identically zero over Z, so over every commutative ring | Corollary 1.3 |
| the source's field C with A'_i = conjugate of A_i (squared phases, n-point phirotope formula) | yes, for realizable uniform rank-2 phirotopes; nothing claimed for non-realizable ones | Remark 1.6 |
| sum over all of S_n (no restriction π_1 = 1) | yes: n·S_n for odd n; trivially 0 for even n | follows from Theorem 1.2 and a cyclic shift |
| n = 1, 2 | not covered: undefined for n = 1, false for n = 2 | Remark 1.5(a) |

## Results in the paper
- **Theorem 1.2** (main). S_n = 0 for every n ≥ 3 and every field, under [A'_iA'_j] ≠ 0 (i ≠ j).
- **Corollary 1.3.** The polynomial P_n = D_n S_n (D_n = Π_{i<j}[A'_iA'_j], explicit formula (3)) is zero in
  Z[coordinates].
- **Theorem 1.4.** For traceless 2 × 2 matrices M_2, …, M_n and n ≥ 4:
  Σ_σ sgn(σ) PT(1σ) M_{σ_2}⋯M_{σ_n} = 0 as a matrix. For n = 3 the sum is PT(123) tr(M_2M_3) I. Hence the trace
  identity with any M_1 (n ≥ 4), or any traceless M_1 (n = 3). Theorem 1.2 is the case M_i = J A_i A_iᵀ, B_i = A'_i.
- **Remark 1.5.** n ≤ 2; the reversal symmetry and why n ≡ 0, 3 (mod 4) is trivial for any symmetric weights; in
  characteristic 2 the reversal pairing proves the identity for every n ≥ 3.
- **Remark 1.6.** The phirotope (squared-phase) form.
- **Lemma 2.1, Lemma 2.2.** Lie polynomials annihilate proper shuffles (coproduct argument); a linear form vanishing
  on proper multilinear shuffles is the coefficient function of the Lie polynomial Σ_Q φ(xQ) ℓ[xQ]. This is the
  multilinear case of Ree's theorem, proved here over every field.
- **Lemma 3.1** (shuffle identity for Parke–Taylor factors; known, equivalent to the Kleiss–Kuijf relations).
  Self-contained proof: GL_2 and rescaling normalisation to B_1 = (0,1), B_i = (1, t_i); the character identity
  R(u)R(v) = Σ_{u⧢v} R by induction and partial fractions; a degree argument in K(z).
- **Corollary 3.2.** Σ_σ PT(1σ)σ = Σ_Q PT(12Q) ℓ[2Q] (the Del Duca–Dixon–Maltoni form).
- **Proposition 4.1.** If the commutators of y_2, …, y_n commute with all y_d, then Σ_σ PT(1σ) y_{σ_2}⋯y_{σ_n} = 0 for
  n ≥ 4 (and = PT(123)[y_2, y_3] for n = 3).
- **Lemma 4.2** (numerator as a trace), **Lemma 4.3** (PQ + QP = tr(PQ) I for traceless P, Q over any commutative ring).
- **Section 5.** Proofs; with y_c = M_c ⊗ e_c in M_2(K) ⊗ Λ(e_2, …, e_n) the commutators are tr(M_bM_c) I ⊗ e_be_c,
  which are central. **Remark 5.1**: open-chain form with arbitrary end points X, Y (n ≥ 4). **Proposition 5.2**:
  Σ_σ sgn(σ|_T) PT(1σ) = 0 for every T ⊆ {2, …, n} (n ≥ 4); for n = 3 it fails exactly for T = {2, 3}.
  **Remark 5.3** (new in the second revision): Proposition 5.2 is not new in substance. For T ≠ {2, …, n} it follows
  directly from Lemma 3.1, because Σ_σ sgn(σ|_T) σ = (Σ_τ sgn(τ) τ) ⧢ (Σ_ρ ρ), or Σ_σ σ = x ⧢ Σ_ρ ρ when |T| ≤ 1. The
  case T = {2, …, n} says that the antisymmetriser lies in Sh_C; over fields of characteristic 0 this is equivalent to
  the classical fact that the sign representation does not occur in Lie_{n−1} for n − 1 ≥ 3 (Klyachko 1974; compare
  Kol–Shir 2014, Section 3.2, for the cyclic analogue).
- **Section 6.** The same proofs work for every linear form vanishing on Sh_C. With constant denominators the signed
  numerator sum is tr(M_1 s_{n−1}(M_2, …, M_n)) for the standard polynomial s_{n−1}; it vanishes for n ≥ 5 by the
  Amitsur–Levitzki theorem and for n = 4 because s_3 is scalar on traceless 2 × 2 matrices, so controls must use
  nonconstant denominators.

## Computations (exact or modular; scripts and outputs in reproducibility/)
- **Author** (`lead/check_bkrg.py`, standard library only, a few seconds; 140 checks, then ALL CHECKS PASSED).
  - S_n = 0 in exact rational arithmetic at random integer points for 3 ≤ n ≤ 11 (term by term for n ≤ 8, subset
    recursion for all these n); modulo 2^61 − 1 and 2^31 − 1 for 3 ≤ n ≤ 15 (three instances each); over F_3, F_5, F_7, F_11,
    F_13 for n ≤ min(p + 1, 12).
  - Nonzero controls: n = 2; the unsigned sum (3 ≤ n ≤ 14); for n ≡ 1, 2 (mod 4), n ≤ 14: random symmetric weights,
    bracket numerators over random antisymmetric denominators, random antisymmetric numerators over bracket
    denominators. For n ≡ 0, 3 (mod 4) these three vanish, as the reversal symmetry predicts.
  - Theorem 1.4 exactly for 3 ≤ n ≤ 9 (with a nonzero control for general matrices); Remark 5.1 for n ≤ 8;
    Proposition 5.2 for all T, n ≤ 8; Lemma 3.1 (all 480 pairs (u, v) for n = 6, 400 random pairs for n = 7),
    Corollary 3.2 coefficientwise and Lemma 4.2 (on at most 50 orderings for each n) for n ≤ 7; P_n = 0 at degenerate
    integer points (A'_1 = A'_2, and A'_1 = A'_2 with A'_3 = −2A'_4) for 4 ≤ n ≤ 7.
- **Finder** (`claimant/`). S_n ≡ 0 modulo 2^61 − 1 for 3 ≤ n ≤ 13 and modulo 2147483629 for
  n ∈ {5, 6, 9, 10, 13, 14, 17, 18}; the shuffle identity and the orthogonality of the signed numerator to random
  left-normed brackets for 4 ≤ n ≤ 7; brute-force controls for n = 5, 6.
- **First independent verification run** (`verifier/`, AI-assisted, separate code written before the finder's scripts
  were read). A C subset recursion over GF(2^61 − 1): S_n = 0 for 3 ≤ n ≤ 21, while random symmetric weights and bracket
  numerators over random antisymmetric denominators give nonzero values for n ≡ 1, 2 (mod 4) (rerun here in this
  layout; the recorded values for n = 17, 18 are reproduced exactly). Python checks for n ≤ 8: Σ_σ PT(1σ)σ is a Lie
  polynomial (Dynkin–Specht–Wever criterion); the signed trace form vanishes on all left-normed brackets for general
  traceless matrices; Theorem 1.4(b); a nonzero control with matrices of nonzero trace. Proposition 5.2 for n ≤ 7.
- **Second independent verification run** (`independent_run_2/`, AI-assisted, separate code written from the statements
  of the note before the other programs were read; details in `independent_run_2/README.md`).
  - S_n exactly (Fractions) at random integer points for 2 ≤ n ≤ 12 (term by term for n ≤ 9; one degenerate
    instance): S_n = 0 for n ≥ 3, S_2 ≠ 0. Modulo 2^61 − 1 for 2 ≤ n ≤ 26, modulo 2^31 − 1 for n ≤ 18 and modulo
    10^9 + 7 for n ≤ 16 (a C subset recursion). Over GF(q) for q ∈ {2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 25, 27, 32} and
    n ≤ min(q + 1, 12) (GF(11) with n = 12 skipped: no 12 pairwise independent points were found by sampling).
  - Controls as predicted: random symmetric weights, bracket numerators over random antisymmetric denominators and
    random antisymmetric numerators over bracket denominators are nonzero exactly for n ≡ 1, 2 (mod 4) (up to n = 26
    modulo 2^61 − 1); unsigned sums are nonzero. In characteristic 2 every symmetric-weight control is 0, as the
    reversal pairing forces. Over very small fields some random controls vanish by chance.
  - P_n expanded over Z: the zero polynomial for n = 3, …, 7 (273,067,200 monomial products at n = 7), with nonzero
    controls.
  - Lemma 3.1 for all splits (u, v) with n ≤ 7; Corollary 3.2 and Lemma 2.1(b) for n ≤ 7; Theorem 1.4(a), (b) exactly
    for 3 ≤ n ≤ 9, and Theorem 1.4(a) over seven finite fields; Remark 5.1 and Proposition 5.2 (all T) for n ≤ 8; Proposition 4.1
    in an algebra of 3 × 3 matrices with central commutators, n ≤ 8; Corollary 1.3 at generic and degenerate points,
    n ≤ 8; Lemma 2.2 over F_2, F_3, F_5, F_7 for |C| ≤ 6 (annihilator of Sh_C of dimension (m − 1)!, expansion (5));
    the T = {2, …, n} case of Proposition 5.2 as a statement about Sh_C over F_3, F_5, F_7 and F_10007; Remark 1.6
    exactly in Q(i) for n ≤ 9.
  - It reran every release program from the source archive; all outputs were reproduced apart from timing lines
    (`independent_run_2/rerun_of_release_scripts/`). In this revision the run's programs were rerun from the copies in
    `independent_run_2/` and reproduced the recorded outputs apart from timing values.

## Independent verification
### First independent verification run
Verdict of the first independent verification run (2026-09-30):

| Item | Verdict |
|---|---|
| Source fidelity (OWR 2/2009, Problem 6, re-read) | CONFIRMED |
| Correctness (every step re-derived; own code as above) | CONFIRMED (`correct: true`) |
| Answers the question as posed | CONFIRMED (proved for every n ≥ 3; n = 2 fails and n = 1 is undefined; suggested status: solved) |
| Novelty | apparently new; required an extended check in the amplitudes literature |
| Classification | PAPER_CANDIDATE, with three required fixes |

Required fixes and how they were applied:
1. *Amplitudes literature.* **Applied.** The introduction now says that Lemma 3.1 is the known shuffle identity of
   Parke–Taylor factors, equivalent to the Kleiss–Kuijf relations (Kleiss–Kuijf 1989; proofs by Del Duca–Dixon–Maltoni
   2000 and, directly for Parke–Taylor factors, Kol–Shir 2014, Section 4). It points to Frost–Mafra–Mason (Section 2.1)
   and Mafra–Schlotterer (Section 5.1) for the link with shuffles, Lie polynomials and Ree's theorem, and names
   Corollary 3.2 as the Del Duca–Dixon–Maltoni decomposition. The BCJ relations and KLT orthogonality are cited as
   examples of signed Parke–Taylor relations. The self-contained proofs are kept. The novelty check was extended (see
   below).
2. *Conventions and hypotheses.* **Applied.** One-line notation with π_1 = 1; the action π(A) = (A_{π(1)}, …) of the
   source; the other reading gives the same sum since sgn(π⁻¹) = sgn(π); the hypothesis [A'_iA'_j] ≠ 0 for i ≠ j,
   shown to be exactly the condition for all terms to be defined when n ≥ 3; n ≤ 2 discussed.
3. *Ree / Reutenauer and the sign identities.* **Applied.** Ree, Ann. of Math. (2) 68 (1958) 210–220 and Reutenauer,
   Free Lie Algebras (1993), Theorem 3.1, were cited. (The second revision replaced the latter by Reutenauer's
   Handbook of Algebra chapter, Theorem 3.1(iv), and the final readiness check by the 1993 book without a theorem
   number; see below.) In addition, the multilinear case needed is now proved in the note (Lemmas 2.1–2.2), over every
   field, so the argument no longer depends on the characteristic-zero theorem. The finder's sub-identities are
   Proposition 5.2, with a proof.

Further changes in this revision: Theorem 1.4 (the matrix form, with M_1 arbitrary for n ≥ 4), Proposition 4.1 (the
general central-commutator principle), Remark 5.1 (open chains), the polynomial form (Corollary 1.3), the phirotope
form (Remark 1.6), and small-characteristic checks.

### Second independent verification run
A second independent verification run (AI-assisted, 2026-09-30) checked the version described above. It read the
sources, re-derived every step of the proofs, wrote its own programs (`reproducibility/independent_run_2/`) and reran
all release programs.

| Area | Verdict |
|---|---|
| Statement fidelity | CONFIRMED against OWR 2/2009, Problem 6 (p. 108), and BKR 2003, Conjecture 6.1, read in the originals. Proved for every n ≥ 3 over every field; n = 2 is false and n = 1 undefined, as the note says. |
| Proofs | CONFIRMED: every step correct and complete, in every characteristic. One slip in a side remark (Ree's theorem needs "with zero constant term"); no proof used it. |
| Computations | CONFIRMED: its own programs found no failure, and all release programs reproduced their recorded outputs. One wrong count in the documentation (140 checks, not 141). |
| Novelty and credit | No earlier proof of the conjecture, and no earlier occurrence of Theorems 1.2 and 1.4. Proposition 5.2 is not new in substance and was overclaimed. The Reutenauer theorem number was unsupported. |
| Presentation | Minor revision. |
| Fatal problems | none |

Required fixes and how they were applied (second revision, 2026-09-30):
1. *Verification wording.* **Applied.** The Verification paragraph now says that the proofs are complete and
   self-contained and that no step depends on a computation. It describes the independent checks as AI-assisted
   verification runs that re-derived each step of the proof. The same change was made in `reproducibility/README.md`
   and in this report.
2. *Proposition 5.2.* **Applied.** New Remark 5.3: for T ≠ {2, …, n} the identity follows directly from Lemma 3.1,
   because Σ_σ sgn(σ|_T) σ = (Σ_τ sgn(τ) τ) ⧢ (Σ_ρ ρ) (or Σ_σ σ = x ⧢ Σ_ρ ρ when |T| ≤ 1) lies in Sh_C. The case
   T = {2, …, n} is equivalent, over fields of characteristic 0, to the classical fact that the sign representation does
   not occur in Lie_{n−1} for n − 1 ≥ 3 (Klyachko, Siberian Math. J. 15 (1974) 914–920, doi:10.1007/BF00966559); the
   remark also points to the cyclic analogue in Kol–Shir, Section 3.2. Proposition 5.2 was removed from the "we did not
   find" sentences of the introduction and of the Scope paragraph, which now say that it is not new in substance. The
   abstract (and the Zenodo description) now says that the argument "recovers the vanishing of the same sum without
   the matrices" instead of "gives a family of sign identities for Parke–Taylor factors".
3. *Reutenauer citation.* **Applied, then changed in the final readiness check.** The second revision cited
   Reutenauer's chapter "Free Lie algebras" in Handbook of Algebra, Vol. 3 (Elsevier/North-Holland 2003), pp. 887–903,
   doi:10.1016/S1570-7954(03)80075-X, for Theorem 3.1(iv), which is the number Mafra–Schlotterer cite (Section 4.3.1 of
   the arXiv version), and no longer cited the 1993 book. Its Scope paragraph said that the chapter was not accessed and
   that the theorem number was the one cited by Mafra–Schlotterer. The final readiness check replaced this citation by
   the 1993 book without a theorem number (see "Final readiness check" below).
4. *Ree's theorem.* **Applied.** The remark after Lemma 2.2 now says "an element with zero constant term that is
   orthogonal to all u ⧢ v …", and notes that the empty word is orthogonal to all these shuffles.
5. *OWR page range.* **Applied.** The open-problem section spans pp. 106–110, in the bibliography and twice in this
   report; Problem 6 is on p. 108.
6. *Documentation numbers.* **Applied.** `check_bkrg.py` performs 140 checks (its output has 140 PASS lines and then
   ALL CHECKS PASSED), in `reproducibility/README.md` and in this report. The "Corpus record" bullet now quotes the
   title of the v1.6.0 record and describes the later copy separately.
7. *"For every n".* **Applied.** Conjecture 1.1 in the paper now reads "For all n ≥ 3 …", with a pointer to the
   hypothesis (2) and to Remark 1.5 for n ≤ 2, and this report says "every n ≥ 3" throughout. The notice to the corpus
   maintainers must also say every n ≥ 3 (n = 2 fails and n = 1 is undefined); see "Suggested corpus update" below.

Optional suggestions applied: the five-point formula (8) is identified with the case n = 5 "up to relabelling and a
factor 2"; Remark 1.5(a) attributes the list of cases starting at n = 3 to OWR only; Remark 1.5(b) notes that in
characteristic 2 the reversal pairing already proves the identity (paired terms are equal and 2α = 0); the method
paragraph credits Rosset's proof of the Amitsur–Levitzki theorem (Israel J. Math. 23 (1976) 187–188,
doi:10.1007/BF02756797) for the device of tensoring with exterior generators; Section 6 explains why the controls must
use nonconstant denominators (Amitsur–Levitzki, Proc. Amer. Math. Soc. 1 (1950) 449–463,
doi:10.1090/S0002-9939-1950-0036751-9); the BCJ and KLT sentence now says that these relations hold only under momentum
conservation or on the support of the scattering equations, and that the note uses only the Kleiss–Kuijf relations;
item 1 of the Verification paragraph now says that Lemma 3.1 is checked on all pairs for n ≤ 6 and on 400 random pairs
for n = 7, and Lemma 4.2 on at most 50 orderings for each n. The numbering of the results did not change; Remark 5.3 is
new. The misprint in BKR's definition of π(A_1, …, A_n) is recorded above but not in the paper.

### Final readiness check
A final readiness check (AI-assisted, 2026-09-30) read the revised note and re-checked each proof, confirmed that the
required fixes above had been applied, rebuilt the paper from the source archive and reran `lead/check_bkrg.py` from a
fresh extraction (140 PASS lines, then ALL CHECKS PASSED). It changed two things in the paper:
1. *Reutenauer citation.* The bibliography of Mafra–Schlotterer (arXiv:2210.14241) lists the Handbook chapter under
   the key they use for "Theorem 3.1 (iv)". The same paper also cites "section 8.6.7", "(1.5.12)" and "Proposition
   1.9" of that key. An earlier paper of the same authors (arXiv:1812.10969) cites "section 8.6.7" of the 1993 book.
   The chapter has 17 pages. By the publisher's chapter summary, Chapter 3 of the book ("Logarithms and exponentials")
   extends the characterizations of Lie polynomials of Chapter 1 to Lie series. So "Theorem 3.1 (iv)" very probably
   refers to the book, not to the chapter. Since neither was accessed, the remark after Lemma 2.2 now cites the book
   (London Math. Soc. Monographs, New Series 7, 1993, doi:10.1093/oso/9780198536796.001.0001) without a theorem number.
   The second verification run had allowed this option. The Scope paragraph now lists Ree 1958 and the book among the
   works not accessed, and it no longer mentions a theorem number.
2. *Verification paragraph.* Item 3 gives the ranges of the second run's modular computations separately: modulo
   2^31 − 1 for n ≤ 18 and modulo 10^9 + 7 for n ≤ 16. Before, it said "modulo two further primes for n ≤ 18".

The numbering of the results and the abstract did not change.

## Relation to the literature, novelty and scope
- **Read.** OWR 2/2009, pp. 106–110 (Problem 6 on p. 108); the full author version of Below–Krummeck–Richter-Gebert
  2003; Kol–Shir, JHEP 11 (2014) 020, Sections 2, 3.2 and 4; Frost–Mafra–Mason, Commun. Math. Phys. 402 (2023),
  Section 2 (arXiv version); Mafra–Schlotterer, Phys. Rep. 1020 (2023), Sections 4.3 and 5.1 (arXiv version). The
  verification runs searched Anderson–Delucchi (DCG 48 (2012)) and Ruiz (arXiv:1807.07552) for the conjecture: neither
  mentions it. The second run also read Del Duca–Dixon–Maltoni 2000 (arXiv version).
- **Searches (September 2026, anonymous, logged in `queries.log`).**
  - zbMATH: the only work citing the 2003 paper is Anderson–Delucchi (found by both verification runs); a search for
    "phirotope*" gives only 4 documents (second run).
  - Semantic Scholar: eight citing works (Anderson–Delucchi; Ruiz 2018; Nisse–Sottile 2013; four papers of J. A. Nieto
    on oriented matroids and phirotopes in physics, 2005–2018; a 2003 work "On generalizing oriented matroids to a
    complex setting"). The full texts of Anderson–Delucchi and Ruiz were searched in the verification runs; the arXiv
    abstracts of the other five that are on arXiv are unrelated to the conjecture. The 2003 work was not seen.
  - arXiv API (first revision): Parke–Taylor with alternating/signed/Grassmann/shuffle/Lie/trace/supergroup;
    Kleiss–Kuijf with Lie/Grassmann/supergroup; U(1) and photon decoupling with nilpotent/central/Heisenberg; bracket
    syzygies; alternating sums over Hamiltonian cycles; the phirotope five-point formula. No hit contains the identity
    of the paper or a proof of the conjecture.
  - arXiv API (second verification run): phirotopes and phased matroids; Parke–Taylor with antisymmetric, alternating,
    sign representation, Grassmann, fermionic, supertrace, Clifford, inverse or reciprocal; Kleiss–Kuijf with Clifford,
    alternating, sign, antisymmetric or Grassmann; Amitsur–Levitzki with Lie or shuffle; free Lie algebra with sign
    representation; bracket polynomials with syzygy or cyclic; all abstracts mentioning Richter-Gebert. No proof of the
    conjecture was found.
  - Crossref: all DOIs of the bibliography verified. The second revision verified Reutenauer 2003
    (10.1016/S1570-7954(03)80075-X, "Free Lie algebras", Handbook of Algebra, pp. 887–903; no longer cited), Klyachko
    1974 (10.1007/BF00966559, Siberian Math. J. 15(6), 914–920), Amitsur–Levitzki 1950
    (10.1090/S0002-9939-1950-0036751-9, Proc. Amer. Math. Soc. 1(4), 449–463) and Rosset 1976 (10.1007/BF02756797,
    Israel J. Math. 23(2), 187–188). The final readiness check verified these four again, and the DOI of Reutenauer's
    1993 book (10.1093/oso/9780198536796.001.0001, "Free Lie Algebras", Oxford University Press) with its chapter records.
  - One web search in the first revision (used to locate the author PDF of the 2003 paper), one in the second
    verification run (no proof of the conjecture found) and one in the final readiness check (for the contents of
    Reutenauer's Handbook chapter; nothing found). OpenAlex was rate-limited during both verification runs.
- **Not accessed.** Reutenauer's book (1993) and his Handbook of Algebra chapter (2003). The paper cites the book for
  Ree's theorem, without a theorem number. The first version cited "Theorem 3.1" of the book, and the second revision
  cited "Theorem 3.1(iv)" of the chapter, following Mafra–Schlotterer (Section 4.3.1: "see Theorem 3.1 (iv) in [152]",
  where [152] is the chapter in their bibliography); see "Final readiness check" above for why that number very
  probably belongs to the book. The note does not depend on it: the multilinear case is proved in Lemma 2.2. Also not
  accessed: Kleiss–Kuijf 1989, Ree 1958, Klyachko 1974 (cited for the classical fact on the sign representation;
  Kol–Shir cite it for the description of Lie_n as induced from a faithful character of a cyclic subgroup of order n,
  from which that fact follows by Frobenius reciprocity), Amitsur–Levitzki 1950 and Rosset 1976.
  Del Duca–Dixon–Maltoni 2000 was seen in full only by the second verification run.
- **Novelty.** The ingredients (Parke–Taylor shuffle identity, Ree's theorem, the DDM decomposition) are classical, and
  the paper says so. We found no earlier proof of the conjecture, and no occurrence of Theorems 1.2 and 1.4 in the
  amplitudes literature. Proposition 5.2 is not new in substance (Remark 5.3): it follows from Lemma 3.1 for
  T ≠ {2, …, n}, and for T = {2, …, n} it is equivalent in characteristic 0 to a classical fact about free Lie algebras.
  The new step is the homomorphism c ↦ M_c ⊗ e_c into M_2 ⊗ Λ, which realises the signed bracket numerators as colour
  factors with central commutators; tensoring with exterior generators is also the device of Rosset's proof of the
  Amitsur–Levitzki theorem, which the paper credits. This negative search is not a proof of priority.
- **Scope.** The note proves Conjecture 6.1 of Below–Krummeck–Richter-Gebert = OWR 2/2009 Problem 6 for all n ≥ 3 over
  every field (and in polynomial form over every commutative ring). It does not treat non-realizable phirotopes, and it
  does not consider analogues for d × d determinants with d ≥ 3.

## Suggested corpus update (not performed)
Mark OWR-3385-008 as solved (proved: the alternating sum vanishes for every n ≥ 3 over every field, whenever the
brackets [A'_iA'_j] are nonzero; the bracket-polynomial form holds identically), citing the eventual DOI of this note.
The notice should say "every n ≥ 3", not "every n": for n = 2 the sum is ([A_1A_2]/[A'_1A'_2])², which is not zero in
general, and for n = 1 it is undefined.

## 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/