# Verification report — OWR-16164-020 (Teufl–Wagner: a determinant related to set partitions)

Verification date: 2026-10-11.

**Verdict.** The note gives a **complete proof** of the divisibility asked for in the problem, for every n; the
scope is as follows.
- **Settled.** For an n × n matrix A of indeterminates and the matrix T = T(A) of Teufl and Wagner, indexed by the
  set partitions of an n-set, (det A)^(C_n) divides det T in Z[a_ij] for every n ≥ 1, C_n being the n-th Catalan
  number (Theorem 1.2). More precisely det T = (det A)^(C_n) · R_n, where R_n = det(T(A)|K) for a T(A)-invariant
  subspace K of dimension B_n − C_n which does not depend on A; the map induced by T(A) on the quotient V/K, which
  has the non-crossing partitions as a basis, has determinant exactly (det A)^(C_n); R_n has integer coefficients,
  is homogeneous of degree n(B_n − C_n), and R_n(I) = 1.
- **Also proved.** det T = (det A)^(B_n) for n ≤ 3 and det T = (det A)^14 · per A for n = 4 (Corollary 1.3; both are
  stated in the source). The exponent C_n is exact for 4 ≤ n ≤ 8 (Proposition 1.4); for 5 ≤ n ≤ 8 this is a
  computer-assisted proof (exact computations at explicit integer matrices, Table 1).
- **Not new mathematics.** The Grassmann-algebra representation of forests is due to Caracciolo, Jacobsen, Saleur,
  Sokal and Sportiello. The statement that the image of the representation has the non-crossing partitions as a
  basis (dimension C_n) was announced without proof by Caracciolo, Sokal and Sportiello (2007), and a draft of a
  proof is in commented-out lines of the TeX source of their preprint (see "Relation to the
  literature"); the note proves the statement for the case needed and does not claim it as new. The subspace K
  (span of the "Rule 1" relations = radical of the tree pairing), the dimension C_n of V/K, its planar basis and
  the integrality of the projection are due to Kenyon and Wilson (2011), with the meander determinant of
  Di Francesco, Golinelli and Guitter (1997).
- **What the note adds.** The determinant of the map induced by T(A) on V/K, hence the divisibility; the
  factorisation over Z[a_ij]; the proof of the formula for n = 4; the exactness of the exponent for n ≤ 8.
- **Only tested.** On random lines R_5 is ± det P_4 times an irreducible factor of degree 30 (P_4 = matrix of the
  permanents of the 4 × 4 submatrices); on random lines modulo primes det P_4 and per A divide R_6 (Section 7.4,
  observations (O1) to (O3)).
- **Open.** Exactness of the exponent for n ≥ 9; a description of the cofactor R_n for n ≥ 5.

No priority is claimed: the divisibility is close to the published structure theory and may be known to
specialists. The note is unrefereed.

## Statement checked
- **Primary source.** E. Teufl, S. Wagner, "A determinant related to set partitions", in the problem session of
  *Enumerative Combinatorics*, Oberwolfach Report 23/2018, Oberwolfach Reports 15 (2018), no. 2, pp. 1457–1458
  (the report: pp. 1381–1464, doi:10.4171/OWR/2018/23).
  - Read in the file of the report (84 pages; sha256 beginning 6512c6b5): the two pages as rendered images, at the
    first results, by the verification runs A and B, at the writing, and again by the verification run on the final
    text, which fetched the file anew from the publisher's page of the report (the same file).
  - p. 1457: the origin (transfer matrices in the enumeration of spanning trees of ladder-like graphs); the
    definition of a transition from a partition of X to a partition of Y (spanning trees on the blocks of the
    partition of X plus a set S of edges x_i y_j form a spanning forest of X ∪ Y in which every component contains
    a vertex of Y and whose components induce the partition of Y); the weight w(S) = ∏ a_ij; the matrix T of size
    B_n with entries t_{P,Q} = Σ w(S), rows and columns ordered in the same way; the matrix for n = 2, whose first
    row and column belong to the partitions into singletons.
  - p. 1458: the name A of the matrix; det T = (det A)^(B_n) for n = 1, 2, 3; det T = (det A)^14 · per A for n = 4;
    as an experimental observation, det T = (det A)^42 · P_1 · P_2 for n = 5 with homogeneous P_1, P_2 of degrees 20
    and 30; Problem 1: whether (det A)^(C_n) is always a factor of det T; experimental evidence for n = 6 and
    nothing beyond. No reference is cited; exactness of the exponent is not asked.
  - Directly after Problem 1, on the same page, the next contribution begins: "Conjectured Uniform Presentation
    for Pure Braid Groups" by J. McCammond and N. Williams. It is unrelated to T.
- **Corpus record.** ulamai/UnsolvedMath, OWR-16164-020 (dataset version 1.6.0; upstream status `open`). Its
  statement places the matrix T in a conjectured uniform presentation of the pure braid group and prints the
  exponent on the line ("(\det A)C_n"). Both points are defects of the record: the phrase is the title of the next
  problem of the report, and the exponent C_n was flattened. The record has no proposers; they are E. Teufl and
  S. Wagner. The matrix of the record is the matrix T(A) defined above, and the question is the divisibility by
  (det A)^(C_n).

## Readings
| Reading | Answer | Where |
|---|---|---|
| Problem 1 as posed: is (det A)^(C_n) a factor of det T for every n, as polynomials in the a_ij? | yes | Theorem 1.2 |
| "Factor" read in Z[a_ij] or in Q[a_ij] | the same, by Gauss's lemma (det A is primitive); proved in Z[a_ij] | Section 1.1, Theorem 1.2(c) |
| Divisibility after specialising A in a commutative ring | follows from the identity in Z[a_ij] | Remark 5.3 |
| Another identification of X with Y than x_i ↔ y_i | permutes the columns of T; det T changes at most by a sign | Section 1.1 |
| Is the exponent C_n best possible? (not asked in the source) | yes for 4 ≤ n ≤ 8 (computer-assisted for n ≥ 5); for n ≤ 3, C_n = B_n and det T = (det A)^(C_n); open for n ≥ 9 | Proposition 1.4 |
| What is the cofactor? (not asked in the source) | R_n = det(T(A)\|K); R_4 = per A; for n ≥ 5 only tested observations | Corollary 1.3, Section 7.4 |
| The wording of the corpus record (pure braid groups) | not the problem of the source; see above | Section 1.1 |

## Results in the paper
- **Lemmas 2.1, 2.2.** For f_uv = (ψ̄_u − ψ̄_v)(ψ_u − ψ_v): the product over the edges of a graph is 0 if the graph
  has a cycle; it is the same for all trees on a vertex set (f_B); for a forest it is the product over the
  components; integrating out a vertex v of a component C gives f_(C∖v), and 0 if C = {v}. These are the case λ = 0
  of Caracciolo–Sokal–Sportiello (2007), Lemma 4.1, Corollaries 4.3 and 4.4, and the case t_i = λ = 0 of their
  Lemma 5.1(b) (numbers of arXiv version 2).
- **Proposition 3.1 (intertwining).** With Φ(P) = ∏ f_B over the blocks and G(A)g = ∫_X g ∏(1 + a_ij f_(x_i y_j)):
  G(A)Φ(P) = Σ_Q t_{P,Q} Φ(Q). Hence K = ker Φ is invariant under T(A).
- **Proposition 3.2 (inverse).** G(−A^T) G(A) = (−1)^n (det A)^2 · id, an identity between matrices of polynomials.
- **Lemma 4.1, Propositions 4.2 and 4.3.** The images Φ(P) of the non-crossing partitions are linearly independent
  (a triangular matrix of coefficients, with diagonal ±1, indexed by the sets of left endpoints and of last points
  of the arcs); the image W of Φ has dimension at most C(2m, m) − C(2m, m+2) = C_n, m = n − 1 (W is annihilated by
  two derivations E, F which generate a copy of sl_2). So dim W = C_n. This is the statement announced in
  Caracciolo–Sokal–Sportiello (2007), end of Section 4, for λ = 0.
- **Corollary 4.4.** The coefficients π_{Q,P} of Φ(Q) in the non-crossing basis are integers; an integral basis of K.
- **Lemma 4.5.** The relation between four vertices (their (4.22) at λ = 0), proved in the note, and the
  vectors of Rule 1 of Kenyon–Wilson lie in K.
- **Theorem 1.2.** From Propositions 3.1, 3.2 and 4.3: the determinant p_n of T(A) on V/K satisfies
  p_n(−A^T) p_n(A) = (−1)^(n C_n) (det A)^(2 C_n); det A is irreducible; so p_n = c (det A)^k with k = C_n, c^2 = 1;
  c = 1 because T(I) is unipotent. Integrality from the integral basis; homogeneity from the number
  n + |P| − |Q| of edges of a transition.
- **Proposition 6.1.** The tree pairing M of Kenyon–Wilson is the pairing "integrate all vertices but one":
  M_{P,Q} = ∫ Φ(P)Φ(Q). Hence K ⊆ rad M, and K = rad M = span of the Rule 1 vectors (using Kenyon–Wilson and the
  meander determinant; Remark 6.2 gives rad M = K without the meander determinant, by the invariance of the pairing
  under the derivations E, F, H and Weyl's theorem).
- **Lemmas 6.3, 6.4, Proposition 6.5; second proof.** T(A) M = M T(A^T)^T and
  T(A) M T(−A)^T = (−1)^n (det A)^2 M, proved with a gluing lemma for forests, the matrix-tree theorem, a bordered
  determinant and a sign-reversing involution. With rank M = C_n (Kenyon–Wilson, Di Francesco–Golinelli–Guitter)
  this gives Theorem 1.2 again, without Grassmann variables. Found by verification run B.
- **Lemma 7.1, Corollary 1.3.** A formula for t_{P,Q} after contraction of the blocks; n ≤ 3: K = 0; n = 4: K is
  spanned by κ and κT(A) = per(A)κ, by an explicit computation with the coordinate at 12|34.
- **Proposition 1.4, Table 1.** Exactness for 4 ≤ n ≤ 8.

## Computations (programs and outputs in reproducibility/)
Theorem 1.2, Corollary 1.3 and the statements of Sections 2 to 6 are proved without computation. Proposition 1.4
for 5 ≤ n ≤ 8 is a computer-assisted proof: exact computations which prove a finite statement, namely
v_p(det T(A_0)) = C_n for an explicit integer matrix A_0 with |det A_0| = p prime. Everything else is a test.
All arithmetic is exact (integers, polynomials, residues modulo primes or prime powers).
- **Table 1 (computer-assisted proofs).**
  - n = 4, 5, 6: det A_0 = 137, 47, 251; T(A_0) over Z, exact determinant; v_p = 14, 42, 132
    (`original/scripts/exactness_certificates.py`, output `original/outputs/exactness_certificates.out`; repeated by
    `writing_stage/check_note.py` and by all three verification runs with their own programs; the run on the final
    text took T(A_0) from its literal enumeration of all forests).
  - n = 7: det A_0 = −15299. (i) Block form: invariance of K exactly, determinant on V/K = (det A_0)^429 exactly,
    det(T|K) ≡ 12090 mod 15299 (`original/scripts/verify_quotient.py 7 1 20261011 --nodetT --modp`, output
    `verify_quotient_n7.out`; the same residue was found by run A and by the run on the final text). (ii) Without
    theory: staged elimination of the 877 × 877 matrix modulo p^4, pivots per stage 580, 165, 132, so
    v_p = 165 + 2·132 = 429 (`verification_run_A/outputs/vA_test5_theorem_n7.out`; the same pivots modulo p^3 in
    `independent_run_2/outputs/r2_t6_exactness_n7.out`). Further certificates for n = 7: det A_0 = −17 and
    det A_0 = −2·2309 (run A, staged elimination), det A_0 = 1543 (run B, block form), det A_0 = −19 (run on the
    final text, staged elimination).
  - n = 8: det A_0 = 607 and det A_0 = −137: staged elimination of the 4140 × 4140 matrix T(A_0) modulo p^3,
    pivots per stage 3139, 572, 429 in both cases, so v_p = 572 + 2·429 = 1430 = C_8
    (`verification_run_A/programs/vA_test8_n8.py run 1`, `run 2`; outputs `vA_test8_n8_run1.out`, `_run2.out`).
    Both computations were repeated at the writing of the note from a copy of the package, with identical output
    (`writing_stage/rerun_n8/`). T(A_0) modulo p^3 comes from the recursion (18) of the note; it was compared with
    a second implementation on 40 random rows and with an exhaustive enumeration of forests on 7 rows for each of
    the two matrices (up to 5.8·10^8 forests per row). The run on the final text recomputed the row with
    det A_0 = 607 with its own programs: the same pivots 3139, 572, 429; R_8(A_0) ≡ 286 modulo 607 in the block form;
    12 rows of T(A_0) modulo p^3 equal to its own enumeration of all forests
    (`independent_run_2/outputs/r2_t6_exactness_n8.out`). A third matrix, det A_0 = 4513, by the block form modulo
    primes (run B; this one depends on Theorem 1.2).
- **Programs with which the results were first obtained** (`original/`, 13 programs and a library). T(A) from the
  definition (all 2^(n^2) edge sets, n ≤ 4) and by the recursion; the matrix of the source for n = 2;
  det T = (det A)^(B_n) symbolically for n = 2, 3; the recursion against a formula with tree determinants on all
  entries for n ≤ 6 (41,209 entries for n = 6) and 3000 entries for n = 7; Lemmas 2.1, 2.2 (584 graphs with a
  cycle, all labelled trees on at most 6 vertices); Proposition 3.1 (symbolic n ≤ 2, integer matrices n ≤ 6);
  Proposition 3.2 on all 4^n monomials (symbolic n ≤ 2, integer n = 3) and on random dense vectors (n = 4, 5);
  formula (7) for the 75 partitions with n ≤ 5; Lemma 4.1 and the triangular matrix for n ≤ 8; E, F annihilate
  Φ(P) (278 partitions, n ≤ 6); dim W = C_n with integer π for n ≤ 7; at integer matrices (6 for each n ≤ 6, 2
  for n = 7): invariance of K, determinant (det A)^(C_n) on V/K, Proposition 3.2 on V/K, and for n ≤ 6
  det T = (det A)^(C_n) det(T|K); κT(A) = per(A)κ as 15 polynomial identities; rad M = K for n ≤ 6; the
  divisibility without theory at 172 random integer matrices (vacuous for one matrix with |det A| = 1;
  det T(A) ≠ 0 in all cases); the observations of Section 7.4.
- **Verification run A** (`verification_run_A/`, tests 1 to 8). Its own enumeration of all edge sets (three choices
  of spanning trees; 54,696 transitions for n = 4), an exhaustive enumeration of forests in C, its own recursion.
  Recursion = exhaustive enumeration on all rows for n = 5 (4 matrices) and n = 6 (2 matrices; 2.8·10^9 and
  5.3·10^8 forests), on 365 rows for n = 7 (1.9·10^10 forests). Propositions 3.1, 3.2 as polynomial identities
  for n ≤ 3 and at integer matrices for n ≤ 5 (all partitions, all monomials; singular matrices included).
  Lemma 4.1 for n ≤ 8 (30,694 pairs for n = 8); ranks and integer π ∈ {0, ±1} for n ≤ 7; Theorem 1.2 at 59
  integer matrices, n ≤ 7. Lines over the field with 2^31 − 19 elements, n ≤ 6: det T of degree up to 1218,
  divisible by the power C_n of det A and not by the next one (n = 4, 5, 6). Staged eliminations for n = 7
  (five matrices) and n = 8 (two matrices).
- **Verification run B** (`verification_run_B/`, programs b1 to b13). Three own constructions of T(A) (literal;
  search over the edges, 8,631,300 transitions for n = 5; recursion over the vertices). det T = (det A)^(B_n)
  symbolically for n ≤ 3; n = 4: 200 integer matrices and 3 lines. Divisibility without theory: 3900 integer
  matrices for n = 5 (1500 with prime determinant, 1500 general, 900 structured) and 294 for n = 6, no violation;
  det T = 0 at 127 singular matrices. Divisibility as an identity of polynomials on two lines for n = 5 (over Q,
  272 points each, degree bound 260) and one line for n = 6 (modulo 40009, 1230 points, degree bound 1218).
  K from Rule 1 (1, 15, 155 relations for n = 4, 5, 6): invariance as polynomial identities for n = 4 (14) and
  n = 5 (420). Propositions 3.1, 3.2 with its own exterior algebra for n ≤ 6 (203 partitions, 4096 monomials for
  n = 6). Φ kills the Rule 1 vectors for n ≤ 7 (1400 for n = 7). Identities (13), (14) exactly for n ≤ 6, modulo
  two primes for n = 7, by a randomised test modulo two primes for n = 8. rank M = C_n for n ≤ 8. All original
  programs re-run with identical outputs (two of them up to warning lines).
- **Verification run on the final text** (`independent_run_2/`, tests 1 to 6; every output ends with TOTAL
  failures: 0). Its own literal enumeration of all forests (C) and of all 2^(n²) edge sets (n ≤ 4: 1, 16, 630,
  54,696 transitions; 8,631,300 for n = 5), its own form of the recursion (18), exterior algebra and linear
  algebra.
  - Theorem 1.2 exactly (invariance of K, determinant (det A)^(C_n) on V/K, det T = (det A)^(C_n) det(T|K)) at 340
    integer matrices with n ≤ 6, of which 322 with T(A) from the literal enumeration (n ≤ 5; 4.96·10^8 forests for
    the 133 matrices with n = 5): 125 with prime determinant, 88 singular of all ranks (det T = 0), identity,
    all-ones, permutation, diagonal, triangular and zero-one matrices. det T = (det A)^(B_n) as polynomials for
    n ≤ 3; det T = (det A)^14 per A at 113 integer matrices.
  - Every statement of Sections 2 to 4 and 6.1 on small cases: Lemmas 2.1, 2.2; Propositions 3.1, 3.2 for n ≤ 5
    (all partitions, all 4^n monomials, singular matrices; polynomial identities for n = 2); formula (7) for n ≤ 5;
    Lemma 4.1 (part (b) for all partitions) and the triangular matrix for n ≤ 7; Corollary 4.4 for n ≤ 6;
    Proposition 4.3 for n ≤ 6; Lemma 4.5 with each rule of its proof; Proposition 6.1(a) for all pairs with n ≤ 5
    and 7816 pairs with n = 6; rank M = C_n for n ≤ 8; Remark 6.2 for n ≤ 5.
  - The second proof step by step: Lemma 6.3 (4000 pairs of forests), Lemma 6.4 (n ≤ 3), (13), (14) and Section
    6.3 for n ≤ 5, Step 1 with independent matrices A and B, Step 2, (15) at 400 instances, ϑ_{P,Q} = M_{P,Q} as
    polynomials (n ≤ 4), Step 4 by enumeration of spanning trees (n ≤ 5); det M_t for n ≤ 7 equals the value
    (5.18) of Di Francesco–Golinelli–Guitter.
  - Lemma 7.1 literally (n ≤ 4), (18) against the enumeration on all rows for n ≤ 6, Section 7.2 as identities of
    polynomials.
  - Table 1: rows n = 4, 5, 6 with T(A_0) from the literal enumeration (4.04·10^8 forests for n = 6); rows n = 7
    (K: residue 12090; S: pivots 580, 165, 132) and n = 8 with det A_0 = 607 (S: pivots 3139, 572, 429), as
    printed.
- **Program written with the note** (`writing_stage/check_note.py`, standard library; TOTAL failures: 0, about 11
  seconds). The statements added or reformulated at the writing: Lemma 7.1 in its literal form against the
  definition (285 entries, n ≤ 4); the relation (8) and the Rule 1 vectors (9) for n ≤ 6 (1, 15, 155 vectors);
  Proposition 6.1(a) for all pairs with n ≤ 6 (6844 products) and rank M = C_n over Q for n ≤ 5; Lemma 6.4 from
  the definition for n ≤ 3; (13), (14) at 19 integer matrices, n ≤ 6; the bordered determinant (15) at 60
  instances; ϑ_{P,Q} = M_{P,Q} for n ≤ 4; the polynomial identity for R_4 = per A (24 monomials) and
  κT(A) = per(A)κ at 20 matrices; rows n = 4, 5, 6 of Table 1; Lemma 5.2 for n ≤ 6.
- **Re-runs.** At the writing of the note (2026-10-11) all programs of `original/` were run again from the layout
  of the package; the outputs are identical to the first recorded ones up to running times, except for the two
  corrected programs (see below). `reproducibility/run_quick.sh` repeats the quick part of all five parts and
  compares with the recorded outputs. The verification run on the final text ran it from an extracted copy of the
  archive of the writing stage (0 differences) and again from the final archive; the log of the latter is
  `reproducibility/RERUN_LOG.txt`.

## Independent verification runs
The results were first obtained with the Grassmann proof and the programs in `original/`, with exactness for
n ≤ 7. Three independent verification runs, all AI-assisted, followed on 2026-10-11, each with its own programs.
Runs A and B examined the first written version of the results, and neither saw the other: run A examined the
statement and the proofs line by line; run B recomputed everything from the definition, examined the original
programs and the literature. The third run (second round; folder `independent_run_2/`) examined the final text of
the note and the package; its result is given after the table.

| Item | Run A | Run B |
|---|---|---|
| Statement against the source; corpus remark | CONFIRMED_WITH_FIXES (one sentence about the pages) | CONFIRMED |
| Lemmas 2.1, 2.2; Proposition 3.1 with all signs | CONFIRMED (re-derived) | CONFIRMED (own implementation, n ≤ 6) |
| Proposition 3.2, sign (−1)^n; determinant on W | CONFIRMED (re-derived) | CONFIRMED (own implementation, n ≤ 6) |
| Lemma 4.1, Propositions 4.2, 4.3 (non-crossing basis, dimension C_n) | CONFIRMED (re-derived; integrality of π added) | consistent with its computations (n ≤ 7); attribution required |
| Theorem 1.2 (factorisation, integrality, degree, R_n(I) = 1) | CONFIRMED | CONFIRMED by a second proof (Section 6) and by computation |
| n ≤ 3 and n = 4 | CONFIRMED | CONFIRMED |
| Exactness n = 4, …, 7 | CONFIRMED; n = 8 added | CONFIRMED; n = 8 added (third matrix) |
| Original programs | outputs compared with its own | all re-run, identical; CONFIRMED_WITH_FIXES (two programs) |
| Tested observations on R_5, R_6 | spot checks at integer points | reproduced on its own lines and primes |
| Novelty | not its part (two web searches: nothing) | CONFIRMED_WITH_FIXES: no prior answer found; attribution of the dimension statement and of K corrected |

Neither run found a wrong statement or a gap in a proof.

**The verification run on the final text: what it examined, and the result.**
- *Statement.* The two pages of the source again (file fetched anew); the corpus record. CONFIRMED.
- *Parts added when the note was written* (until then examined by no independent run): Proposition 6.1 and
  Remark 6.2; the proofs of Lemma 4.5 and of R_4 = per A without a computer; Lemma 7.1 and the recursion (18); the
  statement that Lemma 2.2 is the case t_i = λ = 0 of Lemma 5.1(b) of Caracciolo–Sokal–Sportiello; the second
  proof as written (Sections 6.2, 6.3), every line and every sign. All re-derived and tested with its own
  programs. CONFIRMED; in Remark 6.2 a reason was missing for one of three cases (the operator H) and was added.
- *First proof on the final text* (Sections 2 to 5): well-definedness of the representation, Proposition 3.1,
  Proposition 3.2 with the sign (−1)^n, the triangularity argument, the sl_2 bound, and the step from "divides a
  power of det A" to "equals (det A)^(C_n)" (irreducibility, homogeneity, the value at the identity matrix,
  Gauss's lemma). CONFIRMED.
- *Computations.* Its own programs (see "Computations"); Table 1 recomputed for n ≤ 7 and for det A_0 = 607;
  the quick part of the package run again from the archive. CONFIRMED.
- *Sources and credits.* Every sentence of the note about Caracciolo–Sokal–Sportiello, Kenyon–Wilson and
  Di Francesco–Golinelli–Guitter compared with the TeX sources of the arXiv versions; all 20 DOIs resolved through
  Crossref. CONFIRMED_WITH_FIXES: the citations and numbers are right; the draft in the source of
  Caracciolo–Sokal–Sportiello (see "Relation to the literature") had to be disclosed.
- *Novelty.* A further search (arXiv interface, zbMATH, OpenAlex, one web search): no answer found.
- *Presentation.* All pages rendered and looked at, before and after the revision; house style. CONFIRMED_WITH_FIXES.
- It found no wrong theorem, proposition, lemma or corollary and no gap in a proof.

**Corrections required by the verification run on the final text**, all applied:
1. The note says that the TeX source of the arXiv versions of Caracciolo–Sokal–Sportiello contains a draft of the
   announced appendix in commented-out lines, what it contains, that it
   was found only in this run, and how Section 4 relates to it: abstract (one clause), Section 1.3(ii), "Scope and
   priority".
2. The further search for the announced proof is recorded (later papers of the group on arXiv; the doctoral thesis
   of A. Bedini, now cited; the paper of 2017 still not accessible): "Scope and priority".
3. The paragraph "Verification" was rewritten to the final state.
4. "One of the two independent verification runs" (Section 6.2) and "four parts" of the package (Section 8) were
   adapted to three runs and five parts.
5. "Scope and priority": what was read (TeX sources of the arXiv versions), the searches (four stages, nine web
   searches), the paper of 2017 (abstract and reference list seen).
6. Section 1.3(iii): the sentence on the versions is impersonal.
7. Remark 6.2: the reason for the operator H (H = EF − FE) was added.
8. Section 7.3: the further certificate for n = 7 (det A_0 = −19), and the statement which rows of Table 1 were
   computed at least twice with independently written programs.
9. Length: the note was shortened from 23 to 21 pages. Section 8 was reduced to a summary, with the lists in this
   report and in the README of the package; remarks and the closing paragraphs were tightened; the two
   observations (O2), (O3) on n = 5 were joined, so that the observations are now (O1) to (O3); eleven items which
   were not read and are not used in the note are no longer in the list of references: seven are named in the
   paragraph "Scope and priority" with their arXiv identifiers, and four are named below under "Not read". No
   proof and no disclosure was removed.
10. Package: the programs and outputs of this run were added (`reproducibility/independent_run_2/`), the README
    and `run_quick.sh` were extended, and the archive and the hashes were rebuilt.
11. This report and the README were corrected where the findings of this run change them.

**Corrections required by runs A and B**, all applied in the note or in the package:
1. (Run A) Which parts of the statement are on p. 1457 and which on p. 1458: Section 1.1.
2. (Run A, optional) The identification x_i ↔ y_i and what another identification changes: Section 1.1. The sum
   over an interval and the products of the χ are denoted differently (ξ_I and χ_S): Section 4.1. Integrality of π
   from the triangular matrix, without a citation: Corollary 4.4. The staged elimination for n = 7, the further
   certificates and the remark on the elementary divisors p^2: Section 7.3. Exactness for n = 8: Proposition 1.4,
   Table 1, after a repetition of the computation from a copy.
3. (Run B) The note says that the non-crossing basis and the Catalan dimension are announced, without proof, in
   Caracciolo–Sokal–Sportiello (2007), end of Section 4, around (4.22), that the announced paper was not found,
   and that Section 4 is a proof of an announced statement: abstract, Section 1.3(ii), Section 4, "Scope and
   priority".
4. (Run B) The note says precisely what Kenyon–Wilson (with Di Francesco–Golinelli–Guitter) contain, and narrows
   what is new to the determinant on the quotient: abstract, Section 1.3(iii), Proposition 6.1, "Scope and
   priority". The second proof found by run B is included: Section 6.2, 6.3.
5. (Run B) Citation "Lemma 4.1 and Corollaries 4.3–4.4" (the statement on forests is Corollary 4.4): Section 2.
6. (Run B) The output of `explore_cofactor2.py` for n = 4 contained artefacts of an interpolation from too few
   points. The program now interpolates from max(deg R_n, max_k k·C(n,k)) + 1 points and compares with 12 further
   points; the outputs for n = 4, 5, 6 were regenerated. The former output is not part of the package.
7. (Run B) `divisibility_test.py` prints the numbers of vacuous cases: |det A| = 1 in 1 of the 172 matrices,
   det T(A) = 0 in none.
8. (Run B) The references missing from the first search are cited or named, with a statement of what was read:
   "Scope and priority" (after the last revision seven of them are named in the text and four in this report, see
   item 9 of the list above).
9. (Run B) A reading of the literature by a person with access to the journals, and a question to the proposers,
   before any formal publication: **not done** (no contact was made at any stage). The note states which texts
   were read and which were not, and that the status of the announced paper could not be determined.
10. (Run B) The corpus record should restore the exponent, drop the phrase about the pure braid group and name the
    proposers; this concerns the corpus and is reported here, see "Statement checked".

**Changes made when the note was written, after runs A and B** (tested by `check_note.py`; examined afterwards by
the verification run on the final text, see above):
- Proposition 6.1(a) (the tree pairing as an integral) with the identification K = rad M = span of the Rule 1
  vectors, and Remark 6.2 (rad M = K without the meander determinant).
- A proof of the relation (8) between four vertices and of R_4 = per A which needs no computer; before, both
  rested on symbolic computations (`verify_grassmann.py`, part L6, and `verify_n4_per.py`), which are still in the
  package.
- Lemma 7.1 (contraction) stated and proved; before, it was a comment in the programs.
- The observation that Lemma 2.2 is the case t_i = λ = 0 of Lemma 5.1(b) of Caracciolo–Sokal–Sportiello.
- The second proof was written out again and checked line by line; the meander determinant was read in the arXiv
  text of Di Francesco–Golinelli–Guitter, (5.6) and (5.14)–(5.18).

## Relation to the literature, novelty and scope
- **Searches (11 October 2026, at the four stages).** arXiv interface (queries for the proposers; for
  combinations of set partitions, transfer matrices, spanning trees and forests, determinants, Catalan and Bell
  numbers, non-crossing partitions, permanents, Grassmann variables, electrical networks; for the announced paper
  of Caracciolo–Sokal–Sportiello), zbMATH, Crossref, OpenAlex (including the works citing the Teufl–Wagner paper
  of 2010, the five works citing the report and the 27 works citing Caracciolo–Sokal–Sportiello 2007), nine web
  searches in all.
  - No paper, preprint or note answering Problem 1 was found, and no statement of the determinant of T(A) on the
    quotient or of the factorisation det T = (det A)^(C_n) R_n.
  - The paper announced as reference 39 of Caracciolo–Sokal–Sportiello (2007), "in preparation", was not found:
    the arXiv interface returns nothing for its title; the reference list of their paper of 2017 (as given by
    Crossref) does not contain it; a web search found a seminar announcement of 13 January 2009 with nearly the
    same title and slides of 2007 which list the work as in progress, and no article. The arXiv listing of the
    joint papers of two of the authors (27 items) and zbMATH contain no such paper. The sources of four later
    papers of the group on arXiv on spanning forests and hyperforests (0802.1506, 0906.4503, 0903.4432,
    1608.02916) were searched for the terms: they contain neither the statement nor a proof. The doctoral
    thesis of A. Bedini (Università degli Studi di Milano, academic year 2008–2009, advisor S. Caracciolo),
    whose Section 3.2 reproduces Section 4 of the paper of 2007, contains neither the announcement nor a proof.
  - **A draft in the source of the preprint.** The TeX source of both arXiv versions of
    Caracciolo–Sokal–Sportiello (2007) contains, in lines which are commented out and do not appear in the paper, a
    draft of an appendix on the subalgebra: a proposition that the products for different partitions are
    different; the relation for four vertices (for four disjoint sets), verified by expansion with a table of
    coefficients; the theorem on the non-crossing basis with a proof in two steps (reduction of a crossing
    partition by the relation at a primitive crossing of least diameter; linear independence by a leading
    monomial of every non-crossing partition); and the assertion, without proof, that these relations span all
    linear relations. The draft was found by the verification run on the final text; the first written version of
    the results did not know the announcement, and the note was written from the compiled text. Proposition 4.2 of
    the note is close in idea to the second step of the draft (a distinguished monomial for every non-crossing
    partition) and is a complete proof; the upper bound of Proposition 4.3 (derivations E, F generating sl_2) is
    obtained in another way; the first step of the draft is the reduction also used by Kenyon and Wilson.
- **What was read.** The two pages of the source. In the TeX sources of the arXiv versions:
  Caracciolo–Sokal–Sportiello (2007), version 2: Section 4, Section 5 up to Theorem 5.4, Section 7 (and the
  commented-out draft described above); Kenyon–Wilson (2011), version 4: §1.2 (Rule 1, Theorem 1.2) and §2.1–2.3
  (the pairing, Lemmas 2.1–2.5, Theorem 2.6); Di Francesco–Golinelli–Guitter (1997): §5.2 and §5.3 up to (5.18).
  The numbers of lemmas, corollaries and equations used in the note were checked there. The journal versions were
  not compared; the proofs of the statements quoted from Kenyon–Wilson and Di Francesco–Golinelli–Guitter were
  not checked. Section 3.2 of the thesis of A. Bedini.
- **Not read.** Caracciolo–Sokal–Sportiello (2017; J. Phys. A 50, 114001, doi:10.1088/1751-8121/aa59bc) was not
  accessible to us (its abstract and its reference list were seen; neither mentions the subalgebra). Seen in
  abstract only, and cited in the note: Lam (Adv. Math. 338 (2018), arXiv:1402.6261), Chepuri–George–Speyer
  (arXiv:2106.15418), Bychkov–Gorbounov–Kazakov–Talalaev (Moscow Math. J. 23 (2023), arXiv:2109.13952). Seen in
  abstract only, not used, and named in the note with their arXiv identifiers: Lam–Pylyavskyy (Algebra Number
  Theory 9 (2015), arXiv:1103.3475), Bychkov–Gorbounov–Guterman–Kazakov (J. Geom. Phys. 207, arXiv:2406.03021),
  Gao–Lam–Xu (Canad. J. Math. 77 (2025), arXiv:2208.12798), Jekel (arXiv:1601.00247), Kogan (arXiv:2610.07539,
  October 2026), Jacobsen–Salas–Sokal (J. Stat. Phys. 119 (2005), arXiv:cond-mat/0401026),
  Brankov–Poghosyan–Priezzhev–Ruelle (J. Stat. Mech. (2014) P09031, arXiv:1405.0436). The **four further items**
  to which the note refers: Sabot, "Electrical networks, symplectic reductions, and application to the
  renormalization map of self-similar lattices" (arXiv:math-ph/0304015), Carroll–Speyer, "The cube recurrence"
  (Electron. J. Combin. 11 (2004), R73, arXiv:math/0403417), Bauerschmidt–Crawford–Helmuth–Swan, "Random spanning
  forests and hyperbolic symmetry" (Comm. Math. Phys. 381 (2021), arXiv:1912.04854), seen in abstract only, and
  Curtis–Ingerman–Morrow, "Circular planar graphs and resistor networks" (Linear Algebra Appl. 283 (1998),
  115–150), seen in its bibliographic data only. Teufl–Wagner (2010): a preprint version was searched for the
  terms of the problem by run B, without result. Kogan: the text was searched by run B for the names of the
  proposers and the terms of the problem, without result. Caracciolo–Jacobsen–Saleur–Sokal–Sportiello (2004) is
  cited on the authority of the paper of 2007. All bibliographic data were checked with Crossref (20 DOIs, at the
  writing and again by the run on the final text) or with the arXiv interface.
- **Caveats.** Full texts cannot be searched through these interfaces; an answer in a paper whose abstract does
  not mention the problem, or a private communication to the proposers, would not have been seen. The proposers
  were not contacted. The papers on electrical networks and Lagrangian Grassmannians suggest a
  representation-theoretic reason for the exponent C_n; whether one of them contains a statement from which the
  determinant on the quotient follows could not be decided from the abstracts. A search that finds nothing is not
  a proof of novelty.
- **Scope.** The note answers Problem 1 of the source completely. It does not determine the cofactor R_n for
  n ≥ 5 and does not decide the exactness of the exponent for n ≥ 9. No priority is claimed.

## 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/