# Verification report — OWR-4990374-015 (Tohăneanu: free arrangements with exponents 1, 2 and at most one 3 are supersolvable)

Verification date: 2026-10-12.

**Verdict.** The note proves the conjecture: over a field of characteristic zero, every central free hyperplane
arrangement whose exponents are 0's, 1's, 2's and at most one 3 is supersolvable (Theorem 1.1), in every rank.
- **Complete proof.** The proof is written out in the note and is checked by reading; no computation enters it.
  It rests on two cited theorems (Terao's factorization theorem, used only for one coefficient, and Saito's
  criterion), on Yuzvinsky's theorem that free arrangements are formal, for which a complete proof following
  Brandt and Terao is included (Appendix A), and on a statement which does not mention freeness (Theorem 1.2, the
  core statement, valid over every field).
- **New, to our knowledge and relative to the literature found.** The case without a quadruple point (v = 0) in
  rank k ≥ 4, and therefore the conjecture in general; the other cases are proved again. The new case is
  Lemma 4.3(ii) (the support of a linear dependency among the k + 1 local relations); the same count of
  dimensions gives v ≤ 1, and the induction runs inside the classes of Definition 3.1.
- **Not new, proved again.** Exponents (1,2,…,2), and exponents (1,2,…,2,3) with a quadruple point (v = 1), follow
  from DiPasquale, Math. Ann. 385 (2023), Proposition 6.1 with Corollary 4.10 (numbering of arXiv:1806.05295v1).
  Rank three follows from Tohăneanu, Discrete Math. 339 (2016), Theorem 2.4. The combinatorial analogue of the
  case of exponents 1 and 2, with the same induction step, is in Denham–Garrousian–Tohăneanu, Ann. Comb. 18
  (2014), Theorems 5.6 and 5.11.
- **Neighbouring result.** For arrangements whose underlying matroid is regular, freeness is equivalent to
  supersolvability by a preprint of W. Liang and S. Wang (arXiv:2608.01331v1, 2 August 2026; abstract and
  introduction read). It does not contain the conjecture: the matroid of a free arrangement with these exponents
  is not known to be regular in advance. Afterwards it is graphic when there is no quadruple point (Remark 6.1),
  and it is not regular when there is one.
- **Credit.** The conjecture and the strategy (formality as the tool; the modular coatom obtained from two
  hyperplanes in a single dependent triple; induction on the rank; the counts n = 2k and u + 3v = k + 1; the
  pigeonhole count for exponents (1,2,…,2)) are due to Ş. O. Tohăneanu: Oberwolfach Report 5/2021, pp. 283–286, and
  the preprint arXiv:1707.07091 (version 3). The preprint was withdrawn by its author on 16 January 2022
  (version 4); the withdrawal notice names Sections 4.2.1 and 4.2.2 and case (a) of the proof of
  Proposition 4.8. No result of the preprint is used in the proofs.
- **Limits.** The theorem is claimed for characteristic zero. The extension to other fields is stated only in
  Remark 6.5, for fields over which Saito's criterion holds. The book of Orlik and Terao was not accessible; its
  theorem numbers are given only as cited in papers that were read. No proof and no counterexample of the
  conjecture was found in the literature accessible to us; a search that finds nothing is not a proof of
  novelty, and no priority is claimed.

The note is unrefereed.

## Statement checked
- **Primary source.** Ş. O. Tohăneanu, "Free arrangements with low exponents", abstract in Oberwolfach Reports 18
  (2021), Report No. 5/2021 "Logarithmic Vector Fields and Freeness of Divisors and Arrangements: New
  perspectives and applications", doi:10.4171/OWR/2021/5 (the report occupies pp. 225–300; the abstract
  pp. 283–286).
  - Read in the publisher's open file (76 pages). Fetched again on 2026-10-11 (writing) and on 2026-10-12 (third
    verification run).
  - Standing hypotheses of the abstract (p. 283): a central essential arrangement in a vector space of dimension k
    over a field of characteristic zero; D(A) the module of logarithmic derivations; free; exponents written
    (1, d_2, …, d_k), the 1 coming from the Euler derivation; X modular if X + Y ∈ L(A) for all Y; supersolvable =
    a maximal chain V = X_0 < … < X_k = 0 of modular elements.
  - The conjecture (p. 284, displayed): every central free hyperplane arrangement with "exponents 1's, 2's, and
    at most one 3" is supersolvable. The introduction of the report (p. 228) repeats it.
  - The abstract lists the conjecture as verified in the preprint for exponents 1 and 2 only, for inductively
    free arrangements, and for free arrangements of rank 3, 4, 5.
- **The preprint.** arXiv:1707.07091: version 1 of 22 July 2017, version 2 of 9 April 2019, version 3 of
  16 November 2020, version 4 of 16 January 2022, marked as withdrawn by its author. Version 3 was read
  completely (PDF and TeX source): Conjecture 4.1 is the statement for exponents (1,2,…,2,3); the degree of a
  derivation is the degree of its coefficients (Section 1). The withdrawal notice of version 4 names Sections
  4.2.1 and 4.2.2 and case (a) of the proof of Proposition 4.8.
- **Corpus record.** ulamai/UnsolvedMath, OWR-4990374-015 (dataset version 1.6.0; upstream status `open`). Its
  statement is the conjecture in other words; it does not name the field (the source assumes characteristic
  zero). Its background says that the conjecture is proved in several ranks and for some exponent patterns;
  according to the abstract these verifications are those of the preprint, which was withdrawn later.

## Readings
| Reading | Answer | Where |
|---|---|---|
| The conjecture as stated: central, free, exponents 1's, 2's and at most one 3, characteristic zero | true | Theorem 1.1 |
| "At most one 3": the multiset of exponents contains at most one entry 3 ((1,1,2,3) and (1,3) admitted) | this reading is proved | Section 1.3 |
| Irreducible case, exponents (1,2,…,2,3) (Conjecture 4.1 of the preprint) | true | Corollary 5.3(b) |
| No exponent 3: exponents (1,2,…,2) | true; not new | Corollary 5.3(a), Remark 6.4 |
| Reducible arrangements (several exponents 1) | covered: product of irreducible factors | proof of Theorem 1.1 |
| Arrangements which are not essential (exponents 0) | covered: supersolvability of A/C | Section 2.1, proof of Theorem 1.1 |
| Degree convention: the Euler derivation has degree 1 | as in the source | Section 1.3 |
| Two exponents 3 | false: exponents (1,3,3) and (1,2,3,3) occur for free arrangements which are not supersolvable | Section 7 |
| Other fields | only Remark 6.5: every field over which Saito's criterion holds; the core statement over every field | Remark 6.5, Theorem 1.2 |
| Affine arrangements, multi-arrangements, the converse | not part of the conjecture; not treated (the converse only as Remark 6.2, through a cited theorem) | Section 1.3 |

## Results in the paper
- **Theorem 1.1 (the conjecture).** Characteristic zero; exponents 0's, 1's, 2's and at most one 3.
- **Theorem 1.2 (core statement; any field).** An essential, irreducible, formal arrangement of rank k ≥ 2 with
  2k hyperplanes, all m_Z ≤ 4 and Σ_Z C(m_Z − 1, 2) = k + 1 is supersolvable. For k ≥ 3, except for the complete
  quadrilateral (k = 3), there is a triple point such that two of its three hyperplanes contain no other
  multiple flat; the intersection X of the remaining hyperplanes is a modular line, and A_X/X satisfies the
  same hypotheses in rank k − 1.
- **Section 2.** Cited: (T1) Terao's factorization theorem in the form of Palezzato–Torielli, Theorem 3.9 (only
  the coefficient of t^(k−2)); (T2) Saito's criterion, Palezzato–Torielli, Theorem 3.7; Theorem 2.1 (free ⟹
  formal; Yuzvinsky 1993, Corollary 2.5). Proved: Lemma 2.2 (rank two), Lemma 2.3 (exponents 0 and 1),
  Lemma 2.4 (products), Lemma 2.5 (exponents of free sub-arrangements).
- **Definition 3.1, Proposition 3.2.** The classes C_k, C'_k; a free arrangement with exponents (1,2,…,2,3),
  resp. (1,2,…,2), lies in C_k, resp. C'_k.
- **Lemma 4.1.** In an essential, irreducible, formal arrangement every hyperplane contains a multiple flat, and
  the hypergraph of multiple flats is connected.
- **Lemma 4.2.** Two hyperplanes of degree 1 through a triple point: the modular line X; modular elements of
  L(A_X) are modular in L(A); A_X is formal, A_X/X irreducible, with the same multiple flats except one.
- **Lemma 4.3.** Existence of such a triple point: (i) in C'_k; (ii) in C_k: v ≤ 1, and, unless (k, v) = (3, 0),
  a triple point which lies in two hyperplanes of degree 1; for v = 0 through the support of a linear
  dependency among the k + 1 local relations.
- **Lemma 5.1, Theorem 5.2, Corollary 5.3.** The complete quadrilateral; the classes are supersolvable; free
  arrangements with these exponents are supersolvable.
- **Remarks 6.1–6.5.** The modular chain and the step sizes, and (last paragraph, added at the final revision)
  each step as a parallel connection with a three-point line, so that the matroid is that of K_4, or of a
  triangle, with triangles glued along edges when there is no quadruple point; the converse through the theorem
  of Jambu and Terao; the question of Remark 4.7 of the preprint with the example u = 5, s = 2; the derivation of
  the cases covered by DiPasquale's theorem; the field.
- **Section 7.** Where each hypothesis is used; the count of independent dependencies, at most Σ_i C(d_i − 1, 2);
  sharpness: the non-Fano arrangement (exponents (1,3,3)) and nine hyperplanes in rank 4 (exponents (1,2,3,3)),
  both free with explicit bases and not supersolvable.
- **Section 8.** A worked example with all data.
- **Section 9.** A summary of the tests; the details are in this report.
- **Appendix A.** Proposition A.1: free arrangements are formal (any field); the proof follows Brandt–Terao,
  Proposition 4.13 and Theorem 4.15.
- **Appendix B.** Proposition B.1: Σ d_i = n and Σ_Z C(m_Z − 1, 2) = Σ_i C(d_i, 2) without the factorization
  theorem.

## Computations (tests; programs and outputs in reproducibility/)
No proof depends on a computation. A search that finds no counterexample is not a proof. All computations are
exact (rational arithmetic or arithmetic in a finite field).
- **Program of the note** (`writing_stage/check_note.py`, standard library; 74 checks, TOTAL failures: 0; about
  six minutes, quick mode two seconds). (A) the worked example of Section 8: flats, degrees, relation space, the
  dependency of Lemma 4.3 with t = 4, w = 6, c = 1, s' = 2, the modular line, the chain with steps 1, 2, 3, 2, the
  Saito determinant −Q(A) as an identity of polynomials. (B) the non-Fano arrangement (determinant 2·Q(A), no
  modular element of codimension two) and the nine hyperplanes of rank 4 (determinant Q(A), 48 flats, no chain
  of modular elements); a second arrangement of nine hyperplanes; ten hyperplanes of B_5 which satisfy all
  hypotheses of Theorem 1.2 except formality, and eight hyperplanes in rank 4 which satisfy all except
  irreducibility, neither supersolvable. (C) the hypergraph statements of Lemma 4.3: (i) for k ≤ 6 (570,465
  families for k = 6); v = 1 for k ≤ 5 (4, 96, 6,000 families); v = 0 for k = 3 (the six labelled copies of the
  K_4 configuration, the exception), k = 4 (6,750 families, 450 admissible) and k = 5 (4,184,460 families,
  56,700 admissible), with inequalities (6) and (7) for every admissible T. (D) Lemmas 4.1–4.3, 5.1 and
  Theorem 5.2 on all members of the classes among the sub-arrangements of the 13 lines with normals in {0,±1}^3
  (306 members of C_3, 402 of C'_3) and of B_4 (1,968 members of C_4, 2,052 of C'_4). (E) for the three free
  examples: dim M_d and dim (M_Z)_d against formulas (11) and (12), and the injectivity of M_d → ⊕ (M_Z)_d, for
  d ≤ 3 or 4.
- **Programs with which the result was first tested** (`original/`). Exhaustive enumeration of all subsets of
  size 2k of four ground sets, up to signed permutations of the coordinates (Table 1 of the note). Sets which are
  not essential or not irreducible are discarded first (by Lemma 2.3 they are not free with these exponents); no
  other statement of the note is used before the freeness test.

  | ground set | k | subsets | orbits | essential irreducible | free, exponents (1,2^(k−2),3) | (u,v) = (k+1,0) | (k−2,1) |
  |---|---|---|---|---|---|---|---|
  | {0,±1}^3 (13 lines) | 3 | 1,716 | 106 | 106 | 26 | 9 | 17 |
  | {0,±1}^4 (40 hyperplanes) | 4 | 76,904,685 | 415,920 | 410,655 | 2,929 | 913 | 2,016 |
  | B_5 (25) | 5 | 3,268,760 | 3,071 | 2,363 | 134 | 66 | 68 |
  | B_6 (36) | 6 | 1,251,677,700 | 86,764 | 59,779 | 726 | 382 | 344 |

  All free arrangements found are supersolvable and formal, with m_Z ≤ 4 and v ≤ 1. Subsets of size 2k − 1 with
  exponents (1,2^(k−1)): 2,056, 78 and 333 free orbits for k = 4, 5, 6, all supersolvable. Among the essential
  irreducible orbits with the numerical conditions, the formal ones are supersolvable and the others are not
  (2,929 / 0; 134 / 40; 726 / 759; for size 2k − 1: 2,056 / 0; 78 / 29; 333 / 414). Random arrangements generated
  from the coordinate hyperplanes by relations among three forms, in ranks 4–7: 18,180 / 19,978 / 12,000 / 6,000
  distinct arrangements; 5,824 / 3,452 / 1,166 / 337 free with exponents (1,2^(k−2),3), all supersolvable. A
  brute-force check of the hypergraph statement of Lemma 4.3 (v = 0 for k ≤ 5, v = 1 for k ≤ 6).
- **Verification run A** (`verification_run_A/`, its own library). 966,449 sets: over Q, all 6-subsets and
  5-subsets of 25 lines in rank 3, and in ranks 4, 5, 6 a basis together with further hyperplanes of the form
  x_i ± x_j, x_i − 2x_j; over prime fields, a basis together with further points of the projective space, in rank 3
  over F_p for p ≤ 11, in rank 4 over F_2 and F_3, and in rank 5 over F_2 (exhaustive up to linear equivalence).
  For each set: freeness with the exponents, the numerical conditions, irreducibility, formality,
  supersolvability; for members of the classes Lemmas 4.1 and 4.3 and, at every admissible triple point,
  Lemma 4.2(a), (b), (d). No deviation; the sets free with the exponents are exactly the members of the classes,
  all supersolvable. In ranks 5 and 6 over Q one class member in eight was tested for freeness (8,258 of 65,867,
  all free); every set outside the classes was decided exactly. Theorem 1.2 over F_2 in rank 6: all 36,288,252
  sets "standard basis + 6 vectors"; 137,760 irreducible formal sets with seven triple points (379 orbits), all
  supersolvable; 47,410 irreducible non-formal ones (199 orbits), none supersolvable; freeness on a sample of
  40 + 40 representatives. Lemma 2.5 on 72,888 pairs. Sub-arrangements of B_4 with 7, 8, 9 hyperplanes:
  irreducible free ones with exponents (1,2,2,2): 2,052, (1,2,2,3): 1,968, all supersolvable; (1,2,3,3): 2,204
  supersolvable and 180 not. Hypergraph statements of Lemma 4.3 up to isomorphism: (i) k ≤ 8; v = 1, k ≤ 7;
  v = 0, k ≤ 6, with (6) and (7) for every admissible T.
- **Verification run B** (`verification_run_B/`, its own library).
  - Reproduction of the table above orbit by orbit (another freeness test; orbits by orderly generation, counted
    again by Burnside's lemma): 26, 2,929, 134, 726 free orbits in common, none in only one list; for size 2k − 1:
    26, 2,056, 78, 333.
  - Theorem 1.1 with reducible and non-essential arrangements: orbits of all subsets of sizes ≤ 6, 7, 10, 11 of
    the four ground sets (275; 131,430; 7,554; 83,289 orbits): 123; 7,628; 1,143; 6,151 free with exponents in
    the range, all supersolvable.
  - Extensions by one hyperplane, allowed by (4), of the free arrangements with exponents (1,2^(k−1)): 46,066
    (k = 4), 2,442 (k = 5), 1,572 (first 40 of 333 arrangements, k = 6): all free with exponents (1,2^(k−2),3) and
    supersolvable.
  - Localizations and restrictions of 13 arrangements, among them the non-Fano arrangement and the Coxeter
    arrangements D_4, B_4, F_4, D_5: 328 arrangements, 153 free in the range, all supersolvable.
  - Random arrangements: over Q, 17,100 samples in ranks 4–7 (1,944 free with exponents (1,2^(k−2),3), 5,189 with
    (1,2^(k−1))); over F_7, F_11, F_13, F_31, 17,500 samples (2,939 and 6,018 free); all supersolvable.
  - Theorem 1.2 over finite fields, exhaustively: PG(2,2), PG(3,2), PG(4,2); {0,±1}^3 modulo 3 and 5; {0,±1}^4
    modulo 3, 5 and 7.
  - Free ⟹ formal, Terao's factorization and Lemma 2.5 on all subsets of the 13 lines and of B_4 (2,088 and 20,948
    free subsets; 181,720 and 8,043,045 pairs): no exception.
  - Outside the range the same program finds free orbits which are not supersolvable: 5 of 24 with exponents
    (1,3,3) in {0,±1}^3; 4 of 33 with (1,2,3,3) in B_4.
  - Re-run of all 13 original programs: outputs identical to the recorded ones up to timing marks; comparison
    of the two libraries on 1,500 random arrangements: no disagreement.
- **Independent run 2, the third run** (`independent_run_2/`, its own library `r2lib.py`; 12 October 2026; all
  programs use the numbering of the note). Over Q the decision "free with these exponents" is exact in both
  directions (reductions modulo a prime only where they are rigorous; every derivation lifted to integers and
  verified); over finite fields all arithmetic is by tables.
  - Sections 7 and 8 (`examples_sec7_sec8.py`, 56 checks, 0 failures): every printed statement; the derivations
    lie in D(A) and the determinants are −Q(A), 2·Q(A), Q(A) as identities of polynomials (sympy); the exponents;
    non-supersolvability from the definition (all chains); six triple points and the degrees 3, 3, 2, 2, 3, 3, 2 of
    the non-Fano arrangement; χ(A,t) = (t−1)(t−2)(t−3)^2, 48 flats and the only modular coatom of the nine
    hyperplanes; the two examples for the hypotheses of Theorem 1.2 found again by search.
  - Hypergraph statements of Lemma 4.3 (`hypergraph_lemma43.py`): (i) k ≤ 6; v = 1, k ≤ 6 (725,760 connected
    families for k = 6); v = 0, k ≤ 5 (4,184,460 connected families for k = 5, 56,700 with an admissible T), with
    (6), (7), t ≥ 4 and s' ≥ (k + 1 − t) + 1 for every admissible T: 0 violations; for k = 3, v = 0 exactly the six
    labelled copies of the complete quadrilateral.
  - Table 1, rows 1–3 (`table1_orbits.py`), with an exact decision of freeness for every orbit (nothing is
    discarded first): 106, 415,920 and 3,071 orbits of size 2k and 81, 103,649 and 2,074 orbits of size 2k − 1; all
    numbers of the table reproduced (essential irreducible 106 / 410,655 / 2,363; free 26 / 2,929 / 134, split
    9 + 17 / 913 + 2,016 / 66 + 68; size 2k − 1: 26 / 2,056 / 78; not formal with the numerical conditions:
    0 / 0 / 40 and 0 / 0 / 29, none supersolvable); free ⟺ member of the class; Lemmas 4.1–4.3, 5.1 and Theorem 5.2
    on every member: 0 failures.
  - All 35,750 sub-arrangements of B_4 with 7, 8, 9 hyperplanes (`q_families.py b4sub`): 2,052 free with
    (1,2,2,2), 1,968 with (1,2,2,3), all supersolvable; 2,384 with (1,2,3,3), of which 180 are not supersolvable.
  - Over Q, rank 4 (`q_families.py pool4`, `random4`): all 183,696 + 18,000 sets "coordinate hyperplanes + 4
    (resp. 3) forms with two or three non-zero coefficients in {±1, ±2}" with the numerical conditions: 107,904
    + 10,320 members of the classes, all statements hold; exact freeness for every 20th set (10,085 sets): free ⟺
    member. 60,000 random sets from the same pool without any filter, exact freeness for all: 160 + 618 free =
    the members, all supersolvable.
  - Over Q, rank 5 (`q_families.py pool5`): every 100th of the 2,719,320 + 260,400 sets "coordinate hyperplanes
    + 5 (resp. 4) two-term forms" with the numerical conditions: 17,678 + 1,629 members, all statements hold;
    exact freeness for 5,960 of the examined sets: free ⟺ member.
  - Finite fields including non-prime fields (`gf_tests.py`): rank 3 over F_4, F_5, F_7, F_8, F_9 (four lines in
    general position + 2, resp. 1; for F_4 also all 54,264 + 20,349 subsets with 6 and 5 lines of PG(2,4));
    rank 4 over F_4 and F_5: all 17,253 + 3,195 and 59,232 + 7,936 sets "coordinate hyperplanes + 4 (resp. 3)
    further hyperplanes" with the numerical conditions: 11,421 + 1,971 and 36,768 + 4,672 members, all free with
    the exponents (exact) and supersolvable, all statements hold; 24,000 random sets without filter: free ⟺
    member.
  - Arbitrary exponents over F_2, F_3, F_4, F_5, F_7, F_8, F_9 in ranks 3–5 (`general_exponents_gf.py`, 11 runs):
    558,955 sets, 52,743 free; for every free set Σ d_i = n and Σ_Z C(m_Z − 1, 2) = Σ_i C(d_i, 2)
    (Proposition B.1), the coefficient used from (T1), free ⟹ formal, Lemma 2.3, Lemma 2.5 (and on 166,669
    pairs); 18,419 free sets with exponents 1, 2 and at most one 3, all supersolvable (Theorem 1.1 over these
    fields, Remark 6.5); 0 violations. 34,324 free sets have other exponents; 13,860 of them are not
    supersolvable.
  - The last paragraph of Remark 6.1 (`parallel_connection.py`): the two statements on independent sets at
    every admissible triple point, and the matroid as the cycle matroid of the graph built along the induction:
    0 failures on the 1,968 + 2,052 members of the classes among the subsets of B_4 and on 4,451 + 384 members in
    rank 5.
  - Self-test of the library: 50 checks on arrangements with known behaviour, 0 failures.
- **Integer range of the original library.** It stores its linear systems in 64-bit integers. Each entry is one
  exactly computed product of absolute value at most (k−1)^d · A^(d+1) (A the largest coefficient of the forms);
  if it does not fit, the library stops with an overflow error and returns no answer. Run B observed this for
  coefficients of size about 10^5. All reported runs are inside the range. This is stated in the library;
  `writing_stage/arrlib_range_test.py` shows both behaviours.
- **Re-runs.** At the writing (11 October 2026) all 13 original runs, the program of the note, and a selection
  of the programs of runs A and B (52 further commands in all) were run again from a copy of the package; every
  new output agrees with the recorded one after removal of running times and progress lines
  (`reproducibility/RERUN_LOG.txt`; what exactly is ignored is stated in the README). One recorded output of the
  original programs (`enum_Bn_k6.out`) contains a progress line which its re-run does not print; such lines
  depend on the order in which worker processes return. Seven enumerations of run B, which had been made with
  earlier states of the library of that run (before the correction of an overflow), were repeated with the
  final library: the lines with results are identical. On 12 October 2026 the third run executed
  `run_quick.sh` and the complete `check_note.py` from an extracted copy of the archive of 11 October: all
  comparisons identical. After the revision `run_quick.sh` (now with 17 comparisons for `independent_run_2/`)
  was run from extracted copies of the new archive: in the first run one program of `independent_run_2/` printed
  six lines in an order which depends on the worker processes; it now sorts them, and the next run gave 57 of 57
  identical comparisons (RERUN_LOG.txt, Part 5).
- **Limits of the tests.** Only the ground sets and families named; ranks ≥ 7 only at random; freeness of one
  class member in eight in ranks 5 and 6 over Q (run A), of one set in twenty and of a sample in rank 5 (third
  run); 40 + 40 representatives for freeness over F_2 in rank 6 (run A); 40 of 333 arrangements in the extension
  search for k = 6; few random examples with v = 0; number fields only through reductions to finite fields; the
  row of B_6 of Table 1 was not recomputed by the third run.

## Independent verification runs
The proof was first written with the programs in `original/`. Three independent verification runs, all
AI-assisted, followed; each wrote its own programs. Runs A and B (11 October 2026) did not see each other and
examined the first written version of the proof. The third run (12 October 2026; folder
`independent_run_2/`) examined the final text of the note; it derived the proofs again from this text and wrote
its own programs, and the reports of runs A and B were available to it.

| Item | Run A | Run B | Third run (final text) |
|---|---|---|---|
| Statement against the Oberwolfach abstract and the preprint | CONFIRMED_WITH_FIXES (wording about the preprint) | CONFIRMED | CONFIRMED (sources fetched and read again; withdrawal notice) |
| Cited theorems: hypotheses, the identity from the factorization theorem, the notion of formality, the field | CONFIRMED_WITH_FIXES (one theorem number, attributions) | not its part; found a published proof of "free ⟹ formal" which can be read | CONFIRMED in the cited texts (Yuzvinsky, Brandt–Terao, DiPasquale, Möller–Mücksch–Röhrle, Palezzato–Torielli, Denham–Garrousian–Tohăneanu, Tohăneanu 2016, Hoge–Röhrle) |
| Appendix A (free ⟹ formal) | CONFIRMED_WITH_FIXES (attribute to Brandt–Terao) | read without finding a gap; tested on 23,036 free arrangements | CONFIRMED (re-derived); tested on 52,743 free arrangements over finite fields |
| Proposition 3.2, Lemmas 2.2–2.5 | CONFIRMED (re-derived) | tested | CONFIRMED (re-derived); tested |
| Lemma 4.1, Lemma 4.2 (a)–(d) | CONFIRMED (re-derived) | tested on every class member met | CONFIRMED (re-derived); tested |
| Lemma 4.3 | CONFIRMED (every inequality re-derived; one sentence recommended) | counts re-derived; tested | CONFIRMED (every inequality re-derived); enumerated again |
| Theorem 1.2 (= Theorem 5.2(b), with 5.2(a)) | CONFIRMED | CONFIRMED on every example, over Q and seven finite fields | CONFIRMED; tested over Q and over F_4, F_5, F_7, F_8, F_9 |
| Theorem 1.1 | CONFIRMED (characteristic zero) | no counterexample in any search | CONFIRMED; no counterexample |
| Proposition B.1 in its present form, Remark 6.5 | (earlier form: identity by multiplicities) | not seen | CONFIRMED: complete proof; identity tested for arbitrary exponents over seven finite fields |
| Remark 6.4 (DiPasquale) | not seen | derivation of this run | CONFIRMED_WITH_FIXES (notation of the complex) |
| Credit to Möller–Mücksch–Röhrle; citations of Yuzvinsky | not seen | not seen | CONFIRMED in the sources (one wording corrected) |
| Sections 7 and 8 (explicit bases, determinants, non-supersolvability) | not seen | not seen | CONFIRMED with own programs |
| Original programs | not its part | CONFIRMED (code read; all outputs reproduced) | quick part and the program of the note re-run from the archive |
| Novelty | no later paper on the conjecture found | CONFIRMED_WITH_FIXES: two of three cases follow from DiPasquale (2023) | not found in print; one neighbouring preprint of August 2026 to be cited |

None of the three runs found a mathematical error or a gap.

**Corrections required by runs A and B.** All are applied in the note, except item 16, which could not be
carried out:
1. (A) Neutral wording about the withdrawn preprint, facts only; credit to the preprint and to the abstract for
   the strategy: abstract, Section 1.4, "Scope and priority".
2. (A) Appendix A is presented as following Brandt–Terao, Proposition 4.13 and Theorem 4.15.
3. (A) Attributions: Σ(m_Z − 2) ≥ n − r(A) to Brandt–Terao, Remark 3.7; Lemma 4.1(a) to Brandt–Terao,
   Proposition 3.9(ii); the bound on m_Z to Lemma 2.1 of the preprint and, according to the abstract, to Schenck
   (2003), Lemma 5.2; the counts and the pigeonhole argument to Lemmas 3.1, 3.2, 4.2 of the preprint.
4. (A) What is new is said as: the bound v ≤ 1, the dependency argument for v = 0, the induction inside the
   classes (superseded by item 2 of the third run: one statement of what is new).
5. (A) The factorization theorem and Saito's criterion are cited through Palezzato–Torielli (Theorems 3.9 and
   3.7), which was read; numbers of Orlik–Terao's book only "as cited in".
6. (A) The proof of Lemma 4.3 says explicitly that T is not the set of all triple points.
7. (A, optional) The identity (4) is proved without the factorization theorem (Appendix B), and Remark 6.5
   states the consequence for other fields.
8. (A) Remark 6.3 states what Remark 4.7 of the preprint says, and gives the example as a fact.
9. (A) The rank-three case and the numbers of Orlik–Terao are cited as they were read.
10. (A) Theorem 1.2 is called the "core statement".
11. (B) DiPasquale's Proposition 6.1 and Corollary 4.10 are cited; the cases covered by them are marked as
    proved again (abstract, Section 1.4, Remark 6.4, "Scope and priority").
12. (B) Denham–Garrousian–Tohăneanu, Theorems 5.6 and 5.11, and the question in Tohăneanu (2016) are cited.
13. (B) "Free ⟹ formal": DiPasquale's Corollary 4.10 is named as a further published proof.
14. (B) The theorem numbers of Orlik–Terao are marked as not checked (Section 2.2, "Scope and priority").
15. (B) The field is named in the statement (Theorem 1.1, abstract).
16. (B) A second citation search with MathSciNet and Google Scholar: not made (no access); this is said in the
    Verification paragraph and in "Scope and priority".
17. (B) The integer range of the original library is documented (docstring, README, Section 9).
18. (B, optional) The tests with reducible and non-essential arrangements, the agreement of the lists and the
    finite fields are reported (Section 9 in summary, this report in detail).

**Corrections required by the third run.** All twelve are applied; none changes a statement or a proof.
1. Abstract shortened to 230 words with the prescribed content.
2. One statement of what is new, the same in the abstract, Section 1.4, Remark 6.4 and "Scope and priority": the
   case without a quadruple point in rank at least 4, and therefore the conjecture in general; the other cases
   are proved again.
3. Date 12 October 2026 (note, metadata, dated statements on the last checks).
4. Section 9 shortened to Table 1 and a summary; the lists of tests are in this report; four implementations
   of the freeness test; arithmetic in finite fields.
5. Verification paragraph: three runs; what was written after the first two; what the third run examined.
6. Remark 6.4: the vanishing modules are the 𝒮^i(A) of DiPasquale's Definition 5.1.
7. Section 2.1: the footnote on Ziegler's priority is a footnote to the introduction of Yuzvinsky's paper.
8. Section 7: freeness is also used through Lemma 2.3 and in the reduction of Theorem 1.1.
9. The preprint of Liang and Wang (arXiv:2608.01331) is cited (Section 1.4, "Scope and priority", bibliography),
   with what it gives and does not give; Remark 6.1 has a last paragraph with proof (each step of the induction
   is a parallel connection with a three-point line; graphic matroid when v = 0).
10. Searches repeated in part on 12 October 2026; nine web searches in all.
11. Appendix B, last paragraph: the tests of the two identities over finite fields.
12. Package: this report; `reproducibility/independent_run_2/`; README, `run_quick.sh`, `RERUN_LOG.txt`; archive
    and metadata.

**Written at the final revision and examined by the third run only.** The last paragraph of Remark 6.1 (three
lines of linear algebra; tested by `parallel_connection.py`), the sentences on the preprint of Liang and Wang,
and the new wording of the abstract, of Section 1.4 and of Section 9.

**Written after runs A and B and examined by the third run.** The proof of Proposition B.1 (through the injection
of Appendix A and the theorem of Hilbert and Serre); Remark 6.4; Remark 6.5; the observation that the formality
of A_X in Lemma 4.2(d) is a special case of Möller–Mücksch–Röhrle, Corollary 5.2, and that Lemma 4.2(b) follows
from the criterion quoted in their Lemma 2.1; the citations of Yuzvinsky's paper (Corollary 2.5 with its
hypotheses, Definition 2.1, Example 2.2, (4.3)′ on p. 239, Section 4 and Remark 4.13); Section 8 and the
explicit bases of Sections 7 and 8.

## Relation to the literature, novelty and scope
- **Searches.** On 11 October 2026, at each stage: arXiv (free and supersolvable arrangements with small
  exponents, formal arrangements, modular coatoms, the author of the conjecture and the list of all arXiv papers
  of this author), zbMATH Open, OpenAlex (no work citing the preprint is listed), Crossref, Semantic Scholar (at
  the first stage), MathOverflow, and eight web searches. On 12 October 2026 (third run): arXiv API (the author
  of the conjecture, newest first; eight topical queries), OpenAlex (the preprint: cited by 0; DiPasquale's
  paper: cited by 4 works, none on the conjecture; the Oberwolfach report: cited by 2 works, unrelated; one
  topical search from 2021 on), zbMATH Open (three queries), Crossref (one topical query), and one web search.
  - No proof and no counterexample of the conjecture was found. The texts found which cite the preprint are the
    Oberwolfach abstract and one paper which only lists it.
  - Closest published results: DiPasquale (2023) for the cases named above; Denham–Garrousian–Tohăneanu (2014);
    Tohăneanu (2016) for rank three. Closest preprint: Liang–Wang, arXiv:2608.01331 (regular matroids).
  - MathSciNet and Google Scholar were not searched; the author of the conjecture was not contacted.
- **What was read.** The Oberwolfach abstract (pp. 283–286; p. 228) in the publisher's file; version 3 of the
  preprint completely and the withdrawal notice of version 4; Yuzvinsky (1993) and Brandt–Terao (1994)
  completely, in the publishers' files; in arXiv versions: DiPasquale (version 1: Sections 2 and 4, the
  definitions of Section 5, Section 6.1), Denham–Garrousian–Tohăneanu (version 1: Section 5), Tohăneanu 2016
  (version 2: introduction and Theorem 2.4), Möller–Mücksch–Röhrle (version 2: Sections 1, 2.1–2.3, 2.7, 5),
  Palezzato–Torielli (version 1: Sections 1–3), Hoge–Röhrle (version 3: Section 2), Liang–Wang (version 1:
  abstract and introduction). All bibliographic data with a DOI were checked with Crossref (again on
  12 October 2026).
- **Not read.** The book of Orlik and Terao (not accessible to us); Terao (1981); Jambu–Terao (1984); Schenck
  (2003); Björner–Edelman–Ziegler (1990); Bruns–Herzog; the journal versions of the papers read in arXiv
  versions (the publisher's file of DiPasquale's paper was not accessible to us). What the note says about them
  is taken from the papers named where they are cited.
- **Caveats.** Crossref lists the authors of the journal version of Palezzato–Torielli in the other order; the
  bibliography follows the arXiv version, which was read. The paper of Yuzvinsky contains nothing on
  supersolvable arrangements or on exponents. The term "complete quadrilateral" is defined in the note
  (Section 1.2, Lemma 5.1). The text of Liang and Wang is a preprint, and only its abstract and introduction
  were read. This negative search is not a proof of priority.
- **Scope.** The note proves the conjecture in characteristic zero. It claims nothing for two or more exponents
  3 (where the statement is false) or for an exponent 4, and it does not claim priority.

## 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/