# Verification report — OWR-1703876-012 (Efrat–Fulek–Kobourov–Tóth: polygons with prescribed turning angles in R^3, conjecture on realizations without self-intersections)

Verification date: 2026-10-10.

**Verdict.** The note gives a **counterexample** to the conjecture and proves a corrected criterion; every
statement it makes is proved, and the scope is as follows.
- **Refuted.** The "if" part of the conjecture is false for every n ≥ 6. For n = 6 the angle sequence
  A* = (5π/6, 5π/6, 5π/6, 5π/6, 5π/6, π/6) is realizable (explicit planar hexagon with coordinates in Q(√3)), n is
  even, and every closed polygon in R^3 with these turning angles is planar with total signed turning ±4π, hence
  not simple (Proposition 3.1). For every n ≥ 6 the sequence A_n = (5π/6 ×5, b ×(n−5)), b = π/(6(n−5)), has the
  same properties; it has Σ(π − α_i) = (n−5)π + 2π/3 ≠ π, so it contradicts each printed form of the conjecture
  (Remark 3.3).
- **Proved for n = 4 and n = 5.** The conjecture holds for n = 4 and n = 5 (Corollary 1.4(a)).
- **Corrected criterion.** With e_S(A) = Σ_{i∈S}(π − α_i) + Σ_{i∉S} α_i − π: a realizable sequence has a
  realization without self-intersections if and only if e_S(A) ≠ 0 for every index set S of odd cardinality at
  least 5 (Theorem 1.3(b)); if Σ α_i > 2π and e_S(A) > 0 for all odd S there is one which is not contained in a
  plane (Theorem 1.3(c)). The sequences for which the conjectured equivalence fails are exactly the points of the
  open simplices Δ_S = {e_S = 0}, S odd, 5 ≤ |S| ≤ n−1 (Corollary 1.4(b)).
- **Not new mathematics.** The inequalities e_S(A) ≥ 0 for closed spherical polygons with prescribed side
  lengths and their equality case (all vertices on a great circle, with a prescribed sign pattern and winding
  number) are known in the literature on spherical polygonal linkages: Galitzer's thesis as quoted by Kapovich
  and Millson (1999, Theorem 3.1; also Theorem 2.9 and Section 3), Biswas (1998), Buckman and Schmitt (2002),
  Mondello and Panov (2016, Theorem 2.22 and Lemma 2.24(i)), and an appendix by D. Mamaev in Mondello and Panov
  (2024). The "only if" part of the conjecture and the realizability criterion are due to Efrat, Fulek, Kobourov
  and Tóth (their Theorems 5 and 2). Fenchel's inequality is classical. The note proves these statements again
  only for completeness and says so.
- **Asserted in the source without proof.** The concluding section of the journal paper (p. 378; the same
  sentences are in the first arXiv version) says that it is "not difficult to see" that crossings can be avoided
  unless every realization lies in a plane, and that the question for the minimum number of crossings then
  reduces, by Theorem 1 of that paper, to a test whether a sequence has a realization not contained in a plane.
  The first assertion is the implication (3) of the note: a realization not contained in a plane ⇒ a simple
  realization. The note credits it (Section 1.5, abstract, Remarks 7.3 and 7.4, "Scope and priority") and gives
  a proof (Proposition 4.1 with Remark 7.4). Granted (3), Theorem 1.3(b) follows from Theorem 1 and Lemma 1 of the
  source and from the equality cases as soon as one knows which sequences have a realization not contained in a
  plane; this is the test which the source leaves open, and its answer is Remark 7.4 (exactly the sequences of
  the open region G_n).
- **What the note adds** (to the author's knowledge; no priority is claimed): the observation that tight sets
  with at least five elements other than the set of all indices occur and are obstructions to simple
  realizations (Theorem 1.2(c) for S ≠ Z_n), so that the conjecture fails; the counterexamples A* and A_n; a
  proof of the implication (3), by an induction which inserts one vertex at a time (Section 4); the answer to the
  test (the "if" part of Remark 7.4, which with (3) is Theorem 1.3(c)); and the resulting characterization,
  Theorem 1.3(b) and Corollary 1.4, with all boundary cases.

The note is unrefereed.

## Statement checked
- **Primary sources.**
  - C. D. Tóth (joint work with A. Efrat, R. Fulek and S. Kobourov), "Polygons with Prescribed Angles in 2D and
    3D", abstract in Oberwolfach Reports 17 (2020), no. 2, Report No. 30/2020 "Discrete Geometry (hybrid
    meeting)", pp. 1512–1514 (the report: pp. 1469–1528, doi:10.4171/OWR/2020/30, published 2021). Conjecture 5 is
    on p. 1514, with n ≥ 3; the abstract says that its "only if" part can be proved.
  - A. Efrat, R. Fulek, S. Kobourov, C. D. Tóth, "Polygons with Prescribed Angles in 2D and 3D", J. Graph
    Algorithms Appl. 26(3) (2022) 363–380, doi:10.7155/jgaa.00599 (arXiv:2008.10192). Conjecture 1 is on p. 378,
    with n ≥ 4, introduced as the converse of Theorem 5; it is stated as open.
  - All these texts were fetched again and read completely by verification run 2 (2026-10-10): the publisher's
    file of the report (sha256 `d31cd6e3…09fb`), the sources of arXiv:2008.10192v1 (sha256 `e93290f3…5c4f`) and v2
    (sha256 `59515dc0…88aa`), and the journal's file (sha256 `5dbe8cf8…9117`).
- **Conventions of the source, used in the note.** Turning angles (not angles between adjacent edges) in the
  open interval (0, π); closed oriented polygons in R^3; no genericity is required of a realization; polygons
  contained in a plane are realizations (Theorem 5 of the source is about such sequences); "self-intersection
  free" means that the closed polygonal curve is simple.
- **Versions of the conjecture.**

  | version | date | range of n | condition printed |
  |---|---|---|---|
  | arXiv:2008.10192v1, Conjecture 1 | 24 Aug 2020 | n ≥ 3 | n odd or Σ(π − α_i) ≠ π |
  | Oberwolfach abstract, Conjecture 5 | workshop 20–26 Sep 2020 | n ≥ 3 | n even or Σ(π − α_i) ≠ π |
  | arXiv:2008.10192v2, Conjecture 1 | 1 Nov 2020 | n ≥ 4 | n even or Σ(π − α_i) ≠ π |
  | J. Graph Algorithms Appl., Conjecture 1 | June 2022 | n ≥ 4 | n even or Σ(π − α_i) ≠ π |

  The word "odd" in the first version is taken for a misprint (the sequences of Theorem 5 have odd n). With n ≥ 3
  the "only if" part of the second form fails trivially for triangles. The statement refuted in the note is the
  "if" part, for every n ≥ 6, in each of these forms.
- **Corpus record.** ulamai/UnsolvedMath, OWR-1703876-012 (dataset version 1.6.0; upstream status `open`). Its
  statement follows the Oberwolfach abstract (n ≥ 3, "n is even or ..."). The record's list of literature
  sources names other authors for the journal paper than the paper itself has; the link given there leads to the
  paper of Efrat, Fulek, Kobourov and Tóth.

## Readings
| Reading | Answer | Where |
|---|---|---|
| The conjecture as printed in the journal version (n ≥ 4, turning angles in (0, π), any closed polygon in R^3 as a realization) | false for every n ≥ 6 ("if" part); true for n = 4, 5 | Corollary 1.4 |
| The Oberwolfach form (n ≥ 3) | in addition the "only if" part fails trivially for n = 3 | Section 1.2 |
| The form of arXiv v1 ("n odd or ...") | also refuted by A_n for every n ≥ 6 | Remark 3.3 |
| The equivalent formulation given in the source (every realization has a self-intersection iff A is realizable in the plane as a thrackle) | fails for A*: all turns of a thrackle polygon have the same sign, every realization of A* has turns of both signs | Remark 7.1 |
| Entries read as angles between adjacent edges | not the source's reading; the same hexagon would then be the counterexample for (π/6 ×5, 5π/6) | not in the note; examined by verification run A |
| Only generic polygons | the hexagons H and H_1 are generic in the planar sense of the source | Section 3.2 |
| Only realizations not contained in a plane | not the source's reading (its Theorem 5 is about planar realizations); the corresponding true statement is Remark 7.4 | Remark 7.4 |
| Generic angle sequences | the criterion of the conjecture is right outside finitely many hyperplanes | Remark 7.2 |

## Results in the paper
- **Lemma 2.1.** Equality in the spherical triangle inequality for a chain with positive steps and total length
  smaller than π.
- **Lemma 2.2.** Fenchel's inequality for polygons (classical; proof included).
- **Lemma 2.3 (known).** For unit vectors u_i with d(u_{i−1}, u_i) = α_i: e_S(A) ≥ 0 for odd S; if e_S(A) = 0, all
  u_i lie on a great circle with signed steps −α_i on S and +α_i off S, and the configuration is unique up to
  isometry. Credits before the lemma.
- **Lemma 2.4.** At most one odd set is tight (e_S + e_S' > 0 for S ≠ S').
- **Lemmas 2.5, 2.6.** Planar polygons with prescribed signed turning angles of total 2πk, k ≠ 0; a simple planar
  polygon has total signed turning ±2π (angle sum of a simple polygon).
- **Theorem 1.2 (obstruction).** If e_S(A) = 0 for an odd S with |S| ≥ 3: A is realizable; every realization is
  planar with signed turning angles +α_i on S and −α_i off S, total (|S| − 1)π, directions determined up to
  isometry; for |S| ≥ 5 no realization is simple.
- **Section 3.2, Proposition 3.1.** The hexagon H: vertices (0,0), (√3/2, 1/2), ((√3−1)/2, (1−√3)/2),
  ((√3−1)/2, (3√3−1)/2), ((√3+2)/2, −1/2), (1,0); directions (30°, 240°, 90°, 300°, 150°, 180°), lengths
  (1, 1, 2√3−1, 3, 1, 1); signed turning angles (−150° ×5, +30°), total −720°; three proper crossings. H_1 (lengths
  (2, 1, 3√3−2, 5, 2, 2)) has one crossing. The self-contained argument: two spherical triangle inequalities hold
  with equality, the directions are forced to be (0°, 210°, 60°, 270°, 120°, 150°), the total turning is ±720°.
- **Proposition 4.1 with Lemmas 4.2–4.4.** Every A in the open region G_n (Σ α_i > 2π, all e_S > 0) is realized
  by a simple polygon whose directions span R^3: skew quadrilaterals for n = 4, the insertion of a vertex, and the
  merging of two consecutive angles with the smallest sum (nine comparisons of lower and upper bounds; an example
  shows that the minimal choice is needed).
- **Lemmas 5.1, 5.2.** Convex polygons for Σ α_i = 2π; pseudo-triangles for a tight set of three elements.
- **Theorem 1.3, Table 2, Corollary 1.4.** The characterization, the table of all combinations of boundary
  conditions, and the consequences for the conjecture.
- **Remarks 7.1–7.6.** The thrackle formulation; a linear-time test and generic sequences; the minimum number of
  crossings (one for A*; in general (|S| − 3)/2 by Theorem 1 of the source, which is cited and not proved again;
  the reduction of this question to a test for realizations not contained in a plane is the source's remark);
  realizations not contained in a plane exist iff A is in G_n; higher dimensions; Remark 7.6 (added by
  verification run 2): for fixed directions in general position a generic choice of the lengths gives a simple
  polygon, but in general the directions must be changed too (explicit pentagons with spanning directions in
  which two edges cross for all lengths), so that this argument does not prove the implication (3).

## Computations (tests; programs and outputs in reproducibility/)
No theorem, proposition or lemma depends on a computation. The properties of the explicit hexagons can be checked
from the printed coordinates; they were also verified exactly. Each number is given with the program that
produced it (O = `original/`, A = `verification_run_A/`, B = `verification_run_B/`, R2 = `independent_run_2/`).
- **Hexagons (exact, Q(√3)).** O `verify_counterexample.py` (sympy; 36 checks passed), A `a1_hexagon_exact.py`,
  B `b1_hexagon_exact.py`, `writing_stage/make_figures.py` (standard library), R2 `r2_01_hexagon.py` (42 checks
  passed): closure, turning angles, total
  −720°, genericity, crossings e_0×e_2, e_1×e_5, e_2×e_5 for H and e_0×e_2 for H_1. A `a6`: 20,000 random
  realizations of A* in exact arithmetic: none simple; 1, 3 or 5 crossings (16,398 / 3,284 / 318).
- **Criterion (a).** B `b2a_grid_exact.py` (integer arithmetic): "all e_S ≥ 0" against an interval recursion for
  the existence of a closed spherical polygon, for all 69,324,911 sequences α_i = k_i π/m with (n, m) = (3,60),
  (4,30), (5,12), (5,30), (6,12), (6,18), (7,12), (8,6), (9,6) and for samples of 6,000,000 and 3,000,000 sequences
  with (n, m) = (8,12), (10,12): 78.3 million sequences, no disagreement; no sequence with two tight sets; the
  greedy rule of Remark 7.2 is always right; 2,772 and 37,128 grid sequences for n = 6 and 9,702 for n = 7 lie in
  a set Δ_S with 5 ≤ |S| ≤ n−1. Floating point: A `a12` (199,986 sequences), O
  `test_against_paper_algorithm.py` (32,000 sequences against the recursion of the source's Theorem 3; no
  disagreement, none skipped), B `b2b` (3,750 sequences: a polygon constructed for each of the 1,875 satisfying
  (a), none found for the others).
- **Simple non-planar polygons in G_n.** O `test_construction.py`: the construction of Proposition 4.1 on 6,741
  sequences (n = 4..11), 50 digits; angles to 1e-43; simplicity and non-planarity decided in rational arithmetic
  on the computed vertices. A (own implementation, 80–140 digits): 36,518 sequences, n = 4..12 (`a3` 5,800, `a5`
  28,002, `a10` 2,716), including e_S down to 1e-16 and sequences near two faces. B (search without the
  construction; exact rational certificate; angles to 1e-38): 11,427 sequences, n = 4..10 (`b3` 6,709 of which
  6,708 by its first method and one by a second method, `b3x`; `b3_v2` 4,718).
- **Faces with |S| ≥ 5.** O `test_bad_faces_exact.py`: 240 sequences with rational directions, turning number
  (|S| − 1)/2, 9,600 polygons with random rational lengths, all with intersecting non-adjacent edges. A `a9`: 500
  sequences, 12,500 polygons, at least (|S| − 3)/2 crossings, attained. B `b4`: 3,600 rational sequences rigid by
  the exact recursion, 270 real ones numerically, 3,240 polygons all non-simple; `b6`: A_n for n = 6..30. Angle
  sequences of random planar polygons: O `test_end_to_end.py` (24,000 polygons, 1,708 on a face: 1,660 with
  |S| = 3 and 48 with |S| = 5), B `b7` (24,000; 1,722 on a face: 1,680 / 41 / 1 for |S| = 3 / 5 / 7), A `a10`
  (3,000 polygons, 171 on a face).
- **Planar constructions, Lemma 4.4, special angles.** O `test_planar_good_cases_exact.py` (480 pseudo-triangles,
  2,700 convex polygons, rational arithmetic), A `a9` (2,000 + 2,000), B `b5` (1,050 + 1,200); O
  `test_lemma43.py` (5,250 pairs (A, s)), A `a4` (11,372 pairs); A `a5`: all sequences with entries in six sets
  of special angles, n = 4..8 (34,061 sequences up to rotation and reflection): the equivalence of the
  conjecture fails for 0, 0, 1 (= A*), 0, 48 and 292 of them.
- **Verification run 2** (`independent_run_2/`; programs written from the statements of the note; all decisions
  "simple", "not planar", "crossing" in exact arithmetic).
  - `r2_01_hexagon.py`: the hexagons H and H_1 from the printed vertices, e_S(A*) and Remark 3.2, the forced
    directions and their rigidity from exact ranges of chains, A_n for n = 6..40 (exact) and the realization of
    Section 3.3 for n = 7..20: 42 checks passed; 3,000 realizations of A* with random rational lengths have 1, 3
    or 5 crossings (1,909 / 905 / 186).
  - `r2_02_faces.py`: 1,200 points of faces with |S| = 5, 7, 9 (n = 5..10): S is the only tight set, every
    distance d(u_i, u_j) is forced and equals that of the planar pattern, total turning (|S| − 1)π; 480 planar
    patterns with rational directions on such faces, 9,600 polygons with random rational lengths, each with
    intersecting non-adjacent edges (smallest numbers of crossings 1, 2, 3).
  - `r2_03_open_region.py`: the construction of Proposition 4.1 as printed (Lemmas 4.2, 4.3, 4.4), 100 digits.
    1,010 random sequences of G_n, n = 4..8, in eight families (margins down to 1e-10): all polygons simple and
    not planar, largest angle error 4.2e-89. All sequences with entries kπ/M up to rotation and reflection on
    fifteen grids (n = 4..8, M = 5..12; two grids sampled), 35,253 classes: 30,684 in G_n (all simple and not
    planar), 862 with Σα = 2π (convex polygon of Lemma 5.1 simple), 285 with a tight set of three elements
    (pseudo-triangle of Lemma 5.2 simple), 33 with a tight set of at least five elements (198 planar polygons
    with the forced signs, none simple), 3,389 not realizable. Theorem 1.3(b) and the conjectured criterion
    differ for 20 classes, all with n = 6, and agree for all classes with n = 4, 5.
  - `r2_04_criterion.py`, `r2_04b_faces_all.py` (integer arithmetic): 4,665,369 sequences on 13 grids, n = 3..9:
    "all e_S ≥ 0" agrees with the chain criterion for closed spherical polygons; the combinations of boundary
    conditions which occur are those of Table 2; the equivalence of the conjecture fails exactly on the sets
    Δ_S, S odd, 5 ≤ |S| ≤ n − 1 (and, for n = 3, for the triangles); 219,560 points of all sets Δ_S, n = 4..10,
    without exception.
  - `r2_05_misc.py`: Lemma 4.4 against the definition (7,000 pairs (A, s)) and the example after it; Lemma 2.4;
    the formulas of Lemma 4.2; the coordinates of Figures 1 and 3 against the printed data; the pentagons of
    Remark 7.6 in Q(√3) and the statement on directions in general position; the case q = −p of Remark 7.4:
    30 checks passed.
  - `r2_06_search.py` (floating point): a direct search finds a polygon for 160 of 160 sequences satisfying
    the criterion of Theorem 1.3(a) with margin ≥ 0.05 and for none of 160 violating it with margin ≤ −0.05.
- **Re-runs.** On 2026-10-10 the script `reproducibility/run_checks.sh` was run on a copy of the package, one
  process at a time: the 48 checks of its quick part (24 minutes) and the 38 checks of its slow part (46 minutes).
  All 86 recorded outputs were reproduced, identically or up to the fields that record running times. In the
  course of verification run 2 the quick part was run once more, on a copy extracted from the archive (48 checks,
  15 minutes): again all outputs identical, or identical up to running times. The
  recorded outputs in `original/outputs/` are those of the first run of the programs after the three
  corrections (nine of the sixteen files are byte-identical to the outputs of the first version); before the
  corrections, verification run B had reproduced all sixteen outputs of the first version byte for byte.
  Details: `reproducibility/README.md` and `reproducibility/RERUN_LOG.txt`. The programs of run 2 have their own
  script, `reproducibility/independent_run_2/run_run2.sh`; its quick part (66 checks, 6 minutes) was run on a
  copy extracted from the archive, with all outputs identical or identical up to running times.

## Independent verification runs
The results were first obtained with proofs and with the programs in `original/`. Three independent verification
runs, all AI-assisted, followed on 2026-10-10; each wrote its own programs. Runs A and B examined the first
written version of the results, and neither saw the report of the other: run A examined the statement and the
proofs; run B made independent computations, read the original programs, and examined the literature. Run 2
examined the final text of the note.

| Item | Run A | Run B | Run 2 (final text) |
|---|---|---|---|
| Statement, conventions, printed versions against the source texts | CONFIRMED (no other reading supported by the source rescues the conjecture) | CONFIRMED; found the first form in arXiv v1 | CONFIRMED (abstract, journal version and both arXiv sources read completely) |
| Counterexample A*, the hexagons, the family A_n | CONFIRMED (proof re-derived; own exact arithmetic) | CONFIRMED (own exact arithmetic; rigidity also by an exact machine computation, n ≤ 30) | CONFIRMED (re-derived; recomputed exactly; n ≤ 40) |
| Theorem 1.2 (obstruction) and Lemmas 2.1–2.6 | CONFIRMED_WITH_FIXES (mathematics correct; Lemma 2.1 to be stated for positive steps; Lemma 2.3(b) to be attributed) | tests agree; Lemma 2.3 found to be known in both parts | CONFIRMED (re-derived) |
| Theorem 1.3(a) | CONFIRMED | CONFIRMED as a test (78.3 million exact comparisons) | CONFIRMED (re-derived; 4.67 million exact comparisons of its own) |
| Theorem 1.3(b), (c): Lemmas 4.2–4.4, Proposition 4.1, Lemmas 5.1, 5.2, the table of cases | CONFIRMED (every proof re-derived; the table found exhaustive) | CONFIRMED as tests (no sequence of G_n without a simple non-planar polygon found) | CONFIRMED (re-derived; constructions carried out as printed on 31,694 sequences and 1,147 planar classes) |
| Corollary 1.4 | CONFIRMED | consistent with all grid counts | CONFIRMED_WITH_FIX ((b) in its present form is correct for every n ≥ 6; "S odd" made explicit) |
| Remark 3.2; the case q = −p of Remark 7.4; the figures | — (added later) | — (added later) | CONFIRMED_WITH_FIXES (range of δ; two steps written out; two labels) |
| Programs of the first version | three of them re-run: identical | all 16 outputs reproduced byte for byte; CONFIRMED_WITH_FIXES (three defects in test programs) | the three corrected programs read: sound; quick part of the package re-run from the archive: 48 of 48 reproduced |
| Cited statements (source; Kapovich–Millson; Mondello–Panov; appendix by Mamaev) | definitions and theorems of the source confirmed | Kapovich–Millson and Mondello–Panov read | CONFIRMED_WITH_FIXES (all citations accurate; one assertion of the source was not credited; two attributions made precise) |
| Novelty | not its part (one web search: nothing found) | CONFIRMED_WITH_FIXES: no disproof, proof or modification of the conjecture found; credits to be corrected | CONFIRMED_WITH_FIXES: nothing found; the list of statements "not in print" narrowed |

No run found a wrong statement or a gap in a proof.

**Corrections required by run A**, all made in the note:
1. Lemma 2.1 is stated for positive steps (the degenerate cases of the first version are excluded).
2. The equality case, Lemma 2.3(b), is attributed (paragraph before Lemma 2.3; Sections 1.5 and "Scope and
   priority").
3. The criterion of Buckman and Schmitt is described as cited, not printed, in the source; references in which
   the inequalities can be read are given (Kapovich–Millson, Mondello–Panov).
4. Lemma 5.2: the reason why the chain lies in the triangle D_x (three half-planes).
5. The example (9π/10, 9π/10, π/10, π/10, π/10) after Lemma 4.4, showing that the minimal choice of s is needed.
6. Remark 7.1 holds for n ≥ 4; Remark 7.4 has a direct proof of the equality case of Fenchel's inequality.

**Corrections required by run B**, all made:
1. Credits and the wording about novelty (abstract, Section 1.5, paragraph before Lemma 2.3, "Scope and
   priority").
2. The history of the versions (Table 1) and the sentence that A_n has Σ(π − α_i) = (n−5)π + 2π/3 ≠ π, so that
   every printed form is refuted (Remark 3.3).
3. `original/scripts/test_against_paper_algorithm.py`: the rule for skipping samples is now symmetric
   (|min e_S| < 1e-9 only); re-run: 32,000 samples, 0 disagreements, 0 skipped.
4. Non-planarity is decided exactly, in rational arithmetic on the computed vertices (`lib_polygons.py`,
   function `is_nonplanar_exact`), in `test_construction.py`, `test_end_to_end.py`,
   `demo_near_counterexample.py`, `normalization_checks.py`; re-run: no failure.
5. `test_end_to_end.py`: a larger sample of planar polygons is classified (24,000), and the number of cases on
   a face is reported (1,708; 48 of them with |S| = 5). The first version had reached a face 17 times.
6. The sources which were not read are listed ("Scope and priority").

**Run 2** examined the text written after runs A and B, and first what no run had examined: the statement of
Corollary 1.4(b), Remark 3.2, the case q = −p of Remark 7.4, the figures, and the three corrected programs. Its
main finding concerns credit: the text which it examined did not mention the assertion of the source described
above ("Asserted in the source without proof"), and counted Theorem 1.3(c) among the statements not in print.

**Corrections required by run 2**, all made:
1. Credit for the assertion of the source (implication (3)): Section 1.5, abstract, Remarks 7.3 and 7.4, "Scope
   and priority"; the list of what is not in print is narrowed accordingly.
2. Remark 7.6 (new): directions in general position; the pentagons; no short proof of (3) is known to us.
3. Remark 7.4: the corollary of the appendix by Mamaev (a polygon not contained in a great circle exists in
   S^3 when all inequalities are strict) and the place where the source notes the equality case of Fenchel's
   theorem are mentioned.
4. Remark 3.2: "for −π/6 < δ < 0" (for δ ≤ −π/6 the last entry is not an angle).
5. Remark 7.4, case q = −p: the application of Lemma 2.1 and the choice of the vector m are written out.
6. Paragraph before Lemma 2.3: Mondello and Panov say that their theorem is essentially contained in the paper
   of Biswas.
7. Corollary 1.4(b): "the sets Δ_S, S odd, with 5 ≤ |S| ≤ n − 1".
8. Lemma 2.5: the source refers to Garg for the criterion; the reference is added (not read).
9. Sections 3.2 and 8: the hexagons are verified by five programs; the computations of run 2 are described.
10. The paragraph "Verification" is rewritten to the present state.
11. "Scope and priority": what was read; the assertion of the source; the searches of run 2.
12. Figure 3: labels moved off the chain.
13. Package: the programs and outputs of run 2 (`reproducibility/independent_run_2/`), this report, the README.

Changes of the text made together with these corrections, to keep the length: some sentences of Sections 1.4,
1.5, 7, 8 and of the closing paragraphs are shortened; the three figures are drawn slightly smaller; the four
citing works are named in "Scope and priority" instead of in the list of references; equation numbers from (3)
on are shifted by one.

**Not examined by a further independent run:** the corrections required by run 2 as they are now worded, and
Remark 7.6. The example of Remark 7.6 is verified in exact arithmetic by `independent_run_2/r2_05_misc.py`.

## Relation to the literature, novelty and scope
- **Searches (10 October 2026; first results and the three verification runs).** arXiv (keyword and author
  queries; titles, abstracts and comments are searched there), OpenAlex (titles, abstracts, full texts, lists of
  citing works), Semantic Scholar and OpenCitations (citing works), zbMATH, Crossref (no correction registered
  for the journal article or the proceedings chapter), DataCite, Zenodo, and eight web searches. Run 2 repeated
  the searches on the same day (12 arXiv queries, the lists of citing works, zbMATH, DataCite and Zenodo for the
  corpus identifier and the title, one web search), with no new result.
  - No publication was found that disproves, proves or modifies the conjecture. The journal version of June 2022
    states it as open; arXiv has no version after v2.
  - Four citing works were found: M. Schaefer, J. Graph Algorithms Appl. 25(1) (2021) 195–218,
    doi:10.7155/jgaa.00557; C. D. Tóth, WALCOM 2024, LNCS 14549, 15–31, doi:10.1007/978-981-97-0566-5_3
    (arXiv:2311.15043); R. A. Moy, J. Schmurr, J. Varlack, Rocky Mountain J. Math. 56(2) (2026) 475–501,
    doi:10.1216/rmj.2026.56.475 (arXiv:2306.01633); A. Baumann, T. Biedl, M. Elashmawi, S. D. Fink, M. Saumell,
    A. Schulz, arXiv:2607.01423. They were read at the place of the citation by the first two runs; none
    discusses the conjecture. The note names them in "Scope and priority".
  - In the source, tight sets other than the set of all indices do not occur. The source does assert, without
    proof, that crossings can be avoided unless all realizations are planar (see the verdict above); this is
    credited in the note.
- **What was read.** The Oberwolfach abstract (pp. 1512–1514); the journal version of the source and the sources
  of arXiv v1 and v2, completely (run 2); Kapovich–Millson, Sections 1–3 (definitions, Theorem 2.9, Theorem 3.1,
  Lemma 3.2, Theorem 3.3); Mondello–Panov 2016 (Theorem 2.22, Lemma 2.24, with their context) and the statements
  of the appendix by Mamaev in Mondello–Panov 2024, both in the arXiv versions.
- **Not read.** The printed chapter in the proceedings of Graph Drawing 2020 (LNCS 12590), to which arXiv v1
  corresponds according to its arXiv comment; the Buckman–Schmitt preprint (the address given in the source did
  not answer); Galitzer's thesis (known only through Kapovich–Millson); Garg 1998 (known only through the
  source); Biswas 1998 beyond its zbMATH review; the journal versions of the two papers of Mondello and Panov;
  Fenchel 1929, Horn 1971, Sullivan 2008 and the book of de Berg et al., cited for classical facts. The proof of
  Theorem 1 of the source, which is used only in Remark 7.3 and in the comments of Section 1.5, was not checked.
- **Bibliography.** All entries with a DOI were checked with Crossref on 2026-10-10, and again by run 2 (fifteen
  DOIs, including that of Garg's paper and those of the four citing works), the arXiv identifiers with the arXiv
  API, and the proceedings volumes and the series of Sullivan's chapter with zbMATH; the requests are logged.
- **Limits.** MathSciNet and Google Scholar were not available; DBLP did not answer automated requests; the
  works citing Kapovich–Millson or Mondello–Panov were not examined. A search that finds nothing is not a proof
  of novelty; the authors of the conjecture are active in this field. No priority is claimed.
- **Scope.** The note refutes the "if" part of the conjecture for n ≥ 6, proves it for n = 4, 5, and proves the
  characterization of Theorem 1.3. It does not claim as new the inequalities on the sphere, their equality case,
  the realizability criterion, the "only if" part, or the statement that a sequence with a realization not
  contained in a plane has a simple realization, which the source asserts and for which the note gives a proof.

## 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/