# Verification report — OWR-3385-011 (O'Rourke's question in Rote's list, Oberwolfach 2009: developments of star-shaped spherical polygons)

Verification date: 2026-10-16.

**Verdict.** The note gives a **partial answer**; every statement it makes is proved or certified, except those
labelled as numerical, and the scope is as follows.
- **Settled, in the every-cut reading.** There is a simple spherical heptagon Q with rational vertices, in the open
  hemisphere about N = (0, 0, 1) and star-shaped with respect to N (N in the interior of its region), whose
  development cut at the vertex v0 is not simple: its first and last edges cross transversally (Theorem 1.1(a),
  (b)). Every point of an open arc of length > 0.118 around v0 is a bad cut point (Theorem 1.1(c)). So star-shaped
  spherical polygons need not develop without self-intersection when the development must be simple for every cut
  point, and also when only cuts at vertices are considered (Corollary 1.2). The theorems of O'Rourke-Schevon
  (1989) and O'Rourke (2003, Sec. 4) name no cut point, and O'Rourke-Vîlcu (arXiv:1102.0823, Sec. 7.1-7.2) single
  out the closed curves whose development is simple for every cut point; this every-cut convention is O'Rourke's
  usage, not a definition stated in the source, and every statement of the note names it.
- **Certified side results.** The developments of Q cut at v1, ..., v6 are simple (Theorem 1.1(d)). A second
  rational heptagon Q', star-shaped in the same sense, has a non-simple development when cut at its vertex farthest
  from N (Proposition 1.3; certified by the program of the note and by the second and third verification
  runs).
- **Not new.** The question is O'Rourke's (Rote's list, Problem 13). The convex case recorded in the source is the
  spherical counterpart of the theorem of O'Rourke and Schevon (1989) on closed convex curves on convex polyhedra,
  extended by O'Rourke (2003) to slice curves; the cone analogue of the star-shaped question (closed curves on a
  cone of cone angle at most 2 pi that enclose the apex and meet every generator once develop simply for every cut
  point) is due to O'Rourke and Vîlcu (arXiv:1102.0823, Sec. 1, 7.1 and footnote 2); MathOverflow question
  100496 (O'Rourke, 2012, unanswered) asks a related question for cones. The note neither uses nor reproves the
  convex case.
- **Open.** Whether every star-shaped spherical polygon has at least one cut point with a simple development
  (Question 5.1; numerical evidence only, Section 5); the open-ended request for conditions that guarantee a simple
  development; and part (2) of the record (Problem 14 of the same list, A. Bobenko), which the note does not treat.

The note is unrefereed. Suggested status of the corpus record: partially solved.

## Statement checked
- **Primary source.** G. Rote (collector), *Open problems in discrete differential geometry*, in *Discrete
  Differential Geometry* (organised by A. I. Bobenko, R. Kenyon, J. M. Sullivan, G. M. Ziegler), Oberwolfach
  Reports 6 (2009), report 2/2009, pp. 106-110, doi:10.4171/OWR/2009/02 (Crossref: Oberwolfach Reports 6(1),
  75-144, the whole report). Problem 13 (Joseph O'Rourke) is on p. 110.
  - Read in the open version on G. Rote's page (https://page.mi.fu-berlin.de/rote/Kram/OWR-DDG09-problems.pdf,
    5 pages) and in the report (EMS Press PDF, 70 pages, md5 20c17311d70bb36a140e79228b3cdaaa, fetched
    2026-10-16). The wording of Problem 13 is identical in both.
  - Problem 13 has four sentences: develop a simple piecewise-geodesic curve on S^2 onto the plane (same lengths
    and angles); find conditions which guarantee that the developed image does not intersect itself; this holds
    for a closed convex polygon on S^2; "What about star-shaped polygons?". No cut-point convention is stated.
  - In the same report, O'Rourke's abstract (joint with J. Itoh and C. Vîlcu, pp. 129-131) develops a closed curve
    as the first step of a source unfolding and recalls the convex theorem of O'Rourke-Schevon.
- **Corpus record.** ulamai/UnsolvedMath, OWR-3385-011 (dataset version 1.6.0; status `open`). Part (1) is a
  faithful paraphrase of Problem 13; part (2) is Problem 14 (discrete integrable systems in dimension four and
  higher), which the note does not treat.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Development simple for every cut point (O'Rourke's usage) | no: Q cut at v0 | Theorem 1.1(b), Corollary 1.2 |
| Development simple for every cut at a vertex | no: the bad cut v0 is a vertex | Corollary 1.2 |
| Strong notion of star-shaped: centre in the interior of the region, polygon in the open hemisphere about the centre (Q is a polar graph about N) | still no | Theorem 1.1(a), Lemma 2.2 |
| Touching instead of crossing | the crossing is transversal; the four orientation determinants are at least 0.034 in absolute value | Section 3.4 |
| One good cut point suffices | open; Q and Q' have good cut points; numerical evidence | Question 5.1, Section 5 |
| Rule "cut at the vertex farthest from the star centre" | fails: Q' | Proposition 1.3 |
| "Find conditions which guarantee ..." (open-ended) | not treated | — |
| Part (2) of the record | not treated | — |

## Results in the paper
- **Lemma 2.2 (polar graphs).** Vertices with consecutive longitude gaps in (0, pi) and winding number 1 give a
  simple polygon whose region containing N is star-shaped with respect to N, with N interior; in the open
  hemisphere if all vertices are. Only this direction is claimed (the converse fails, e.g. for an edge on a
  meridian).
- **Definition 2.3 and Lemma 2.4.** The development cut at a vertex or at an interior point of an edge ("same
  lengths and turning angles"); a development contains a congruent copy of the development of every sub-arc not
  containing the cut point in its interior.
- **Theorem 1.1.** (a) Q is simple, in {z > 0}, star-shaped with respect to N, N interior. (b) The development cut
  at v0 is not simple: segments 0 and 6 (images of e0, e6) cross transversally, at arc-length fractions
  0.066993... of e0 and 0.900760... of e6. (c) Every cut point in the open arc of Q between these two points
  through v0 is bad; the arc has length in [0.11835, 0.11836]. (d) The developments cut at v1, ..., v6 are simple.
- **Corollary 1.2.** The answer to the star-shaped question is no in the every-cut reading and in the every-vertex
  reading.
- **Proposition 1.3.** Q' (Table 3): star-shaped, v5' farthest from N, development cut at v5' not simple; the
  vertex cuts are not simple except at v3'.
- **Remark 4.1 (numerical).** The bad cut points of Q, by floating-point sampling: v0; t in (0, 0.0670) and
  (0.1992, 0.2550) on e0; t in (0.7195, 0.8079) and (0.9008, 1) on e6 (these are the intervals of the rational
  heptagon Q, from the second verification run's 30,001-point scan, confirmed by the third run's scan with
  bisection; the first computations had quoted the intervals of a nearby floating-point polygon). Ten cut points
  are certified.
- **Remark 4.2.** Copies of Q scaled towards N (stereographic data times 1/2, 1/5, 1/20) are certified to have the
  same behaviour; the decrease of the crossing depth like s^4 is numerical.
- **Question 5.1** (open) with numerical evidence: no polygon with every cut point bad was found; best fractions of
  bad cut points 0.965, 0.993, 0.998 for minimum polar distances 0.03, 0.005, 0.001, the near-misses degenerating
  into two lobes joined at N.

## Computations (programs and outputs in reproducibility/)
- **The program of the note** (`check_note.py`, Python standard library only, about 2.5 s). Exact rational
  arithmetic and its own interval arithmetic (dyadic endpoints, every arithmetic result rounded outward to a
  multiple of 2^-320; isqrt; arctan by halving and the alternating series; Machin's formula). It checks Table 1
  against the stereographic formula; Theorem 1.1(a) by Lemma 2.2 and independently by the exact gnomonic image;
  Theorem 1.1(b) (signs, determinants to 20 decimals, distances, crossing fractions), the length of the arc of
  Theorem 1.1(c) and (d); the Gauss-Bonnet consistency check (enclosure width < 1e-90); Proposition 1.3 and
  Table 3 (with the exact vertices of Q', the value of z5' and the bound 0.24 for the determinants of the cut at
  v5'); the ten certified cut points of Remark 4.1; Remark 4.2; a convex control hexagon; and tests of its own
  arithmetic. Every enclosure used has width < 1e-80. Output: `check_note_output.txt`, last line
  `ALL CHECKS PASSED`. The arc length, the vertices of Q' with z5' and the bound 0.24 were added after the third
  verification run; the other output lines did not change.
- **Programs with which the results were first obtained** (`original/`): floating-point developments
  (`dev_core.py`); certificates in Arb ball arithmetic (`certify_arb.py`, python-flint) and mpmath interval
  arithmetic (`certify_iv.py`); a law-of-cosines angle check; the search programs and their saved results.
- **Re-runs at the writing stage** (from a copy of the package): the outputs of `check_note.py`, of the four
  original certificate and check programs (the vertex file generated by `certify_arb.py` is identical to
  `vertices_rational.json`), and of all programs of both verification runs except the time-boxed searches were
  identical to the recorded outputs. A 5-second run of `hfar_search.py` with seed 41 reproduced the
  farthest-vertex polygon exactly. The fractions 0.9851 and 0.9951 of the second run's best polygons were
  recomputed with the first run's `fcuts.py` (`paper_stage/`).
- No mathematical claim depends on a floating-point computation: all certified statements are decided by exact
  rational arithmetic or by interval enclosures; the floating-point results appear only as numerical evidence
  (Remark 4.1, Remark 4.2, Section 5).

## Independent verification runs
The result was first obtained with proofs and programs. Two independent verification runs, both AI-assisted,
followed (2026-10-16); each wrote its own programs. Run A examined the statement, the proofs and the certificate;
run B made an independent computation and the literature and novelty check.

| Item | Run A | Run B |
|---|---|---|
| Statement and readings against the source (Problem 13 read in full) | CONFIRMED; every-cut reading is O'Rourke's usage, record PARTIAL | CONFIRMED; record PARTIAL |
| Lemma 2.2 (polar graph => simple, star-shaped) | PROVED (re-derived; boundary cases) | proof read; correct |
| Development (same lengths and angles; rolling) | CHECKED (rolling-sphere simulation with rotation matrices, 60 digits: mirror image of the same chain) | CHECKED (geodesic-flow development in floating point) |
| Theorem 1.1(a) | CHECKED exactly by two routes (polar graph; gnomonic image) | CHECKED exactly (gnomonic image) |
| Theorem 1.1(b), (d) | CERTIFIED with its own exact rational interval arithmetic (no angles; arcsine series), determinants (-0.467474, +0.051503, +0.034768, -0.484209) | CERTIFIED with its own interval arithmetic on rationals; byte-identical re-runs of the original programs |
| Bad cut points (bookkeeping behind Theorem 1.1(c)) | PROVED (with one imprecise remark, not used in the note) | — |
| Gauss-Bonnet | area by l'Huilier's formula, residual -5.6e-60 | enclosure of width 3e-89 containing 0 |
| Interior cut points (Remark 4.1) | floating point, 2000 samples per edge | 30,001-point scan; ten points certified |
| Proposition 1.3 (farthest vertex) | floating point | CERTIFIED (rational version, stereographic data rounded to 1e-7) |
| Scaled copies (Remark 4.2) | floating point, s down to 0.02 | CERTIFIED for s = 1/2, 1/5, 1/20 |
| Searches (Section 5) | two saved polygons re-evaluated | saved polygons re-evaluated; own search (n = 8..16, two 20-minute runs): best 0.985 and 0.995, none all-bad |
| Deliberately wrong inputs and controls | 300 random convex polygons: no bad cut | off-sphere vertex, swapped vertices, doubled polygon, wrong turning angle, convex hexagons: all detected or behaving as expected |
| Literature and novelty | sources checked; no prior answer found | sources checked; no prior answer found; access limits listed |

Neither run found an error in the counterexample or in the proofs used by the note.

**Corrections required by run A** (all applied in the note): (1) record-level label PARTIAL, with the every-cut
reading named, and the some-cut variant, the request for conditions and part (2) stated as open; (2) only the
direction "polar graph => star-shaped" is claimed for Lemma 2.2; (3) the imprecise remark on bad cut points is
not part of the note; (4) the interior intervals are those of the rational heptagon; (5) the cone result of
O'Rourke-Vîlcu needs every generator to meet the curve once, and their notion of living on a cone does not apply
on S^2; (6) the arXiv version is cited under its own title, with C_x located in Sec. 7.1; (7) the convex case is
described as recorded in the source, with O'Rourke-Schevon and the arm lemma of Abel et al. as the tools; (8)
"arbitrarily small" is marked as numerical; (9) the shrink family and footnote 2 of O'Rourke-Vîlcu were added;
(10) the every-cut convention is described as O'Rourke's usage, not as a definition of the source.

**Corrections required by run B** (all applied): (1) record-level label PARTIAL; (2) the intervals
(0.1992, 0.2550) and (0.7195, 0.8079) of the rational heptagon; (3) the proved direction of Lemma 2.2; (4) the
visibility condition of the cone result, the arXiv title, and the statement that the journal version was not
consulted; (5) the usage line of `certify_arb.py`: its argument is an output file (reproducibility/README.md);
(6) footnote 2 and Sec. 7.1 of O'Rourke-Vîlcu as the motivation.

**At the writing stage**, the program `check_note.py` was written, with its own arithmetic; Theorem 1.1(c) was
stated as the special case of the bookkeeping lemma re-derived by run A; the bibliographic data were checked with
Crossref; the report version of the source was read.

### Third verification run (independent run 2, on the written note; 2026-10-16)
A further independent verification run, also AI-assisted, examined the written note and the package, including
the parts added at the writing stage (Theorem 1.1(c), Lemma 2.4, Proposition 1.3, Remarks 4.1-4.2 and the
arithmetic of `check_note.py`). Its programs and outputs are in `reproducibility/independent_run_2/`.

| Item | Third run |
|---|---|
| Statement and scope | Problem 13 re-read in both versions (open list; report pp. 106-110, Problem 13 on p. 110), O'Rourke's abstract (pp. 129-131), O'Rourke 2003 Sec. 1 and 4, O'Rourke-Vîlcu arXiv v2 (Sec. 1, 7.1, 7.2, 8.3, footnote 2), MathOverflow 100496: CONFIRMED; record PARTIAL |
| Lemma 2.2, Definition 2.3, Lemma 2.4, (D), (X) | PROVED (re-derived line by line) |
| Theorem 1.1(c) | PROVED (re-derived; transversality of the shortened segments); arc length CERTIFIED in [0.11835420, 0.11835421]; rigorous spot checks at 11 points inside the arc and at one point just outside each end |
| `check_note.py` arithmetic | REVIEWED line by line: outward rounding, square-root bounds, alternating-series bounds, halving and reciprocal branches of arctan, Machin's formula, (D)/(X): correct |
| Theorem 1.1(a)-(d), Tables 1-3, Section 3 numbers | CERTIFIED with its own standard-library arithmetic: decimal grid 10^-120; arccos by verified cosine brackets (Taylor polynomial with remainder); developed directions as exact Gaussian rationals over one square root; all printed determinant intervals contain its enclosures |
| Gauss-Bonnet | EXACT identity of Gaussian rationals (multiple of 2 pi fixed by enclosures) |
| Proposition 1.3 | CERTIFIED: z5' = 434355078771/19565644921229 smallest, crossing pairs of Table 3, cut at v3' simple, eight determinants at least 0.2479 in absolute value |
| Remark 4.1, Remark 4.2 | ten points CERTIFIED; floating-point scan with 4000 points per edge and bisection: (0, 0.066994) and (0.199227, 0.254987) on e0, (0.719468, 0.807874) and (0.900760, 1) on e6, 4.572 % of the perimeter; Q_s CERTIFIED for s = 1/2, 1/5, 1/20 |
| Development by rolling | mpmath, 50 digits: mirror image of Definition 2.3 for all seven vertex cuts, to 1e-50 |
| Section 5 | all eleven saved best polygons re-evaluated by direct developments: fractions as stated; one good arc each, of 6.3 to 11.1 times r_min |
| Package | re-run from its archive: all 20 recorded outputs byte-identical; `hfar_search.py` (seed 41) and `hyptest.py 7 1.55 12000 31` reproduced the saved results; after the corrections the final archive was re-run in the same way, with the same result for all 24 recorded outputs (including the new output of `check_note.py` and the four of this run) |
| Literature | bibliography checked with Crossref and arXiv abstract pages; Crossref, zbMATH Open, OpenAlex searches and one web search: no prior answer found |

It found no mathematical error. **Corrections required by the third run** (all applied): (1) the quantifier of
Theorem 1.1(c) made explicit ("for every point p of the open arc ..., the development cut at p is not simple");
(2) `check_note.py` now computes the arc length of Theorem 1.1(c), prints the vertices of Q' and checks
z5' = 434355078771/19565644921229, and checks the bound 0.24 of the proof of Proposition 1.3 (the note had
attributed these to the program); (3) the stereographic coordinates are written xi_i, eta_i, so that u_j, w_j
denote only the developments; (4) the abstract and Sec. 1.1 say that the theorems of O'Rourke-Schevon and
O'Rourke 2003 name no cut point (no corner is singled out) and that O'Rourke-Vîlcu single out the curves with
C_x simple for every x; (5) cones in the sense of O'Rourke-Vîlcu have cone angle at most 2 pi; (6) the
description of the arithmetic in Sec. 3.3 (dyadic end points, rounding to multiples of 2^-320); (7) this run is
recorded in the note and here; (8) the searches of this run are added to "Scope and priority"; (9) the README
gives the exact command of the hypothesis test (r_max = 1.55). Optional changes also made: polar distances below
1.55 named in Sec. 5; "as a connected part" in Lemma 2.4; "end points rounded" in Remark 4.1.

## Relation to the literature, novelty and scope
- **Searches (2026-10-16).** The report containing the problem list, with O'Rourke's abstract; O'Rourke 2003
  (Comput. Geom. 24, doi:10.1016/S0925-7721(02)00044-5; arXiv cs/0006035 v4, Sections 1 and 4 read);
  O'Rourke-Vîlcu, arXiv:1102.0823 v2 (*Conical existence of closed curves on convex polyhedra*, read in full;
  journal version *Development of curves on polyhedra via conical existence*, Comput. Geom. 47 (2014) 149-163,
  doi:10.1016/j.comgeo.2013.08.010, not consulted); the abstract of Abel et al., arXiv:0804.0986; MathOverflow
  through the StackExchange API (question 100496 the only relevant result; no answer as of 2026-10-16); the works
  citing O'Rourke 2003 (21), O'Rourke-Vîlcu 2014 (7) and arXiv:1102.0823 (4) listed by Semantic Scholar (titles and
  abstracts; later papers of O'Rourke, Vîlcu and Itoh partly searched in full); the open-problem list on the web
  site of the book *Geometric Folding Algorithms*; queries to the arXiv API, Crossref, OpenAlex and zbMATH Open;
  eight web searches.
  - No printed answer to the star-shaped question, and no statement of the some-cut variant for spheres, was found.
  - Limits: the arXiv export API and OpenAlex refused part of the requests (rate limits); the works citing
    O'Rourke-Schevon 1989 (seven in OpenAlex) were not listed; the journal version of O'Rourke-Vîlcu and the
    book *Geometric Folding Algorithms* were not read. A search that finds nothing is not a proof of novelty, and
    no priority is claimed.
- **What was read.** Problem 13 in both versions; O'Rourke's abstract in the report; O'Rourke 2003, Sections 1 and
  4; O'Rourke-Vîlcu arXiv v2 (Sections 1, 7.1, 7.2, 8.3 and footnote 2 are cited); MathOverflow 100496; the
  abstract of Abel et al. O'Rourke-Schevon 1989 was not read; its theorem is quoted as stated in O'Rourke 2003,
  Sec. 4, and O'Rourke-Vîlcu, Sec. 7.1. Van Oosterom-Strackee (1983) was not read; its formula for the area of a
  spherical triangle is used only in the consistency check. All DOIs were checked with Crossref.
- **Scope.** The note answers the star-shaped question in the every-cut reading only. The some-cut variant, the
  open-ended request for conditions and part (2) of the record remain open. The contribution of the note is the
  certified star-shaped counterexample on the sphere for the every-cut version, the certified failure of the
  farthest-vertex rule, and numerical evidence on the some-cut version.

## 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/