# Verification report — OWR-13498-009 (Amenta, Oberwolfach Report 13/2015, Problem 4: are face angles of a nonconvex polytope determined by its dihedral angles?)

Verification date: 2026-10-20.

**Verdict.** The note answers **clause 1** of the record (Problem 4, N. Amenta) **in the negative, by one explicit pair**;
every statement it makes is proved, and the scope is as follows.
- **Settled.** Two embedded nonconvex polyhedra P and P' realize the same labelled triangulation of the sphere (the
  boundary of the triangular bipyramid, 5 vertices, 9 edges, 6 triangles). No two adjacent faces are coplanar, all
  nine interior dihedral angles are equal (edge a2a3 reflex in both, about 273.8239°; the other eight convex), and
  12 of the 18 face angles differ (all at a1, a2, a3; the 6 apex angles agree). Coordinates in Q(√D), D a 55-digit
  non-square; the proof is elementary lemmas plus an exact finite certificate (Theorem 1.1, Corollary 1.2).
- **Consequence.** The pair refutes, for this mesh, Conjecture 1 of Amenta and Rojas (CCCG 2018, arXiv 1810.01546,
  Section 8; restated in Section 7 of the author manuscript of the journal version, Comput. Geom. 90 (2020) 101657)
  (Corollary 1.3).
- **Not claimed.** No family of pairs (a 7-dimensional count is only a heuristic); nothing on continuous dihedral
  flexes; no statement on convex polyhedra (none can occur in such a pair, by the theorem of the Wang–Xie
  preprint, Remark 6.1; no journal version of that preprint was found).
- **Not addressed.** Clause 2 of the record (Problem 5, G. Rote, offset polytopes); the record's pointer to
  Aurenhammer–Walzl (DCG 56 (2016) 743–801) was not read in full and not checked against every requirement.
- **Novelty not established.** Stoker 1968 and Karcher 1968, the closest prior sources, could not be read
  (publisher bot challenge, no open copy); nor the later papers of Milka (1970) and Pogorelov (2002) on Stoker's
  problem, described in the literature as convex-case results. No printed example was found in the sources searched; a search that
  finds nothing is not a proof of novelty. No priority claim.

The note is unrefereed.

## Statement checked
- **Primary source.** A. I. Bobenko, R. Kenyon, P. Schröder (organisers), *Discrete Differential Geometry*,
  Oberwolfach Rep. 12(1) (2015) 661–729, doi:10.4171/OWR/2015/13; open problems collected by G. Rote, pp. 719–723.
  - Read in the open PDF of the publisher (content.ems.press; 70 pages, 4,345,044 bytes, sha256
    `e3ba6ef5fc327cfe1ca2c3c52a30d8107fbe487f832ae799f6ccd3e8e3da25ca`), volume p. 721 = PDF page 61.
  - PROBLEM 4 (Nina Amenta), "Are face angles determined by dihedral angles?": Stoker's conjecture for convex
    3-polytopes with given combinatorics and all dihedral angles specified (different from 0 and π); question:
    is this also true for a non-convex polytope? One may assume the polytope triangulated and homeomorphic to a
    sphere.
  - PROBLEM 5 (Günter Rote), existence of offset polytopes P_ε for small ε (not treated).
- **Corpus record.** ulamai/UnsolvedMath, OWR-13498-009 (dataset version 1.8.0, status `partially_solved`). The
  record merges Problems 4 and 5 of the same page; the note treats Problem 4 only.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Global determination (two different polyhedra, same combinatorics, same dihedral angles), the quantifier of Stoker's conjecture | no: one explicit pair | Theorem 1.1, Corollary 1.2 |
| Triangulated and homeomorphic to a sphere (allowed by the source) | the pair is triangulated, spherical, embedded, bounds a ball | Theorem 1.1(a), Lemma 3.1 |
| Dihedral angles different from 0 and π | no two adjacent faces coplanar (exact determinants) | [C3] |
| Interior angle in (0, 2π) versus angle between face planes, exterior angle, rotation angle of normals (Amenta–Rojas) | equal interior angles imply equality in all of these (same orientation for the rotation convention) | Remark 2.2, Section 1.3 |
| Up to similarity or reflection | P' is not similar to P (face angles differ); mirror image and similar copies of P' are also valid partners | [C9], [C10] |
| Continuous dihedral flex | not addressed | Section 1.4 |
| Non-triangulated polytopes | elementary examples exist; those known to the note have straight face angles, or dihedral angles π after triangulation | Remark 6.3 |
| Clause 2 (offsets) | not addressed | Scope and priority |

## Results in the paper
- **Lemma 3.1** (embedded bipyramids): a1, a2, a3 affinely independent and x, y strictly on opposite sides of their
  plane imply an embedded polyhedral sphere bounding T_x ∪ T_y.
- **Lemma 3.2** (outward normals): interior angle π − φ or π + φ, far-vertex reflex test. Valid only for an embedded
  surface with its true outward orientation, supplied by Lemma 3.1 and the positive signed volume.
- **Lemma 3.3** (orientation-free): in-face vectors orthogonal to the edge and exact cone membership.
- **Lemma 3.4** (apices): the trihedral apex cones are fixed by their dihedral angles (polar triangle).
- **Theorem 1.1**, **Corollaries 1.2, 1.3**; Remarks 6.1 (no convex member), 6.2 (Euler: 5 vertices, 6 faces is the
  bipyramid), 6.3 (non-triangulated examples), 6.4 (Cho–Kim examples have equal face angles).

## Computations (programs and outputs in reproducibility/)
- **`check_note.py`** (standard library only, 0.3 s, `ALL CHECKS PASSED`): checks [C1]–[C12] of Section 4 in exact
  arithmetic in Q(√D) — certificate checksum, D and the quadratic of λ*, Lemma 3.1 hypotheses, non-coplanarity, an
  embeddedness test not using Lemma 3.1, orientation, all nine dihedral angles by Lemma 3.2 and by Lemma 3.3,
  12 differing face angles, non-similarity, other partners, controls, construction data.
- **`mutation_test.py`** (1 s, `ALL MUTATIONS REJECTED`): six wrong certificates, each rejected.
- **Original programs** (`original/`): `verify_pair.py` + `qd.py` (exact, `ALL CHECKS PASSED`), `crosscheck_mp.py`
  (120 digits, ray-parity interior test, segment–triangle self-intersection test; maximal dihedral difference
  7.7e-121 rad; `CROSSCHECK OK`), four controls, the search and construction scripts.
- All outputs were reproduced from an extracted copy of the source archive on 2026-10-20 (Python 3.13), byte for
  byte, in the third verification run. One line of `verification_run_B/outputs/own_factor_check.txt` that
  `own_factor_check.py` does not print was moved to `own_factor_check_note.txt` in the same folder.

## Independent verification runs
Three independent verification runs, all AI-assisted, each with its own programs. Runs A and B examined the
finder's write-up; the third run (folder `independent_run_2/`) examined the text of the note.

| Item | Run A | Run B |
|---|---|---|
| Statement against the OWR PDF | CONFIRMED (read p. 721) | CONFIRMED (read p. 721) |
| Lemmas 3.1, 3.2, 3.4 | CONFIRMED (re-derived) | CONFIRMED (re-derived) |
| Certificate, exact recomputation | CONFIRMED: own orientation-free program (in-face vectors, cone membership), signs cross-validated at 80 digits | CONFIRMED: own verifier written before reading the claim's code, signs confirmed by 400-digit interval arithmetic |
| Embeddedness without Lemma 3.1 | — | CONFIRMED: exact edge/triangle tests; winding-number typing of every edge |
| Wrong inputs | controls rejected | 8 of 10 mutated inputs rejected; mirror image and similar copy of P' accepted as partners (correct) |
| Construction data and factorization | plausibility only | CONFIRMED (own sympy recomputation) |
| Further pairs (evidence) | — | 21 nontrivial pairs from 3000 nonconvex starts; none with a convex member in 3000 convex starts |
| Literature | Pak's notes added | Amenta–Rojas Conjecture 1 found; Crossref data |
| Overall | CONFIRMED_WITH_FIXES | CONFIRMED (fixes required) |

All required fixes were made; see the note's Verification paragraph. The fixes concerned wording (Karcher's class,
Amenta–Rojas Theorem 2, the heuristic 7-dimensional count), the explicit orientation caveat of Lemma 3.2 with an
orientation-free check, Corollary 1.3, journal data, and the reading of the closest sources.

### Third run (on the text of the note, 2026-10-20)
- **Statement and sources re-read.** OWR p. 721 (open PDF, same sha256); Amenta–Rojas arXiv 1810.01546 (§§1, 2, 3, 6,
  8); the Amenta–Rojas 2020 author manuscript (§1, §2, §7 on the rendered p. 9); Wang–Xie (Theorem 1.6, previous
  work); Cho–Kim (§1.1, §3); Mazzeo–Montcouquiol (introduction, Theorem 1); Biedl et al. (§3, Figure 1 inspected
  as a drawing); Pak's notes (§26.7–26.9, Exercises 26.1, 26.2, 27.3, 32.2 with hints). CONFIRMED.
- **Proofs.** Lemmas 3.1–3.4, the angle test (1), Remarks 6.1 and 6.2 and the proof of Theorem 1.1 checked line by
  line. CONFIRMED (one wording fix in Remark 6.2).
- **Own exact verifier** (`independent_run_2/run2_verify.py`, standard library, written before reading
  `check_note.py`): embeddedness by an exact triangle–triangle test for all 15 pairs of triangles (neither Lemma 3.1
  nor the segment–triangle test); type of every edge by an exact point-in-polyhedron test (ray parity), agreeing
  with Lemma 3.2 and Lemma 3.3 at all 18 edge instances; all 9 dihedral angles equal, a2a3 the only reflex edge
  (273.823852°); exactly 12 face angles differ, the 6 apex angles agree; P′ not similar to P; all 35 printed values
  of Theorem 1.1 and Tables 1–2 match; the line xy meets H beyond a2a3 (barycentric (−0.425, 0.758, 0.667) for P).
  CONFIRMED.
- **Mutations** (`run2_mutations.py`): 10 wrong or trivial inputs rejected by geometric checks alone (P against
  itself, its mirror image and a similar copy; x′ moved by 1e-30; a3′ moved by 1e-6; x moved by 1e-20; D + 1; labels
  a1, a2 and x, y exchanged; x′ reflected); 4 valid inputs accepted (mirror image and similar copy of P′, roles
  exchanged, the certificate).
- **Construction data** (`run2_construction.py`, sympy): R from the quaternion (100, 5, −18, 7); rays with positive
  factors; A_kl(x) = 0; the factorization (λ, three printed linear factors, the printed quadratic); x(λ*) equals the
  certificate's x; the rational roots are degenerate. CONFIRMED.
- **Package re-run** from the source archive: all outputs identical (including both runs of `own_search2.py` and
  `own_refine.py`), except one line of a recorded output that its program does not print (moved, see above).
- **Fixes required and made:** name Milka 1970 and Pogorelov 2002 (unread); describe the Wang–Xie theorem as a
  preprint result; restrict a sentence on non-triangulated examples to the examples known; correct the recorded
  output; update the search record. Overall: CONFIRMED_WITH_FIXES (no mathematical error).

## Relation to the literature, novelty and scope
- **Searches (19 and 20 October 2026).** arXiv API (Stoker's conjecture; dihedral and face angles; nonconvex polyhedra
  with given dihedral angles; dihedral rigidity and dihedral flexes; Amenta), Crossref, zbMATH (no review text for
  Stoker 1968, Karcher 1968, Milka 1970 or Pogorelov 2002), OpenAlex (works citing Stoker 1968: 49 titles),
  Semantic Scholar (citing works of Stoker 1968: 77 titles; Karcher 1968: 9 titles; Amenta–Rojas 2018: 3 titles;
  scanned by title), and ten web searches. No printed example of two embedded triangulated nonconvex polyhedra with
  the same combinatorics, equal dihedral angles and different face angles was found; no source claims that no such
  pair exists.
- **What was read.** The OWR page; Amenta–Rojas 2018 in full (arXiv); the author manuscript of Amenta–Rojas 2020 in
  the NSF Public Access Repository (Sections 1, 2, 7; the version of record was not compared); Wang–Xie
  (introduction, Theorem 1.6); Cho–Kim (Sections 1, 3); Biedl et al. 2013; Alexandrov (Section 1); Mazzeo–Montcouquiol
  (introduction, Theorem 1); abstracts or introductions of Winter 2026, Belletti 2023, Hempel 2015 (arXiv),
  Fillastre–Izmestiev; summary and introduction of Hempel's ETH thesis; the AIM workshop summary "Rigidity and
  polyhedral combinatorics"; Pak, *Lectures on Discrete and Polyhedral Geometry* (draft of 20 April 2010),
  Sections 26.7–26.9, Exercises 26.1, 26.2, 27.3, 32.2 with hints, Sections 32.6–32.7.
- **Not read.** Stoker 1968 (CPAM 21, 119–168) and Karcher 1968 (CPAM 21, 169–174): Wiley returned 403 with a bot
  challenge, not worked around; OpenAlex and Semantic Scholar list no open copy; scholar.archive.org showed a bot
  check, not worked around. Milka 1970 (Ukr. Geom. Sb. 9, 85–86; Zbl 0218.52004) and Pogorelov 2002 (Dokl. Akad.
  Nauk 385(1), 25–27; Dokl. Math. 66(1), 19–21; Zbl 1229.52010): no open copy found. The version of record of Amenta–Rojas 2020
  (ScienceDirect refused automated access). Any thesis or later work of Rojas. Aurenhammer–Walzl beyond its abstract.
  Schlenker 2000 (cited as reported).
- **Secondary descriptions of the unread papers.** Wang–Xie: Karcher verified Stoker's conjecture "for a class of"
  convex polyhedra with 5 vertices and 6 faces (by Euler, convex bipyramids). Pak's hints attribute to Karcher the
  cases of simple and circumscribed polytopes. Pak: the conjecture is in Stoker's Section 6. Cho–Kim: Stoker's
  Theorem III concerns nonconvex polyhedra with saddle vertices, data dihedral angles and edge lengths.
  Mazzeo–Montcouquiol list Karcher, Milka and Pogorelov among the papers on the problem, with results for specific
  classes; Amenta–Rojas: Pogorelov proved the conjecture for triangulated convex polyhedra; Pak (Exercise 32.2 and
  hint): Pogorelov's result on dihedral-angle-preserving deformations of simplicial convex polytopes.
- **Wang–Xie.** arXiv 2203.09511v2; no journal version found in Crossref or OpenAlex (2026-10-20). The note calls
  their Theorem 1.6 a preprint result.
- **Circumstantial.** Amenta–Rojas conjectured the opposite in 2018 (and in the 2020 manuscript); this suggests that no
  example was known to them, but it is not evidence of novelty.
- **Scope.** Clause 1 answered in the negative by one explicit pair. Clause 2 not addressed. No priority claim.

## 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/