# Verification report — OWR-13498-012 (Rote: a curious identity on self-stresses, Oberwolfach Report 13/2015, Problem 8)

Verification date: 2026-10-20.

**Verdict.** The note gives a **complete answer in a stated scope**; every statement it makes is proved.
- **Question 1 (other graphs with n vertices and 2n − 2 edges).** Answered **yes at generic positions**, which is
  how the note reads the problem's "general position" (Theorem C): every graph that is dependent in the generic
  2D rigidity matroid, in particular every graph with n vertices and 2n − 2 edges, carries at generic positions a
  self-stress ω with Σ ω_ij [a p_i p_j][b p_i p_j] = 1 for all a, b; the set of such stresses is an affine
  hyperplane of the stress space, a single stress exactly when the graph (with 2n − 2 edges) is generically rigid.
  Exact rational certificates for the two named graphs (prism + edge, K_{3,3} + edge). "No three points collinear"
  does not suffice (Proposition 4.1: explicit exceptional positions).
- **Question 2 (meaning, general phenomenon, (co)homology).** Open-ended; answered by proved theorems that contain
  the identity: Theorem A (independence of a, b for every graph, every position, every self-stress; the constant
  is N(ω) = Σ ω_ij det(p_i,p_j)² in the determinant convention, N(ω)/4 with Rote's area brackets), Corollary 2.2 (one class in coker R(p), the dual of the stress space),
  Theorem A_h (projective form), Theorem B (N(ω) = 6 × volume under the Maxwell–Cremona lifting; Rote's line
  integral is Green's formula). Whether these readings are the meaning intended is a judgment.
- **Open.** The reading of Question 1 as a request for a closed formula: Proposition E excludes monomials in polygon
  areas for the two named graphs; sums of products of areas do occur (Remark 6.2). A link with stress homology of
  cell complexes is not attempted.
- **Not new.** The questions and the wheel identity are Rote's; the four-point case is Rote–Santos–Streinu (2003),
  Lemma 3.10; the graph-independent f′ argument is theirs (p. 21); the projective transfer of stresses is classical
  (Rankine 1863; Izmestiev's survey, Lemma 4.5); the Maxwell–Cremona correspondence is cited. Theorems A and B are
  elementary and may be folklore.

Result type: COMPLETE_SCOPED_PROOF. The note is unrefereed.

## Statement checked
- **Primary source.** G. Rote, Problem 8 "A curious identity on self-stresses", in: A. I. Bobenko, R. Kenyon,
  P. Schröder (eds.), Discrete Differential Geometry, Oberwolfach Reports 12(1) (2015) 661–729, Report No. 13/2015,
  pp. 722–723, doi:10.4171/OWR/2015/13. Read in the publisher's PDF (EMS, open access; 70 pages, sha256
  `e3ba6ef5fc327cfe1ca2c3c52a30d8107fbe487f832ae799f6ccd3e8e3da25ca`), both as text extraction and as a rendering
  of pp. 722–723 (macOS Core Graphics, 2× scale).
  - Wheel W_n (n ≥ 3): hub p_0, rim cycle p_1 … p_n, general position, 2n edges. Printed:
    ω_{i,i+1} = 1/([p_i p_{i+1} p_0][p_1 … p_n]), ω_{0,i} = [p_{i−1} p_i p_{i+1}]/([p_{i−1} p_i p_0][p_i p_{i+1} p_0][p_1 … p_n]),
    brackets = signed areas; f_ij := [a p_i p_j][b p_j p_j]; identity (1) Σ ω_ij f_ij = 1, "used (and proved) for
    n = 3" in the cited RSS paper; f′_ij = (3/2)|p_i − p_j| ∫ |x|² ds = (1/2)|p_i − p_j|²(|p_i|² + |p_j|² + ⟨p_i,p_j⟩)
    "fulfills (1)" [RSS, Lemma 3.10].
  - Question 1: other graphs with n vertices and 2n − 2 edges for which a self-stress satisfying (1) "can be
    defined"; next candidates: triangular prism plus an edge, K_{3,3} plus an edge. Question 2: meaning of (1), a
    more general phenomenon, connections to homology and cohomology.
- **Corrections to the printed statement** (numbered in the note's statement card):
  (a) the spoke stress needs a minus sign — absent in the rendered page as well as in the text extraction (other
  minus signs of the page are rendered); the half-plane argument (convex counter-clockwise rim, hub inside: all
  printed values positive, while at a rim vertex the three edge vectors lie in an open half-plane) shows the
  printed function is not a self-stress, and Proposition 2.4 shows it fails at every rim vertex in general position,
  while the corrected function is a self-stress and satisfies (1);
  (b) f_ij must read [a p_i p_j][b p_i p_j] (RSS (9)); the printed [b p_j p_j] ≡ 0;
  (c) (1) is the same in both bracket conventions (ω scales by 4, f by 1/4), but Σ ω f′ = 1 only in the determinant
  convention of RSS; with signed areas Σ ω f′ = 4.
- **Corpus record.** UnsolvedMath v1.8.0, OWR-13498-012, listed as open; its text omits the formulas and the
  quantifier "for all a, b"; the source decides.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Q1: a self-stress with (1) for all a, b simultaneously, at generic positions ("general position") | yes for every dependent graph, in particular every graph with n ≥ 2 vertices and 2n − 2 edges; unique iff the generic stress space is one-dimensional, i.e. for 2n − 2 edges iff generically rigid | Theorem C |
| Q1 for the two named graphs | yes, unique, exact certificates | Table 1, Section 4 |
| Q1 with "general position" = "no three points collinear" | no: explicit positions with unique stress and N = 0 (W_4, prism + 04, K_{3,3} + 01) | Proposition 4.1 |
| Q1 as "the stress space is nonzero" or "(1) at one pair a, b" | trivial; not the question (RSS p. 37 already notes uniqueness up to scale for wheels) | Section 1.1 |
| Q1 as a request for a closed formula of the wheel type | monomials in polygon areas excluded for the two named graphs; rational expressions with sums occur; open | Proposition E, Remark 6.2 |
| Q2: general phenomenon | (a,b)-independence for every graph and stress; projective form for every line | Theorems A, A_h |
| Q2: meaning | N(ω) = 6 × volume of the Maxwell–Cremona lifting; f′ = Green's formula | Theorem B, Corollary 5.1 |
| Q2: homology/cohomology | one class κ_p in coker R(p) = (stress space)*; (1) ⇔ ⟨ω, κ_p⟩ = 1; stress homology of cell complexes not attempted | Corollary 2.2, Section 5 |

## Results in the paper
- **Theorem A.** For every framework (any graph, any points, coincident points allowed) and every self-stress:
  Σ ω_ij D(a,p_i,p_j)D(b,p_i,p_j) = N(ω) for all a, b; Σ ω L L^T = N(ω) e_3 e_3^T; Σ ω f′ = N(ω). Proof from
  Σ ω d d^T = 0 and Σ ω c d = 0 (ordered-pair bookkeeping).
- **Lemma 2.1, Corollary 2.2.** f(a,b) − f(o,o) = R U and f′ − f(o,o) = R V with explicit velocity fields; one class
  κ_p in coker R(p). Only membership in im R(p) is used.
- **Corollary 2.3.** Affine covariance N(ω; Ap) = det(A_0)² N(ω; p); collinear subdivision keeps the stress and N.
- **Proposition 2.4.** Proof of Rote's wheel identity (corrected sign) for all n: equilibrium from the
  three-vector identity, N = 1 by evaluating at base point p_0.
- **Theorem A_h, Lemma D.** Σ ω L L^T/(h_i h_j) = C_h(ω) l l^T with C_h(ω) = det(S)² N(μ; p′), μ = ω h_i h_j (the
  transfer is classical); C_h(ω) ≠ 0 for some admissible h when ω ≠ 0 and edge directions at each vertex are
  pairwise non-parallel (points distinct).
- **Theorem C.** Existence at generic positions for every dependent graph (polynomial Ψ_G); uniqueness iff d_G = 1
  (for n ≥ 2 vertices and 2n − 2 edges: iff generically rigid).
- **Proposition 4.1.** Exceptional positions with no three points collinear and N = 0.
- **Theorem B, Corollary 5.1.** Volume interpretation for non-crossing drawings; drawings with crossings in general
  position (no edge through a non-incident vertex, no overlapping collinear edges, no three edges through one
  point) reduce to it by subdivision. K_4 in convex position planarises to W_4 (RSS Lemma 3.10 = Rote's W_4 formula).
  Pyramid picture for rims star-shaped from the hub.
- **Lemma 6.1, Proposition E, Remark 6.2.** Q_P, Q_K irreducible over C (proof written out); no monomial area
  formula for the normalised stress of the two named graphs; sums of area products occur.

## Computations (programs and outputs in reproducibility/)
Only finite exact statements depend on computation (Table 1, Proposition 4.1, (13), the witnesses in Lemma 6.1
and Proposition E).
- **`check_note.py`** (written with the note; standard library, exact rationals, under one second; last line
  `ALL CHECKS PASSED`, 237 checks): [C1] Proposition 2.4 and corrections (a)–(c), n = 3..8, one convex and three
  random wheels each, including the residual 2 det(w,u) z of the printed sign; [C2] (8), (9), the three-vector
  identity and the segment integral as polynomial identities; [C3] Theorem A on 48 random frameworks (175 basis
  stresses; collinear triples, coincident points, all points collinear) with a negative control; [C4] the two
  certificates: isomorphism uniqueness (all 6 one-edge extensions isomorphic), no three collinear, rank 9,
  dimension 1, nonzero on all edges, every 9-edge subgraph of rank 9, the 20 values of Table 1, F(a,b) ≡ 1,
  Σ ω f′ = 1, non-crossing prism drawing, 4 crossings of the K_{3,3} drawing; [C5] liftings: prism faces and heights
  77/23550, 187/23550, 119/23550, volume 1/6; K_{3,3} + 01 subdivided at its 4 crossings, N unchanged, 6·Vol = 1; K_4
  convex = Rote's W_4 edge by edge, volume 1/6; convex and star-shaped wheels: pyramid with apex height 1/D_rim;
  [C6] Proposition 4.1 and C_h(σ) ≠ 0 for h = x_1 + 2x_2 + 30; [C7] Theorem A_h on 71 random triples and
  collinear subdivision; [C8] Table 2 stresses as polynomial identities on the slice, (13), the 7-term area
  expression of Q_P, the decompositions and witnesses of Lemma 6.1, Q = 0 and w_e ≠ 0 at the exceptional points,
  consistency with the certificates.
- **Original programs** (`original/`, sympy and networkx): wheel formula n = 3..6; (I1), (I2); Theorem A, A′ on 60
  random graphs (174 basis stresses); certificates and the planar lifting; Theorem A_h and transfer (101 triples);
  zero-area wheel; K_4 planarisation; symbolic stresses on the slice; factorisations over Q (Proposition E, with the
  W_5 control); exceptional positions; **TESTED corroboration only**: all 149 graphs of the networkx atlas with
  n ≤ 7 and 2n − 2 edges have N ≢ 0 on the stress space at a random rational position.
- All programs of the package were re-run from a copy of the package for this report; see
  `reproducibility/README.md` for the command lines, running times and the comparison with the recorded outputs.

## Independent verification runs
Two independent verification runs, both AI-assisted, examined the first written version of the result; each read
the sources and wrote its own programs.

| Item | Run A | Run B |
|---|---|---|
| Statement against the source (formulas, n ≥ 3, 2n edges, both questions, the two graphs) | CONFIRMED (text extraction) | CONFIRMED (own PDF extraction) |
| Correction (a), spoke sign | PROVED (half-plane argument) and computed (n = 3..6) | computed (n = 3..8): printed sign 0/24 stresses, corrected 24/24; other minus signs survive in the extraction |
| Corrections (b), (c) | CONFIRMED | CONFIRMED (Σ ω f′ = 4 with areas, 1 with determinants) |
| Reading of Q1 (all a, b; generic) | agreed: the intended reading, not a trivialisation | agreed |
| Theorem A, A′, (I1), (I2), cokernel class | PROVED (own proof of A) | PROVED; 778 stresses of 120 frameworks, 4 position modes, negative control |
| Theorem A_h, Lemma D, Theorem C | PROVED; A_h 12/12 exact | PROVED; A_h 99/99; Lemma D control (collinear ⇒ C_h ≡ 0); Theorem C TESTED on 149 + 60 graphs |
| Certificates (Table 1) | CHECKED, all 20 values | CHECKED, all 20 values; altered value caught |
| Exceptional positions | CHECKED | CHECKED |
| Theorem B | PROVED (Maxwell–Cremona cited); own lifting of the planar prism, volume 1/6 | PROVED (distributional Laplacian route); own face tracing: 187/187 plane, 80/80 with crossings (682 crossing points); wrong sign convention fails 187/187 |
| Proposition E | PROVED (logic); factorisations recomputed with own stress | PROVED; factorisations recomputed; quartics irreducible also over Q(i); found the 7-term area expression of Q_P |
| Re-run of the original programs | outputs agree where they overlap | all outputs byte-identical |
| Bibliography | — | Crossref and zbMATH: all DOIs correct |
| Overall | CONFIRMED (minor fixes) | CONFIRMED_WITH_FIXES (citations and wording) |

**Fixes required by the runs, all made:** A1/B5 general-position hypothesis for
drawings with crossings (Corollary 5.1); A2 half-plane argument and the observation that other minus signs survive
(now also: the rendered page shows no minus sign); A3 pyramid picture only for rims star-shaped from the hub;
A4 "lie in im R(p)" instead of "span"; A5 "general position" = generic, and "no three collinear" does not suffice,
in the abstract, the status and the scope; B1 classical projective invariance cited (Rankine; Izmestiev, Lemma 4.5),
only the constant C_h and Lemma D presented as not found elsewhere; B2 RSS's f′ argument is graph-independent
((I2) credited to RSS); Cooperband–Ghrist–Hansen and Izmestiev added; B3 sums of area products occur (Remark 6.2),
the closed-formula reading stays open; B4 bibliography re-verified.

**Writing stage.** `check_note.py` was added and re-checks every finite statement of the note (above). New in the
text compared with the first written version: the proof of Rote's wheel identity for all n (Proposition 2.4, by the
three-vector identity and the base point p_0), a written-out irreducibility proof for Q_P and Q_K (Lemma 6.1, with
exact witnesses, so that Proposition E does not depend on a computer-algebra factorisation), and the slice stresses
in triangle-area form (Table 2). The rendering of p. 722 of the report confirms correction (a).

## Verification run of the written note (programs in reproducibility/independent_run_2/)
A further independent verification run, also AI-assisted, checked the text of the note (the runs A and B had
examined the first written version). It read the sources again and re-derived every proof line by line; its
programs were written from scratch (exact rationals, sympy).

| Item | Result |
|---|---|
| Statement against the source | CONFIRMED: OWR p. 722–723 re-read (text and a new rendering of p. 722: no minus sign on ω_{0,i}, [b p_j p_j] printed); RSS v2 Lemmas 2.4–2.6, (3), (9), (10), Lemma 3.10 and pp. 20–21, Lemma 5.1, pp. 36–37; RSS survey Theorem 5.1 and Section 8; Izmestiev Section 4.1 (Theorem B, Lemma 4.5, the hedged attribution to Rankine); Cooperband–Ghrist–Hansen abstract and introduction (no volume or area identity) |
| Corrections (a)–(c) | CONFIRMED: half-plane argument re-derived; n = 3..6, 24 wheels: printed sign fails at every rim vertex, corrected sign a self-stress; Σ ω f′ = 4 (areas) and 1 (determinants) |
| Theorem A, Lemma 2.1, Corollaries 2.2, 2.3, Proposition 2.4 | PROVED (re-derived); (8), (9) symbolic; 150 stresses of 60 frameworks with degenerate positions |
| Theorem A_h, Lemma D | PROVED; 35 random cases exact; Lemma D limit O(t) at the exceptional positions; C_h ≠ 0 for h = x_1 + 2x_2 + 30 |
| Theorem C, certificates, Proposition 4.1 | PROVED; both certificates recomputed from the positions alone (all 20 values, F(a,b) ≡ 1, circuits), mutation test; exceptional positions confirmed (rim area 0 by the shoelace formula) |
| Theorem B, Corollary 5.1, wheel paragraph | PROVED (Maxwell–Cremona cited; jump-rule sign checked against the survey's statement); own lifting: prism heights 77/23550, 187/23550, 119/23550, volume 1/6; K_{3,3} + 01 subdivided, volume 1/6; wheels H = 1/D_rim; 27 random plane frameworks N = 6 Vol |
| Lemma 6.1, Proposition E, Remark 6.2 | PROVED; normalised slice stresses by a kernel computation over Q(x_3,…,y_5) agree with Table 2; (13), Q_P (13 terms), Q_K (24 terms), irreducible quartic in every reduced denominator; W_5 control; 7-term area expression; Lemma 6.1 witnesses |
| Re-run of the package from source.zip | identical outputs (up to running-time lines) |
| Bibliography | Crossref: 8 DOIs correct; arXiv API: 9 identifiers correct |
| Overall | CONFIRMED WITH FIXES (wording) |

**Fixes required by this run, all made:** (1) the abstract and Section 1.2 said, of every dependent graph, that the
stress is unique exactly when the graph is generically rigid; uniqueness means a one-dimensional generic stress
space, which for 2n − 2 edges is generic rigidity (a rigid graph with 2n − 1 edges has a two-dimensional space);
(2) the abstract now says that the constant N(ω) and the single cokernel class refer to the determinant convention
(with area brackets the left side is N(ω)/4); (3) n ≥ 2 added (the one-vertex graph has 2n − 2 = 0 edges and no
normalised stress); (4) the attribution of the projective invariance to Rankine is given as Izmestiev's (Rankine
was not read). No theorem, proof or computed value changed.

## Relation to the literature, novelty and scope
- **Searches (20 October 2026).** arXiv API (self-stress with identity/volume/wheel/cohomology; equilibrium stress
  and volume; Maxwell–Cremona; pseudo-triangulations and stress; author Rote; several 429 responses, retried);
  zbMATH Open keyword searches (self-stress 2015–2026, 28 hits; Maxwell–Cremona 2015–2026, 17 hits; titles
  screened); Semantic Scholar: 116 works citing RSS 2003, titles screened; OpenAlex: records found, citing lists
  refused (HTTP 429), so the 4 works citing the report could not be listed; zbMATH has no citation search; seven
  web searches in all; Crossref and zbMATH for the bibliography.
  The verification run of the written note repeated arXiv API queries (self-stress with volume/wheel/identity;
  equilibrium stress, volume and lifting; Maxwell–Cremona and volume; Rote and stress; rigidity circuit; cokernel),
  zbMATH Open searches (title self-stress 2015–2026: 9 hits; abstract self-stress and volume: 2 hits; all unrelated)
  and one web search, with nothing new; OpenAlex again refused the citing list (HTTP 429, retried after 60 s).
  - No printed answer to Problem 8 was found, and no source with Theorem A for general graphs, the volume
    interpretation, or Theorem C.
  - Closest: RSS 2003 (K_4 only, with the wheel remark on p. 37 and the graph-independent f′ argument on p. 21);
    the RSS survey (four-point statement only); Cooperband–Ghrist–Hansen, arXiv 2311.12946 (cosheaf theory of
    reciprocal figures and lifts; a text search finds no volume or area identity); lifting papers arXiv 2312.09891,
    2505.18841, 2410.12729 (abstracts), 1501.03549 (title).
- **What was read.** OWR pp. 722–723 (PDF text and rendering); RSS arXiv v2 (Lemmas 2.4–2.6, Theorem 3.9, Lemma 3.10
  with proof, pp. 20–21, Lemma 5.1, pp. 36–37), book version not compared; RSS survey arXiv (Sections 5.1 and 8);
  Izmestiev arXiv v1 (Section 4.1, Lemma 4.5, proof of Theorem B, where the projective invariance is attributed
  to Rankine with "It seems"); abstracts as listed. Not read: Maxwell 1864,
  Whiteley 1982, Hopcroft–Kahn 1992, Rankine 1863, Crapo–Whiteley 1994, Rybnikov 1999.
- **Caveats.** The search was limited (arXiv rate limits; OpenAlex citing lists refused; older rigidity literature
  such as Crapo–Whiteley and the journal Structural Topology not checked). Theorems A and B are elementary and may
  be folklore in rigidity theory. A negative search is not a proof of priority, and none is claimed.
- **Scope.** Q1 at generic positions; Q2 open-ended, answered by the volume and cokernel interpretations; the
  closed-formula reading of Q1 open.

## 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/