# Verification report — OWR-13497-006 (Anderson's four-agent formation problem, Oberwolfach Report 12/2015)

Verification date: 2026-10-01. The first independent verification run took place on 2026-09-30, the second on
2026-10-01; the note was revised after each run.

**Verdict.** The answer is yes, for every planar target.
- The setting is four agents in the plane with all six distances prescribed by a planar target, under the standard distance-based gradient law p_i' = Σ_j e_ij (p_j − p_i).
- At every incorrect equilibrium of this law, the Hessian of V = (1/4) Σ e_ij² has a negative eigenvalue. Equivalently, the complete-graph s-stress of four points in the plane has no spurious second-order critical points, for every ground truth.
- Every incorrect equilibrium except the collocated one is a saddle in the strict sense (indefinite Hessian).
- The collocated equilibrium is a nondegenerate local maximum modulo translations. The source itself counts collocated equilibria among the saddle points in the rectangle case.
- At a spanning incorrect equilibrium, λ_min(Hess V) ≤ −κ‖λ‖²/2 ≤ −√(8V/3). Both constants are sharp.
- Consequently, almost every initial condition converges to a formation congruent to the target. Global convergence in the literal sense fails for every target that is not a single point: incorrect equilibria are fixed points, and collinear starts stay collinear.

The note is unrefereed. Neither verification run found a mathematical error.

## Statement checked
- **Primary source.** B. D. O. Anderson, "Open problem: Is there global convergence to a four-vehicle formation shape?". It appears in F. Allgöwer, U. Helmke, N. Leonard (organisers), *Control Theory: A Mathematical Perspective on Cyber-Physical Systems*, Oberwolfach Reports 12 (2015), Report No. 12/2015, pp. 647–649, doi:10.4171/OWR/2015/12.
  - The PDF of the report was read. The copy used has SHA-256 f4f7b2e89da38ad98d7c478837b662446c31f58d0e75ecbe456afa09b88b1e77. Both verification runs fetched it anonymously through the DOI and found it byte-identical to the copy the finder used. The second run read pp. 647–649 on rendered page images.
  - **Law.** The errors are e_ij = d_ij²(p) − dbar_ij² (actual minus desired), and the motion is p_i' = Σ_{j≠i} e_ij (p_j − p_i).
  - **Normalisation.** The report writes V = (1/2)‖e‖² and p' = −∇V. For that V, −∇V is twice the right side of the law. The paper uses V = (1/4) Σ e_ij², for which the law is exactly p' = −∇V. Equilibria and the signs of Hessian eigenvalues do not depend on this factor.
  - **Question** (p. 648): "Are all incorrect equilibria again saddle points?" It is asked for a general quadrilateral, or for three agents forming a triangle with the fourth inside it.
  - **Rectangle case.** The report's summary counts the incorrect rectangles, the collocated and collinear equilibria, and further equilibria spanning the plane among the saddle points.
  - **Remark (1), p. 649.** Six prescribed distances determine the shape up to congruence.
- **Corpus records** (ulamai/UnsolvedMath, status `open`).
  - OWR-13497-006 asks whether, for a general quadrilateral or a triangle with an interior agent, all incorrect equilibria are saddle points.
  - OWR-13497-004 asks for a proof of global convergence to the desired formation for such targets, "as is known for rectangular targets".
  - OWR-13497-005 asks whether the convergence properties known for rectangles continue to hold globally for such targets.
  - OWR-13497-001 is the heading record: global convergence of the four-vehicle system to a formation shape.

## Readings
| Reading | Answer | Witness |
|---|---|---|
| "Saddle point" as in the source, i.e. an unstable equilibrium (the source counts collocated equilibria as saddles) | yes, for every planar target | Thm 1.1 |
| Strict saddle in the optimization sense (λ_min(Hess V) < 0) | yes | Thm 1.1(a), (b) |
| Saddle with an indefinite Hessian | yes, except the collocated equilibria, which are nondegenerate local maxima modulo translations | Thm 1.1(c) |
| Almost-global convergence to the target shape, as known for rectangles (records -001, -004, -005) | yes | Cor. 1.2 |
| Convergence from every initial condition (literal "global") | no (for every target that is not a single point) | the incorrect equilibria themselves; collinear initial conditions for a non-collinear target (Cor. 1.2) |

## Results in the paper
- **Theorem 1.1.** Let the target be any configuration of four points in the plane, and let P be an incorrect equilibrium. Then:
  - (a) If P is collinear or collocated, λ_min(Hess V(P)) ≤ λ_min(L(e)) < 0.
  - (b) If P spans the plane, e_ij = κλ_iλ_j with κ > 0, where λ is the affine dependency of P, and λ_min(Hess V(P)) ≤ −κ‖λ‖²/2 ≤ −√(8V(P)/3).
  - (c) If P is not collocated, the Hessian also has a positive eigenvalue. The collocated equilibrium is a nondegenerate local maximum modulo translations, with Hess V = −L(dbar²) ⊗ I₂.
- **Proposition 6.1** (sharpness). Take the rectangle target with vertices (0,0), (4,0), (4,√6), (0,√6), and the incorrect rectangle p = (2,−1), (−2,1), (−1,2), (1,−2).
  - Here λ = (1,1,−1,−1), κ = 4 and V = 24.
  - The Hessian has characteristic polynomial x³(x+8)(x−56)(x−80)(x²−160x+1728), so λ_min = −8. This equals both bounds in (b).
- **Corollary 1.2** (almost-global convergence).
  - Every solution is bounded, keeps its centroid, and converges to a single equilibrium (Łojasiewicz).
  - The set of initial conditions that converge to an incorrect equilibrium is Lebesgue-null. The proof uses the centre-stable manifold theorem for the time-one map, Shub Thm III.7 in the form used by Lee et al. 2019, together with a Lindelöf countable cover.
  - If the target is not a single point, the exceptional set contains the incorrect equilibria (for instance the collocated configurations) and, for non-collinear targets, all collinear initial configurations. For a one-point target there are no incorrect equilibria.
- **Ingredients and credit.**
  - Lemma 2.2: at an equilibrium with L(e) ⪰ 0 one has e = 0. This is the convex-lift argument, standard for low-rank factorizations; for the s-stress, see Criscitiello–McRae–Rebjock–Boumal, Lemma 3.4.
  - Proposition 3.1 (collinear/collocated case): the instability is known from Sun–Helmke–Anderson (CDC 2015), Lemma 6, as used in Park–Sun–Anderson–Ahn (TAC 2018), Lemma 3, for targets that are not degenerate. For every target it follows from Criscitiello–McRae–Rebjock–Boumal, Lemma 3.4 (rank-deficient second-order critical points of the complete-graph s-stress are global minimizers). The paper's short proof also gives the bound in (a).
  - Summers–Yu–Anderson–Dasgupta (CDC 2009), according to its abstract, prove that every incorrect equilibrium is locally unstable for formations whose internal angles satisfy an acuteness condition. The paper was not seen, so it is not known which targets the condition covers. The paper credits this result and qualifies its novelty claim accordingly.
  - Lemma 3.2: e is a multiple of the classical self-stress λ_iλ_j of four points spanning the plane, and κ > 0 by Lemma 2.2.
  - Section 5 (test direction, target Gram matrix, Schur complement). The planar target Gram matrix has rank ≤ 2, so a 2×2 Schur complement is singular. This is the mechanism of the Schur-companion directions of Criscitiello (arXiv:2608.16799, §6.2), which there serves the relaxed case k ≥ ℓ + 1.
- **New ingredient: Proposition 4.1.** λ_max(NK) ≤ ‖λ‖⁴/2 for two 2×2 matrices depending only on λ, with equality iff η3 = 0, i.e. λ = (a,−a,c,−c) up to order. This handles the unrelaxed case k = ℓ = 2.
  - Lemma 4.2 gives closed forms of tr(NK) and det(NK) in σ = s2/2, η3 and η4. The paper derives them from power sums and Newton's identities.
  - Identities (7) and (8) together with AM–GM (|η4| ≤ σ²/4) show that both eigenvalues are ≤ s2²/2. Remark 4.3 gives an explicit sum-of-squares form of the certificate.
- **Remark 5.1.** Every centred spanning incorrect equilibrium arises from data (λ, κ > 0, r ∈ R²) with BBᵀ = (κ/2)N + rrᵀ, and every such datum gives a planar target and a spanning incorrect equilibrium.
- **Remark 5.2** (n = ℓ + 2 agents in R^ℓ). The argument proves instability whenever λ_max(NK) < s2² for all λ.
  - For λ = (0,…,0,1,−1) the ratio λ_max(NK)/s2² equals (n − 2)/4 exactly, so the method fails for n ≥ 6.
  - This agrees with the spurious second-order critical points of Criscitiello–McRae–Rebjock–Boumal, Prop. 6.3 (ℓ = 4 non-strict, ℓ ≥ 5 strict), which have two coincident points.
  - Five agents in R³ are not treated: the needed inequality is only tested numerically (≈ 0.75 < 1).
- **Examples 6.2 and 6.3** (exact integer data).
  - Example 6.2 is a triangle-with-interior-agent target, with squared distances (21, 45, 336, 96, 309, 621) and target affine dependency (−5, 1, 3, 1). The incorrect equilibrium is P = [(3,−2), (2,0), (10,0), (−15,2)], with λ = (−1,4,−2,−1) and κ = 4.
  - Example 6.3 is a convex trapezoid target, not a parallelogram, with squared distances (417, 33, 48, 672, 297, 105) and target affine dependency (3, −1, −3, 1). The incorrect equilibrium is P = [(−5,0), (15,5), (−9,−5), (−1,0)], with λ = (−2,−1,−1,4) and κ = 4.
  - Both equilibria have Morse index 1. The exact characteristic polynomials are printed in the paper.

## Computations (scripts and outputs in reproducibility/)
The proofs do not rely on computation; the exact computations confirm the displayed identities, and the numerical tests are evidence only.
- **Lead** (`lead/check_paper_formulas.py`, sympy, about three minutes; ALL PASS). Written for the paper; it imports nothing from the finder's code. It checks in exact arithmetic:
  - Lemma 2.1, Lemma 3.2 and eqs. (9)–(11) as polynomial identities, and (12) at 25 random rational points (it also follows from (11) in one line).
  - Lemma 4.2 (power-sum expansions, Newton identities, eqs. (5), (6), det K, det N) as identities of rational functions. Eq. (6) is also recomputed from the 4×4 projector form, without a basis.
  - (7) and (8), both with σ, η3, η4 as independent variables and in λ, Remark 4.3, and the equality case.
  - Proposition 6.1, Examples 6.2 and 6.3 (every displayed number and polynomial), and the value (n − 2)/4 of Remark 5.2 for n = 4..8.
- **Finder** (`claimant/`).
  - Exact: s1 (twenty identities), s2 ((7), (8) and the sums of squares, computed directly in λ), and s6 (both examples).
  - Numerical tests, which are evidence only:
    - 2,397 incorrect critical points for 320 targets in 8 families, all with a negative Hessian eigenvalue; over the 725 spanning ones, max λ_min/(κ s2) = −0.500000;
    - 97,325 constructed spanning incorrect equilibria plus an adversarial maximization, with supremum −1/2 at η3 = 0;
    - over 19,992 further ones, max λ_min/√(2V) = −1.1547 ≈ −2/√3;
    - 6,000 flow integrations for 150 targets, all reaching the target shape (45 after continuation);
    - the estimate (n − 2)/4 for n = 4..8.
- **First independent verification run** (`independent_run/`, AI-assisted, own code).
  - v1 re-derives Proposition 4.1 by a different route, using a non-orthonormal basis of W and a second certificate μK − KNK ⪰ 0 at 400 exact rational points; the maximum of the ratio over these points is exactly 0.5.
  - v2 runs a damped Newton search on 67 targets: 426 incorrect critical points, all with a negative Hessian eigenvalue, and max λ_min/(κ s2) = −0.5.
  - v3 builds 40,000 equilibria as in Remark 5.1 and runs a hill climb; the supremum is −0.50000000002.
  - The run also re-ran the finder's scripts (s3, s4, s5 with smaller samples) and obtained consistent outputs.
- **Second independent verification run** (`independent_run_2/`, AI-assisted, own code written from the text of the paper; about 12 minutes in all; ALL PASS).
  - `r2_A_invariants.py` (exact): the Newton identities; eq. (5) basis-free from the projector; eq. (6) from the characteristic polynomial of the 4×4 matrix Π(D + DλλᵀD/(2Δ))Π(s₂Π + 4ΠDΠ) = U(NK)Uᵀ, together with det K, det N and the three 2×2 identities of the proof; (7), (8) in (σ, η3, η4) and in λ; Remark 4.3; the AM–GM bound; the equality case. A QR-based floating check at 200,000 random λ gives a maximal relative error of 3·10⁻¹⁵ in (6) and max λ_max(NK)/s2² = 0.500000000000. Remark 5.2: (n − 2)/4 exactly for n = 4..8, and the numerical estimate 0.75 for n = 5.
  - `r2_B_structure.py` (exact): Lemma 2.1 against sympy's own Hessian of V; the proof of Lemma 2.2; Lemma 3.2 on 59 spanning configurations, including collinear triples and coincident pairs; eqs. (9)–(12) with symbolic λ, P, κ, z; V = κ²(s2² − s4)/8 and the √(8V/3) step; Theorem 1.1(c).
  - `r2_C_examples.py` (exact, Hessian by differentiating V): Proposition 6.1 and Examples 6.2, 6.3, including every displayed polynomial and value, the sign counts, and the geometry of the targets (interior point, crossing diagonals, trapezoid, convex position).
  - `r2_D_numeric.py` (evidence only).
    - A Levenberg–Marquardt search, assuming no structure, found 369 incorrect critical points for 56 targets of eleven types (116 spanning, 197 collinear, 56 collocated). All have a negative Hessian eigenvalue, and all non-collocated ones also have a positive one.
    - On the spanning ones, max λ_min/(κ s2) = −0.5000000000 and max λ_min/√(8V/3) = −1.0000000000.
    - For 40,000 equilibria built as in Remark 5.1, with an adversarial search, the suprema of λ_min/(κ s2) and λ_min/√(2V) are −0.5000000000 and −1.1547005384.
    - 420/420 flows reached the target shape.
  - `rerun_of_package/`: the lead program, the finder's programs and the first run's programs, re-run from a fresh extraction of `source.zip`. All outputs are identical to the recorded ones apart from timing, except one numerical test. There, s3 with reduced samples found 359 instead of 358 incorrect critical points, because of floating-point differences in the last digits during deduplication; it has the same conclusions.

## Independent verification
Two independent verification runs, both AI-assisted, checked the note. Neither found a mathematical error.

### First run (2026-09-30)
| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (source read from the DOI; the question, the law and the source's use of "saddle" checked) |
| Proofs | CONFIRMED (every step checked; the λ-inequality re-derived by a different route) |
| Computations | CONFIRMED (all certificates re-run; three independent numerical checks) |
| Answer as posed | CONFIRMED (affirmative, in the source's sense of "saddle"; the stronger strict-saddle statements are proved) |
| Novelty | No prior proof found; no priority claim |
| Presentation | Five required write-up fixes; all applied |

The five required fixes, all applied in the paper:
1. **Credit.** Step 1 is credited to Sun–Helmke–Anderson 2015, Lemma 6, as used in Park et al. 2018, Lemma 3 (Proposition 3.1 and the introduction). Step 2 is credited to the classical self-stress (Lemma 3.2 and the remark after it). The Schur-complement mechanism is credited to Criscitiello 2026, §6 (Schur-companion directions). The new ingredient is stated as the unrelaxed case k = ℓ, handled through λ_max(NK) ≤ s2²/2 (introduction, "Method").
2. **Degenerate targets.** The novelty claim is qualified: for collinear targets the absence of spurious second-order critical points already follows from Criscitiello 2026, Thm 6.1 (introduction; Scope and priority).
3. **Paraphrase of Criscitiello–McRae–Rebjock–Boumal.** The paper now states that they suspect a benign landscape for almost every ground truth when ℓ ≤ 4 and n = 6, and that their construction in the plane yields nothing for n ≤ 5.
4. **Literature sweep.** The Semantic Scholar citing-work sweeps are recorded, and so is the fact that OpenAlex was rate-limited. The Sahebsara–de Queiroz 2024 abstract was confirmed; they use a 3-D embedding with virtual body frames. The first run did not see the 2009 paper of Summers et al. and took it to be superseded by the same authors' 2013 statement that the four-agent question was open; the second run found that its abstract states a conditional result (see below).
5. **Headline.** The answer is stated as: every incorrect equilibrium is unstable (strict saddle), and the collocated one is a local maximum modulo translations, which the source itself calls a saddle. Literal global convergence fails for collinear starts.

Further changes made while writing the paper:
- The invariants of Lemma 4.2 are now derived in the paper, with a new exact check.
- The sharpness of both constants is shown by an exact example (Proposition 6.1).
- The converse construction (Remark 5.1) was added.
- Citations were corrected against Crossref. The IEEE TAC 2018 paper has four authors (Park, Sun, Anderson, Ahn); M. H. Trinh is a co-author of the CDC 2016 paper only.

### Second run (2026-10-01)
| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED (source re-fetched through the DOI, byte-identical; pp. 647–649 read on rendered pages; the question, the law, the factor 2 in V and the source's use of "saddle" checked; records -001, -004, -005, -006 compared) |
| Proofs | CORRECT (checked line by line, including every expansion and Newton identity in the proof of Lemma 4.2, the certificate (7)–(8) with AM–GM and the sums of squares, the Schur-complement step, Remark 5.1 and the corollary; one edge case in the statement of the corollary corrected, see fix 5) |
| s-stress formulation | CONFIRMED (V = g/2 for the s-stress g = ½Σ(‖z_i − z_j‖² − d_ij²)² of both Criscitiello et al. papers; "benign" means every second-order critical point is a global minimizer, so the formulation is exactly equivalent to the first statement of Theorem 1.1, the collocated point included) |
| Computations | REPRODUCED with its own code (`independent_run_2/`), and the whole package re-run from a fresh extraction of `source.zip` (consistent outputs) |
| Novelty | no earlier complete proof found; one missing credit (Summers et al. 2009, conditional result) and one credit placement (Criscitiello et al. Lemma 3.4) |
| Presentation | good; required changes of credit and wording only |

Required fixes, and where they were applied:
1. **Credit Summers–Yu–Anderson–Dasgupta (CDC 2009).** According to its abstract, that paper proves that every incorrect equilibrium is locally unstable for formations whose internal angles satisfy an acuteness condition. Applied in the introduction ("Earlier work"), where the paper states this and says that the paper was not seen. In "Scope and priority" the novelty claim now excepts the targets covered by that condition, and the paper is listed among the works of which only the abstract was read.
2. **Credit Criscitiello–McRae–Rebjock–Boumal, Lemma 3.4, for part (a) for every target.** Applied in "Earlier work"; the paper now says that its short proof also gives the bound in (a).
3. **Abstract.** The s-stress sentence now follows the first statement and says that it is equivalent to it. It no longer follows the convergence sentence, where it read as if it were equivalent to almost-global convergence.
4. **"Method" paragraph.** The Schur complement is singular, not zero, for a target spanning the plane: "that is singular because the target is planar".
5. **Corollary 1.2.** The non-emptiness of the exceptional set, and the failure of literal global convergence, now carry the hypothesis that the target is not a single point, since a one-point target has no incorrect equilibria. Applied in the corollary, the sentence after it and the proof ("The exceptional set").
6. **Paragraph after Proposition 6.1.** "An incorrect rectangle in the sense of [AYDS10]" referred to a paper that was not read. It now refers to the description in the Oberwolfach report, p. 648.
7. **Verification item 3.** The wording that presented the first run's checks as manual was removed; the first run is described as an AI-assisted verification run that checked every step of the proofs.
8. **Verification item 4.** Added for the second run.
9. **"Scope and priority".** The search record was updated: arXiv to 2026-10-01; OpenCitations citing lists; four web searches in all; the abstracts of the 2009, 2010, 2011 and 2015 conference papers read.
10. **Remark 5.1.** Now says "Every centred spanning incorrect equilibrium … with B = UᵀP".
11. **This report, `reproducibility/README.md` and the Zenodo metadata** were updated, and the release was rebuilt.

Optional change applied: the bibliography entry of Dasgupta–Anderson–Yu–Summers (AUCC 2011) now gives pp. 44–49. These are the pages given by Park et al. (2018), by Summers et al. (2013) and by the ANU research portal; the Oberwolfach report gives 94–99. The entry says so.

## Relation to the literature, novelty and scope
- **Formation control.** Later work either states that the non-rectangular case is open or changes the law.
  - Summers, Yu, Anderson, Dasgupta (CDC 2009; abstract read on the ANU research portal) state that they prove local instability of every incorrect equilibrium for formations whose internal angles satisfy an acuteness condition. The full text was not accessible to us (the author-hosted copy is behind a university gate page), so the scope of the condition is unknown. This conditional result is credited in the paper.
  - Summers, Yu, Dasgupta, Anderson (ISIC 2013, arXiv:1307.2089) say in the abstract that even for four agents the existence of locally stable incorrect equilibria was open. They propose semidefinite certificates that can settle this in principle. Their introduction cites only the rectangle results.
  - Park, Sun, Anderson, Ahn (IEEE TAC 63 (2018) 2678–2685; authors' version read) state twice that the general non-rectangular K4 case under the standard law was not known. They obtain almost-global convergence for a modified law with virtual coordinates.
  - Park, Sun, Trinh, Anderson, Ahn (CDC 2016; arXiv abstract read) also use a modified law.
  - Sahebsara, de Queiroz (Systems & Control Letters 185 (2024) 105726; abstract read) embed the formation in 3-D with virtual body frames.
  - Anderson–Yu–Dasgupta–Summers (IFAC NecSys 2010) show, per their abstract, that in some cases the incorrect equilibria are necessarily unstable. Dasgupta–Anderson–Yu–Summers (AUCC 2011) show, per their abstract, that each incorrect equilibrium is attached to at most one desired formation and that all are unstable if the desired formation is a rectangle. Sun–Helmke–Anderson (CDC 2015) show, per their abstract, that degenerate critical formations are unstable. Only the abstracts were read (ANU research portal).
- **s-stress / Euclidean distance geometry.**
  - Song et al. (LAA 727 (2025), arXiv:2408.07256): benign for n ≤ d + 1; a spurious local minimum for d = 1, n ≥ 7.
  - Criscitiello–McRae–Rebjock–Boumal (Math. Program. 2026, doi:10.1007/s10107-026-02380-y; arXiv:2507.15662v2 read):
    - Lemma 3.4: rank-deficient second-order critical points are global minimizers;
    - §6.2: strict spurious second-order critical points for n = max{ℓ+2, 7} and non-strict ones for n = max{ℓ+2, 6};
    - the remark on n ≤ 5 and the suspicion for ℓ ≤ 4, n = 6, as above;
    - Prop. 6.3: two coincident points, strict for ℓ ≥ 5 and non-strict for ℓ = 4.
  - Criscitiello (arXiv:2608.16799v1 read): benign for k ≥ 2(ℓ + 1), and for k = n − 2 when k ≥ ℓ + 1 (Thm 6.1); Schur-companion directions in §6.2.
  - The case (n, ℓ, k) = (4, 2, 2) is not covered by these results.
- **Searches.** All requests were anonymous and are logged.
  - Finder (September 2026): the arXiv export API (49 requests), Crossref (36) and zbMATH Open (6). OpenAlex failed with HTTP 429 on all 10 attempts. One web search.
  - First verification run: Semantic Scholar citing-work sweeps, all without a solution. It checked 14 works citing Park et al. 2018, 17 citing Park et al. 2016, 47 citing Anderson et al. 2010, about 42 citing Sun–Helmke–Anderson 2015, 3 citing Summers et al. 2013 and 13 citing Sahebsara–de Queiroz 2024. It also ran further arXiv and Crossref title and abstract searches (2016–2026) and one web search.
  - Writing the paper: every reference was checked against Crossref or the arXiv API. New arXiv searches found no later work on the landscape of the s-stress or on four-agent formation control; the only newer s-stress hit, arXiv:2609.06284, concerns robust conditional multidimensional scaling. Semantic Scholar citation queries were rate-limited (HTTP 429), and one web search was used to confirm the Sahebsara–de Queiroz abstract.
  - Second verification run (2026-10-01):
    - Crossref: all 16 cited DOIs match title, authors, venue, volume and pages.
    - arXiv API: 17 queries up to 2026-10-01. The recent formation-control preprints found concern local stability, directed sensing or input–output properties.
    - OpenCitations: the lists of works citing Park et al. 2018, Summers et al. 2013 and 2009, Sun–Helmke–Anderson 2015, Anderson et al. 2010, Park et al. 2016, Sahebsara–de Queiroz 2024 and Criscitiello et al. 2026 (99 distinct works). Those from 2024–2026 were identified via Crossref, with abstracts from Semantic Scholar where relevant; all use other laws or other graphs.
    - zbMATH Open: 11 queries.
    - OpenAlex: list queries were again rate-limited.
    - One web search, which led to the abstract of the 2009 paper.
  - No proof of the case of the standard law with a general non-rectangular target was found.
- **Caveats.** The following were not read in full:
  - Sun–Helmke–Anderson 2015, a 2018 book chapter by Z. Sun, the CDC 2016 paper and Sahebsara–de Queiroz 2024;
  - the 2010 and 2011 rectangle papers;
  - the CDC 2009 paper.

  The OpenAlex citing-work sweep could not be done. This negative search is not a proof of priority.
- **Scope.**
  - The note answers OWR-13497-006 for every planar target, and records -001, -004 and -005 in the almost-global sense.
  - It does not treat other graphs, or n ≥ 5 agents. For n = 5 in R³ the method would apply if a five-variable inequality were proved; that inequality is only tested numerically.

## 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/