# Verification report — OWR-11578-001 (Harrach's question on the eddy-current potential map A ↦ φ_A)

Verification date: 2026-09-30 (revised the same day after referee report 2).

**Verdict.** The answer is negative: the map A ↦ φ_A cannot be defined for general nonnegative σ ∈ L^∞(R³).
- *Necklace.* Let σ be the indicator of a cyclic necklace of n ≥ 3 closed balls in which consecutive balls
  touch at one point, or a smooth σ ≥ 0 positive exactly on the open balls. Then for an explicit smooth,
  compactly supported, divergence-free A*, no φ ∈ H¹_loc(R³) solves div σ(A* + ∇φ) = 0.
- *Torus.* The same holds for a solid torus whose σ is positive a.e. and vanishes at least linearly across one
  cross-section: σ ≤ C|sin(θ/2)|χ_𝕋, with θ the azimuth. (Vanishing on a cross-section alone says nothing about an
  L^∞ function: σ = χ_𝕋 lies in the class L^∞_R, where the potential exists.)
- *No eddy-current solution.* In both cases the eddy-current equation with zero initial data and the
  divergence-free source J_t = −t curl(μ⁻¹curl A*), which vanishes near the conductor, has no solution in
  L²(0,T;W(curl)).
- *Positive side.* The product σ(A+∇φ_A) can be defined for every σ ≥ 0 by a weighted projection. With it
  the unified formulation stays uniquely solvable and uniformly coercive. The eddy-current equation is solvable
  exactly when the potential equation can be solved along the solution A of the unified formulation with
  ∇φ(·) ∈ L²(0,T;L²_ρ), and this can fail.

The note is unrefereed.

## Statement checked
- **Primary source.** B. Harrach (joint work with L. Arnold), "Inverse eddy current problems", Oberwolfach
  Report 11/2012, Oberwolfach Rep. 9(1) (2012), pp. 630–633, doi:10.4171/OWR/2012/11. It was fetched
  anonymously from EMS Press and read.
  - On p. 631 the abstract calls it an open problem whether the mapping A ↦ φ_A "can be defined for general
    non-negative σ" in L^∞(R³). The rendered page prints a mapsto arrow, A ↦ φ_A; this was checked at high
    resolution in revision 2. In the PDF text layer the arrow can appear as "→", because the bar of ↦ is a
    separate glyph.
  - Its hypothesis reads: supp σ is a disjoint union of finitely many Lipschitz domains with connected
    complements, and σ is bounded below there.
- **Author-hosted version of the underlying paper.** L. Arnold and B. Harrach, SIAM J. Appl. Math. 72 (2012)
  558–576, doi:10.1137/110831477, https://www.math.uni-frankfurt.de/~harrach/publications/eddycurrents.pdf.
  It was read in full.
  - Its class L^∞_R (p. 566) requires components with **pairwise disjoint closures**, R³∖Ω̄ connected, and
    σ ∈ L^∞_+(Ω).
  - Its Lemma 3.1 maps into H(curl0,R³) = {F ∈ L² : curl F = 0}.
  - It remarks that √σ∇u_E is uniquely determined.
  - Section 2 (uniqueness, and equivalence of the formulations) holds for every σ ≥ 0 with bounded support.
- **Restatements by the poser.** Harrach's talk slides restate the question as a solution theory for
  div σ∇φ = −div σA with general σ ≥ 0:
  - 2013 AIP (KAIST), talks/2013AIP_Eddy.pdf, PDF pp. 18 and 26;
  - 2014 SIAM Imaging Science, talks/2014SIIS_eddy.pdf, PDF p. 15.

  The follow-ups keep the hypothesis: Inverse Problems 29 (2013) 095004; IPDO 2013.
- **Thesis of the poser's coauthor.** Lilian Simon (née Arnold), *Direct and inverse transient eddy current
  problems*, Dissertation, Johannes Gutenberg-Universität Mainz, 2014 (oral examination 5 June 2014),
  urn:nbn:de:hebis:77-39057, DNB 1061142264. Referee report 2 found it in the catalogue of the German National
  Library; its full text is openly available there.
  - It restates the unified formulation with the class L^∞_R of the SIAM paper (its (3.12): disjoint closures,
    connected complement, Ω̄ = supp σ) and the same potential map (its Lemma 3.5 = SIAM Lemma 3.1).
  - Its Chapter 5 extends the theory to bounded domains (the class L_C).
  - The title page, the table of contents, the introduction, Sect. 3.3 and the conclusions were read, and the
    full text was searched. We found no treatment of general σ ≥ 0 and no counterexample in it; it assumes
    L^∞_R from Sect. 3.3 on ("For our results we need stronger assumptions on σ"). It is cited in the
    revised paper as [Sim14].
- **Corpus record.** ulamai/UnsolvedMath 1.6.0, OWR-11578-001 (status `open`). The statement matches the
  source.

## Readings
| Reading | Refuted? | Witness |
|---|---|---|
| φ_A ∈ H¹_loc(R³) (equivalently ∇φ_A ∈ L²(R³), as in SIAM Lemma 3.1) for every divergence-free A ∈ W(curl) | yes | necklace σ = χ_N (Thm 1.1); degenerate torus (Thm 1.2); A = A* |
| the same, for smooth σ | yes | σ = Σ_k β(\|x−c_k\|²) (Thm 1.1); σ = sin²(θ/2)·β (Thm 1.2) |
| φ_A ∈ H¹_loc(O) on some open neighbourhood O of supp σ | yes | Remark 4.4 |
| some definition of φ_A for which A + ∇φ_A solves the eddy-current equation in L²(0,T;W(curl)) | yes | the equation has no such solution at all (Thm 1.3) |
| the OWR hypothesis read with open components (touching closures allowed), and the claim that A+∇φ_A solves the equation | yes | necklace (Thm 1.3); the SIAM class excludes it, so no published theorem is contradicted |
| formulation level: σ(A+∇φ_A) definable so that the unified formulation is uniquely solvable and uniformly coercive | no (answered positively) | σ(A+∇φ_A) := √σ P_σ √σ A (Thm 1.4) |
| per-component Neumann potentials on int(supp σ) | exists for the necklace | equals the projection w_A = −Q_σ√σA; it does not give a W(curl) solution (Thm 1.3) |

## Results in the paper
- **Lemma 2.1.** V := {A ∈ W(curl) : div A = 0} equals W¹_♦. Every E ∈ W(curl) is A_E + ∇φ with A_E ∈ V and
  φ ∈ H¹_loc.
- **Lemma 2.2 / Definition 2.3.** φ solves the potential equation iff √σ∇φ = −Q_σ√σA. The definition
  σ(A+∇φ_A) := √σP_σ√σA agrees with Arnold–Harrach wherever φ_A exists. Its norm is ≤ ‖σ‖_∞C_R.
- **Theorem 1.1 (necklace).**
  - Lemma 4.2: contact points have zero capacity, so P_σ√σA* = 0.
  - Lemma 4.3 (gap lemma): an H¹ function that is continuous on two tangent balls takes equal values at the
    contact point. The proof uses ACL and ∫|∂₃u|² ≥ δ²r/(8|x′|²).
  - A 2π branch mismatch around the cycle then gives a contradiction.
- **Theorem 1.2 (torus, σ ≤ C|sin(θ/2)|χ_𝕋, σ > 0 a.e.).** Explicit profiles with ∫|t|h_j′² ≤ 2/j give
  P_σ√σA* = 0, and ACL across the degenerate half-disc gives a contradiction.
- **Theorem 1.3.** A = tA* solves the unified formulation. By Theorem 1.4(c), any eddy-current solution
  would give φ(t)/t solving the potential equation for A*.
- **Theorem 1.4 (general σ ≥ 0).**
  - (a) consistency;
  - (b) existence by Lions–Lax–Milgram, uniqueness, and σ-independent coercivity;
  - (c) every solution E of the natural formulation has curl E = curl A and √σE = P_σ√σA, and
    E(t) − A(t) = ∇φ(t) with φ(t) solving the potential equation;
  - (d) the converse, for G ∈ L²(0,T;L²_ρ).
  - So the natural formulation is solvable exactly when the potential equation can be solved along A with
    ∇φ(·) ∈ L²(0,T;L²_ρ) (in (c), G = E − A has this property automatically).
- **Remarks 7.1–7.2 (sketches, not used).**
  - For the necklace the exact obstruction is a cycle functional ℓ(A) with ℓ(A*) = −2π.
  - Potentials with ∇φ ∈ L^p for p < 2 should exist.
- **Remark 7.3 (computation, not used).** ℰ(h) = π ln(1/h) + 2π(ln 2 + γ) + o(1) and
  ℰ − LB → π(1 + ln 2 + 2γ) = 8.945932.
- **Remark 7.4.** The set {√σ∇φ : φ ∈ H¹_loc} is closed, so the potential always exists, when σ ≍ 1 on a bounded
  open conductor with finitely many components and the H¹-extension property and σ = 0 elsewhere. Revision 2
  adds the argument: extension gives restriction onto H¹(Ω_c) and a compact embedding, hence Poincaré on each
  component and closedness of ∇H¹(Ω_c).

## Computations (scripts and outputs in reproducibility/)
- **Lead** (`lead/verify_owr11578_lead.py`, sympy/mpmath/numpy/scipy, about 15 s; 73 PASS lines, ALL CHECKS
  PASSED). It checks:
  - Lemma 4.1 for n = 3..12, symbolically and in 60 digits, with atan2 at the actual contact points: mismatch
    (0,…,0,2π) and ℓ(A*) = −2π;
  - the formulas for A* and curl A*, and curl A* = curl curl A* = 0 on U;
  - the gap bounds and the segment estimate;
  - the capacity scaling ‖∇η_ε‖² = nε·384π/35;
  - the torus bounds: the two integrals of Lemma 5.1 symbolically; its bounds for j = 2..200 in 250-digit
    arithmetic, which resolves the smallest margin (1.53e−173 at j = 200); and a quadrature for the weight
    sin²(t/2);
  - ℰ(h) and LB(h) with their limits.
  - *Corrected in revision 2* (two record errors found by referee report 2; the mathematics is unaffected):
    - At 60 digits the comparison 2 arsinh(e^j/4) ≥ 2(j − log 2) was vacuous for j = 70..200 (131 values),
      because the margin is about 8e^{−2j}. It is now made at 250 digits. Each margin is resolved and agrees with
      8e^{−2j}(1 + O(e^{−2j})). The inequality arsinh y ≥ log(2y) is elementary; the check is a numerical
      confirmation, not a proof.
    - The printed upper bounds for ∫|t|h_j′² were divided by an extra factor 4. The corrected values are
      0.3559, 0.1161, 0.05372, 0.0259, 0.01272 and 0.006305 for j = 2, 5, 10, 20, 40 and 80, each ≤ 2/j.
      The check the paper relies on used the correct expression throughout.
    - The earlier README and report said "74 PASS"; the output has 73 PASS lines (the 74th match was the final
      "ALL CHECKS PASSED" line). The docstring runtime now reads about 15 s (it said one minute).
- **Finder** (`claimant/`):
  - `verify_necklace.py` (ALL CHECKS PASSED) checks the geometry for n = 3..8, the A* identities, the gap
    bounds and the exponents;
  - `gap_energy.py` computes ℰ(h) by bispherical series and image charges, which agree for h ≥ 10⁻⁶.
- **Independent referees, verifications 1–2** (`referee/`).
  - The second referee wrote code before reading the finder's. It checks:
    - the geometry for n = 3..12 with atan2;
    - the symbolic identities, including curl A* = 0 near N;
    - ℰ(h) by a multipole linear system (8-digit agreement for h ≥ 10⁻⁴) and by direct summation to
      h = 10⁻¹⁰;
    - the 1D weighted capacities behind the torus example.
  - The first referee checked A* = ∇θ on the balls, the divergence and the circulation numerically, and ran
    a Monte Carlo O(ε) test of Lemma 4.2.
- **Referee report 2** (`referee2/referee2_check.py`; 47 checks, all PASS, about 10 s). It was written from the
  text of the paper alone, before any finder, lead or earlier-referee script was opened. It uses its own
  cutoffs λ, κ, ζ, χ₀ and checks:
  - Lemma 4.1 for 3 ≤ n ≤ 24 in 60 digits (branch values from the defining branch), ℓ(A*) = −2π, N ⊂ U, and
    a Monte Carlo test of the azimuth window;
  - A*: the symbolic curl and divergence identities; numerically div A* ≈ 0, curl A* = 0 on U, curl A* ≢ 0,
    the circulation 2π, and A* = ∇θ_k on the balls;
  - the gap lemma: the identity and bounds for g, segment containment, and the logarithmic lower bound;
  - Lemma 4.2: C_ζ = 384π/35, the scaling for a C^∞ profile, and a deterministic quadrature of
    ‖∇ψ_ε − ∇θ_k‖² near p_k (including the point where the branches jump), which shows error² = O(ε);
  - Lemma 5.1 for a concrete χ₀ (j = 2..80), and its elementary inequalities for 2 ≤ j ≤ 400 at adaptive
    precision;
  - the torus: ∫_D ϱ⁻¹ = 2π(1 − √3/2), explicit energies below 2π³/j, Q̄ ⊂ 𝕋°, continuity of Θ_j across the
    cut, and positive 1D capacity for the weights |t|^γ with γ < 1;
  - Remark 7.3: its own image-charge recursion agrees with the series to 1e−9 (relative) for h = 1e−1..1e−8;
    ℰ ≥ LB; ℰ(1e−10) = 80.31976929; ℰ − LB − π(1 + log 2 + 2γ) = −2.5e−9 at h = 1e−10; the csch expansion;
    and ℰ → 2π as h → ∞.
- **Reruns (2026-09-30, revision 2)** with Python 3.13, sympy 1.14, mpmath 1.3, numpy 2.5 and scipy 1.18:
  - the finder's two scripts, the four second-referee scripts and the first referee's script reproduce their
    recorded outputs byte for byte;
  - `referee2_check.py` reproduces its recorded output except for the timing line (9.6 s recorded, 9.7 s on
    rerun);
  - the corrected lead script gives identical output on two runs. Against the pre-revision output only the two
    corrected lines of Part E differ.
  - As referee report 2 noted, the finder's `gap_energy.py` truncates the series slightly for h ≤ 10⁻⁷
    (80.31976903 against 80.31976929 at h = 10⁻¹⁰). The paper quotes 80.3198, which is unaffected.

## Independent adversarial audit
Three independent verifications were made: two on 2026-09-29/30, before version 1.0, and referee report 2 on
2026-09-30.

| Item | Verifications 1–2 | Referee report 2 |
|---|---|---|
| Statement fidelity | CONFIRMED (OWR p. 631 read; SIAM author version read) | CONFIRMED (primary source fetched and read; two bibliographic nits, fixes 8–9) |
| Proofs | CONFIRMED (every step checked by hand by both verifiers; no gap) | CONFIRMED (every proof checked line by line; no gap; the two sketches are labelled as sketches) |
| Computations | CONFIRMED (independent code; reruns byte-identical) | CONFIRMED (independent code confirms every numerical claim; two record errors in the lead script, fix 6) |
| Answer as posed | CONFIRMED (negative; answers the question as intended) | CONFIRMED (a complete negative answer to the question as posed, not a partial one) |
| Novelty | CONFIRMED as far as can be checked (no prior resolution found; classical ingredients) | CONFIRMED as far as can be checked (no prior solution; the thesis of L. Simon, née Arnold, must be cited, fix 2; it does not contain the result) |
| Presentation | CONFIRMED_WITH_FIXES | CONFIRMED_WITH_FIXES (one abstract sentence false as written, fix 1) |
| Release package | not rated | CONFIRMED (to be regenerated after the fixes, fix 13) |
| Fatal | none reported | no (no mathematical error and no prior publication of the result) |

All required fixes of verifications 1–2 were applied in version 1.0:
1. Sign in the nonexistence proof: u := φ(t)/t (the earlier text had −φ(t)/t).
2. The author-hosted SIAM version is cited. The paper records its class L^∞_R (disjoint closures, connected
   complement), Lemma 3.1 (values in H(curl0,R³)) and the remark on √σ∇u_E. It notes that the OWR wording is
   looser.
3. Harrach's 2013 AIP slides (PDF pp. 18, 26) are cited, and also the 2014 SIAM-IS slides (PDF p. 15).
4. The operator-norm bound is ‖σ‖_∞C_R (the earlier text had ‖σ‖_∞C_R²).
5. The weighted space L²(0,T;L²_ρ) is used in Thm 1.4(d). The natural formulation uses SIAM's test class
   D(R³×[0,T[)³ with E₀ = 0, and the unified formulation uses SIAM's test space H¹_{T0}(0,T;W¹_♦).
6. The paper states explicitly that J_t vanishes on the open set U ⊃ N, because curl A* = 0 there.
7. Novelty framing: the ingredients are called classical. The paper relates the result to PPTW
   Assumption 4.3 / (23), and presents the answer to Harrach's question and Thm 1.3 as the new content.
8. The cycle-functional converse and the W^{1,p} remark are kept, labelled as sketches that are not used.
9. Recommended strengthening: the degenerate-torus example was added as Thm 1.2, so the lower bound cannot
   simply be dropped either (no necessity is claimed).
10. Optional items: atan2 at the contact points (new lead script), and the closed-form asymptotics of ℰ(h)
    (Remark 7.3).

All thirteen required fixes of referee report 2 were applied in this revision (2026-09-30):
1. Abstract and Zenodo description: the torus now "vanishes at least linearly across one cross-section
   (σ ≤ C|sin(θ/2)|, with θ the azimuth)".
2. The thesis is cited as [Sim14]: Lilian Simon (née Arnold), Diss. JGU Mainz 2014, urn:nbn:de:hebis:77-39057.
   The paper says that it keeps the class L^∞_R (its (3.12)) and the potential map (its Lemma 3.5) and does not
   treat general σ, in "Which hypotheses fail" and in "Scope and priority". The sentence "We did not locate a
   thesis of L. Arnold" was removed. This report and RESULT.md are updated.
3. Abstract: "This requires …" now reads "To solve this equation, Arnold and Harrach assume …". It is followed
   by "So neither hypothesis can simply be dropped, although we do not claim that either is necessary."
4. The abstract and the paragraph after Thm 1.4 now carry the qualifier ∇φ(·) ∈ L²(0,T;L²_ρ) of Thm 1.4(c),(d).
   "Measurably in time" was removed, and the paragraph explains why the qualifier holds automatically in (c).
5. The scalar constants in the proof of Thm 1.1 and in Remark 7.1 are now d_k; c_k denotes only the centres.
   The balls in the proof of Lemma 2.1 are now B(0,j). Two optional clashes were also removed: the disc in
   Lemma 5.2 is now D_𝕋, and the gap energy is now ℰ(h).
6. Verification record:
   - (a) the Lemma 5.1 bounds are now checked in 250-digit arithmetic, which resolves every margin, and the
     paper calls the inequalities elementary instead of saying they were checked "exactly";
   - (b) lead line 227 now reads `Eu / Il ** 2`, and the printed label and values (0.3559, 0.1161, …) are correct;
   - (c) the output was regenerated (73 PASS lines), and the README, this report and the docstring (about 15 s)
     are corrected.
7. "Semantic Scholar (twenty-seven)" had no logged request, so it was replaced by the logged lists: OpenAlex,
   20 works citing AH12, and zbMATH, 13 documents, both queried on 2026-09-30. queries.log records the decision.
   The further citing works that referee report 2 screened are named below.
8. Har12 pages: 630–633.
9. Quotation. The rendered p. 631 prints "A ↦ φ_A" with a mapsto arrow, so the earlier quotation was in fact
   verbatim; "A→ϕA" is how the PDF text layer can render it. Following the referee's alternative, the symbol
   is now outside the quotation marks: whether the mapping A ↦ φ_A "can be defined for general non-negative σ".
10. PPTW: the necklace violates both the disjoint-closure condition and (23) of Assumption 4.3. The paper says
    that the numbering is that of arXiv:1810.07974v2, and so does the bibliography entry.
11. Maz'ya's *Sobolev Spaces* [Maz11] is now cited for capacities and Sobolev extension domains. The paper
    adds that the failure of the extension property for two tangent balls also follows from its Lemma 4.3.
12. The Verification paragraph and this report record referee report 2: its independent code (47 checks) and
    the image-charge confirmation of ℰ(h) to 1e−9 for h ≥ 1e−8. The table above lists all three
    verifications.
13. Rebuilt with tectonic 0.17 (no warnings, no overfull or underfull boxes; 16 pages, each rendered at 1.4×
    and inspected). main.tex, references.bib and paper.pdf were recopied into release/, and source.zip and the
    three zenodo/ copies were rebuilt. The sha256, md5 and sizes were recomputed, and the Zenodo description is
    the new abstract.

Optional suggestions of referee report 2:
- O1 applied: the citations are ordered so that the labels print in order ([AH13a, AH13b], [AGLR20a, AGLR20b]).
- O2 not applied: no classical source for the bispherical series was checked in this revision, and no
  unchecked reference was added. Remark 7.3 still cites Keller (1963) for the logarithmic growth.
- O3 applied: Acevedo–Gómez–López-Rodríguez, arXiv:2005.05429 (well-posedness; σ between two positive constants
  on a fixed conductor and zero in the insulator), was read and added to the screened list as [AGLR20c].
- O4 applied: Remark 7.4 now gives the argument for the "more generally" claim.

## Relation to the literature, novelty and scope
- **Searches (September 2026).**
  - Sources: arXiv API, Crossref, OpenAlex and zbMATH, the catalogue of the German National Library (DNB),
    and one web search per agent. All requests were anonymous and are logged in queries.log.
  - Citing documents of Arnold–Harrach screened: OpenAlex lists 20 works and zbMATH 13 documents (both queried
    on 2026-09-30).
  - No work treats div σ∇φ = −div σA for general σ ≥ 0. None gives a counterexample with touching or
    degenerate conductors, or a nonexistence result of this kind.
- **Closest works.**
  - The thesis of L. Simon (née Arnold), Mainz 2014: the same class L^∞_R and potential map as the SIAM paper,
    extended to bounded domains; no general σ.
  - Pauly–Picard–Trostorff–Waurick, JFA 280 (2021) 108847 (numbering of arXiv:1810.07974v2). Their sufficient
    conditions are disjoint closures, the extension condition (23), and positive definite σ; there is no
    necessity result. The necklace violates the first two.
  - Francini–Franzina–Vessella (2020) and Versaci et al. (2026) assume uniformly positive σ.
  - Reis–Stykel, Chill–Reis–Stykel, Hensel–Yousept, Nicaise, Nicaise–Tröltzsch, Tröltzsch–Valli,
    Kalinin–Tiukhtina–Lavrova, Tyukhtina and Acevedo–Gómez–López-Rodríguez (arXiv:2005.05429) assume a fixed
    conducting region with some regularity.
  - Acevedo–Gómez–López-Rodríguez (arXiv:2005.14339, 2009.02607): numerics.
  - Further works that OpenAlex lists as citing AH12 and that the paper does not name. Referee report 2 read
    the titles and abstracts of all 20 citing works and found none that treats general σ ≥ 0. The eight it
    names assume positive conductivity on a fixed conductor or are numerical:
    - Stratis–Yannacopoulos (2015), doi:10.1002/mma.3323;
    - Tröltzsch–Valli (2016), Optimization, doi:10.1080/02331934.2016.1179301;
    - Kalinin–Tyukhtina (2018), Math. Methods Appl. Sci., doi:10.1002/mma.5259;
    - Römer (2016), doi:10.1007/978-3-319-41294-8_2;
    - Lukáš et al. (2017), doi:10.15598/aeee.v15i2.2262;
    - Kolmbauer–Langer (2012), doi:10.1137/110842533;
    - Axelsson–Lukáš (2019), doi:10.1515/jnma-2017-0064;
    - Mohapatra–Deka (2025/26), doi:10.1016/j.cam.2025.116984.

    Two further items are listed here by title: "Modeling and Control of Low-Frequency Electromagnetic Fields
    in Multiply Connected Conductors" (2016), doi:10.1007/978-3-319-55795-3_48, and "Modified gauge conditions
    for Maxwell equations in quasi-stationary magnetic approximation" (2017),
    doi:10.15507/2079-6900.19.201704.55-67.
- **Classical ingredients.** Zero capacity of points and the failure of the H¹-extension property for tangent
  balls (Maz'ya, *Sobolev Spaces*); the logarithmic gap energy (Keller 1963); degenerate weights.
- **Caveats.**
  - Only E₀ = 0 is treated.
  - We do not characterise the σ for which φ_A always exists.
  - This negative search is not a proof of priority.
- **Scope.** The note answers Harrach's question negatively, in the H¹_loc reading used by Arnold–Harrach, and
  shows that no reading can keep "A + ∇φ_A solves the eddy-current equation". The formulation-level part
  survives for every σ ≥ 0.

## 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/