Simplicial methods in the resource theory of contextuality

Aziz Kharoof; Cihan Okay

REVIEW 1 major objections 5 minor 1 cited by

Simplicial methods in the resource theory of contextuality

T0 review · 1 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Convex maps between simplicial distributions are exactly the images of noncontextual distributions on mapping scenarios.

desk verdict A genuinely new extension of the Closing Bell theorem to simplicial scenarios, with a real but likely repairable gap in the Appendix B naturality proof that underpins the stochastic category. read the letter →

arxiv 2505.24010 v1 pith:WDZX4R36 submitted 2025-05-29 quant-ph math.ATmath.CT

Aziz Kharoof , Cihan Okay This is my paper

classification quant-phmath.ATmath.CT MSC 18M0518C2055U10

keywords contextualitysimplicialdistributionsbundlescenarioseventmappingresourcetheorysymmetricmonoidalcategoriesGrothendieckconstruction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper builds a resource theory of contextuality, the obstruction to explaining empirical probability data as mixtures of global assignments, in the language of simplicial sets and symmetric monoidal categories. It introduces event scenarios as a functorial generalization of presheaf-theoretic measurement scenarios and proves, via the Grothendieck construction, that they are equivalent to bundle scenarios. It then extends the distribution monad to a stochastic category of simplicial scenarios and defines mapping scenarios as internal-hom objects. The main theorem states that a convex map between the spaces of simplicial distributions of two scenarios is a convex combination of maps induced by scenario morphisms if and only if it is the image, under the map $\mu_{f,g}$, of a noncontextual simplicial distribution on the corresponding mapping scenario. If correct, this gives a uniform categorical reformulation of the resource theory of contextuality, recovering the earlier Closing Bell characterization for standard scenarios as a special case.

What carries the argument

The load-bearing mechanism is the simplicial mapping scenario $\mathrm{Map}(f,g)$: its $n$-simplices are triples $(y;x,\alpha)$ consisting of an $n$-simplex $y$ of the target scenario, an $n$-simplex $x$ of the source, and a morphism of simplicial scenarios from the pullback of $f$ along $x$ to the pullback of $g$ along $y$. From this object the paper builds the convex map $\mu_{f,g}:\mathrm{sDist}(\mathrm{Map}(f,g))\to \mathrm{Conv}(\mathrm{sDist}(f),\mathrm{sDist}(g))$, defined by taking limits of a family of maps that send sections and distributions on the mapping scenario to convex transformations between the two distribution spaces. The theorem states that $\mu_{f,g}$ is an exact classifier: a convex transformation is a convex combination of morphism-induced maps precisely when it is the $\mu_{f,g}$-image of a noncontextual distribution.

What would settle it

Calculate both sides of the naturality equation $D(\alpha\times\beta)\circ m_{f,g}=m_{f',g'}\circ(D(\alpha)\times D(\beta))$ for a commutative diagram of shape (52) in which $\gamma$ is not injective; any inequality would falsify the gluing axiom and invalidate the stochastic category $\mathrm{sScen}_D$ that underlies Theorem 4.17. Alternatively, find simplicial bundle scenarios $f,g$ and a convex map $\varphi:\mathrm{sDist}(f)\to\mathrm{sDist}(g)$ that is a convex combination of morphism-induced maps but has no noncontextual $p$ with $\varphi=\mu_{f,g}(p)$.

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Extended reading notes

Core claim

The central discovery is that the stochastic category of simplicial scenarios $\mathrm{sScen}_D$ carries a complete classification of convex transformations between simplicial distributions. Given simplicial bundle scenarios $f:E\to X$ and $g:F\to Y$, the paper constructs the simplicial mapping scenario $\mathrm{Map}(f,g)\to Y$ and a convex map $\mu_{f,g}:\mathrm{sDist}(\mathrm{Map}(f,g))\to \mathrm{Conv}(\mathrm{sDist}(f),\mathrm{sDist}(g))$ that sends a distribution on the mapping scenario to a convex transformation of distribution spaces. Theorem 4.17 asserts that a convex map $\varphi:\mathrm{sDist}(f)\to\mathrm{sDist}(g)$ is a convex combination of maps induced by morphisms in $\mathrm{sScen}(f,g)$ if and only if there exists a noncontextual simplicial distribution $p$ on $\mathrm{Map}(f,g)$ with $\varphi=\mu_{f,g}(p)$. The theorem extends the Closing Bell theorem of [4] from standard measurement scenarios to all simplicial bundle scenarios, and the paper shows in Section D.2 that the presheaf-theoretic result follows as a special case through the nerve and category-of-elements equivalences.

Load-bearing premise

The whole construction assumes the distribution monad satisfies the naturality axiom describing how its gluing operation behaves under relabeling, and the provided verification uses an injectivity condition that is not explicitly stated; if that axiom fails, the stochastic category of simplicial scenarios is not well-defined and Theorem 4.17 has no categorical foundation.

Editorial extensions

If this is right

For standard measurement scenarios, Theorem 4.17 reproduces the Closing Bell characterization: convex maps between empirical models are exactly the images of noncontextual empirical models on the mapping scenario (Section D.2).
The equivalence of event and bundle scenarios (Theorem 2.13) makes the two frameworks interchangeable: any contextuality statement about empirical models translates, through the nerve and category-of-elements functors, into a statement about simplicial distributions.
The symmetric monoidal structure on event scenarios gives a tensor product under which two individually noncontextual scenarios can compose into a contextual one (Example 2.22), so contextuality can be activated by parallel composition.
Because the simplicial distribution functor is represented by the terminal object in the stochastic category $\mathrm{sScen}_D$, the category of elements of this functor is a resource theory in the sense of [3], with stochastic simplicial maps as the free operations.
Every morphism of simplicial scenarios gives a section of $\mathrm{Map}(f,g)$, and Diagram (48) shows that the morphism-induced convex maps are exactly those $\mu_{f,g}$-images that come from sections; thus noncontextual distributions on the mapping scenario are the minimal data needed to generate all morphism-induced convex maps.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: If the naturality axiom for the distribution monad can be repaired or replaced, the same mapping-scenario classification should hold for any monad with a gluing operation satisfying Definition 3.1, yielding resource theories of contextuality over other stochastic processes.
Inference: The failure of closed monoidal structure (Proposition 4.11) suggests that the plain tensor product is not the right composition for a compositional resource theory; restricting to scenarios where the inclusion (40) is a bijection, or choosing a different monoidal product, may be needed to obtain internal homs.
Inference: Theorem 4.17 turns the search for contextual transformations into a search over distributions on a single space: to certify that a convex map $\varphi$ is not a convex combination of morphism-induced maps, it suffices to check that every $p\in\mathrm{sDist}(\mathrm{Map}(f,g))$ with $\mu_{f,g}(p)=\varphi$ is contextual.
Inference: The tensor-product activation example could be developed into a distillation-type statement, where the amount of contextuality is measured by whether a distribution lies outside the noncontextual image and tensor products are used to consume noncontextual resources to produce contextual ones.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

This paper has a genuine contribution and a real gap that looks repairable. The new stuff: event scenarios, the equivalence eScen ≃ bScen via Grothendieck, the symmetric monoidal structure, and the mapping scenario Map(f,g). Theorem 4.17 genuinely extends the Closing Bell theorem [4] to simplicial scenarios, and the recovery of [4] in Appendix D is honest. The paper is also straight about where it falls short: it admits the mapping scenario is not closed monoidal relative to the tensor product, and it leaves several monoidal axiom checks to the reader.

The soft spot is Appendix B. In proving Proposition 3.2(2), the authors state a stronger naturality claim than the axiom requires, and the proof substitutes D(f)(p)(f(x)) for a sum over γ^{-1}(f'(x')) without assuming γ injective. The stress-test counterexample is correct: with γ collapsing two elements, the stronger claim fails, and the left side has mass 0 where the right side has mass 1/4. This is not cosmetic, because Proposition 3.6 uses property (2) to prove π* is a functor on sBund_D, and that is what makes sScenD a category. So the written proof is invalid.

That said, the needed repair is small and likely correct. In the axiom's actual use inside the pullback functor, γ is the unit δ_Z, which is injective; the counterexample does not touch that instance. What the paper needs is a corrected proof that proves the axiom in the instance used, or an explicit statement of the injectivity condition with a note that it holds there. The reader's conditional verdict is fair.

The citation pattern is fine. The heavy reliance on prior work by the same authors is real, but the definitions are restated and the new results do not presuppose what they claim to prove. The main theorem is a new result relative to [4].

This paper is for people working in categorical quantum foundations or simplicial contextuality. The gap makes it not ready as-is, but it is clearly worth a serious referee. Recommend sending to peer review, with a request that the referee check the Appendix B fix and the monoidal axiom gaps.

Referee Report

1 major / 5 minor

Summary. The paper develops a categorical framework for contextuality in the spirit of simplicial distributions. It introduces event scenarios (Definition 2.3), proves via the Grothendieck construction that they are equivalent to bundle scenarios (Theorem 2.13), equips these categories with symmetric monoidal structures, and builds a stochastic category sScenD from a monad-with-gluing structure on the distribution monad (Definitions 3.1, 3.14, and 3.15). Mapping scenarios are introduced for event, bundle, and simplicial scenarios, and the main result (Theorem 4.17) characterizes convex maps between simplicial distributions as images, under the map mu_{f,g}, of noncontextual distributions on the simplicial mapping scenario, extending the 'Closing Bell' result to the simplicial setting.

Significance. If the technical gap discussed below is repaired, this is a significant contribution. The equivalence between event scenarios and bundle scenarios is a clean and useful bridge between presheaf-theoretic and bundle-based approaches, and the mapping-scenario characterization is a natural generalization of Barbosa--Karvonen--Mansfield's Theorem 44 to simplicial scenarios. The constructions are explicit, parameter-free, and illustrated with nontrivial examples (Examples 2.7 and 2.22). The main theorem, if fully established, gives a resource-theoretic statement in the spirit of Coecke--Fritz--Spekkens for contextuality.

major comments (1)

[Appendix B / Proposition 3.2(2)] The proof of the naturality axiom for the distribution monad's gluing operation is not valid as written. The text proves the stronger statement D(alpha x beta) composed with m_{f,g} equals m_{f',g'} composed with (D alpha x D beta) for arbitrary alpha, beta, gamma in Diagram (52). In the fourth equality of Eq. (54), the sum over z with gamma(z)=f'(x') is replaced by D(f)(p)(f(x)); this substitution requires gamma to be injective, but injectivity is neither stated in Diagram (52) nor guaranteed by Definition 3.1. The stronger statement is false in general: take Z={z1,z2} with gamma collapsing both to z', X=Y={x1,x2}, f=g with f(x_i)=z_i, alpha(x1)=x', alpha(x2)=x'', beta(y1)=y'', beta(y2)=y', and p=q uniform. Then the left-hand side has zero mass at (x',y') while the right-hand side has mass 1/4 there. This gap is load-bearing: Proposition 3.6 invokes property (2) to prove that the pullback functor pi* preserves composition, and that functoriality underlies the definitions of sBund_D, sScen_D, and the simplicial distribution functor used in Theorem 4.17. The repair is to prove only the instances of naturality actually used in Propositions 3.6 and 3.7, or to state and prove the axiom under the hypotheses holding there, and to remove the unproved stronger claim.

minor comments (5)

[Section 3.3, Eq. (27); Section 3.4, Eq. (32)] The map delta:sScen_id -> sScen_T is described as a fully faithful embedding; this is generally false, since a stochastic map alpha:E -> D(F) satisfying the bundle condition need not be of the form delta_F composed with a deterministic map. It should be called a faithful embedding, or additional hypotheses should be supplied.
[Proposition 2.21 and Section 3.4] Several symmetric monoidal coherence verifications are delegated to the reader. Please state the coherence isomorphisms explicitly, at least for the tensor product on sScenD, since this structure is used later in the mapping scenario constructions.
[Appendix B, Eq. (55)] The denominator D(f)(g)(g(y)) in the computation for weak multiplicativity appears to be a typo; it should be D(g)(q)(g(y)) or the corresponding expression in terms of D(f)(mu_X(P)).
[Definition 2.5] The direction of the simplicial map pi in a morphism (pi,alpha):F -> G should be reconciled with the Grothendieck construction convention given in Appendix A; as written, pi:Sigma' -> N-hat Sigma appears to conflict with the contravariant convention stated there.
[Proof of Proposition 2.21] The sentence 'it is clear that id_F tensor id_F' should read 'id_{F tensor F}'; please fix this typo.

Circularity Check

1 steps flagged · score 6.0 of 10

Main characterization in Theorem 4.17 reduces by construction to the definition of μ and of noncontextuality.

self definitional [Section 4.3, Theorem 4.17 and its proof, Eqs. (47)–(50); see also Definition 3.18.]
"The top map in Diagram (47) is equal to sDistf,g ◦ ζ−1 f,g. This way we have obtained a convex map µf,g : sDist(Map(f, g)) → Conv(sDist(f),sDist(g)) ... Since the composition µf,g ◦ ΘMap(f,g) ◦D(ζf,g) lies in Conv, by Diagram (49), we get that µf,g ◦ΘMap(f,g) ◦D(ζf,g) = sDist′f,g . (50) Using Equation (50) and the fact that ζf,g is bijective, we obtain the desired result."

By Definition 3.18, a distribution p on Map(f,g) is noncontextual exactly when p lies in the image of ΘMap(f,g). A convex combination of maps induced by morphisms in sScen(f,g) is, by the adjunction defining sDist′, exactly an element in the image of sDist′f,g : D(sScen(f,g)) → Conv(sDist(f),sDist(g)). Equation (50) is not an independent theorem: Diagram (47) defines μf,g precisely so that the top map is sDistf,g∘ζ−1 and (49)/(50) commute. Therefore both directions of Theorem 4.17 are immediate from the defining square: if φ = sDist′(q), take p = Θ(Dζ(q)); if φ = μ(p) with p = Θ(q), then φ = sDist′(Dζ−1(q)). The claimed characterization is true by construction, not derived from an independent property of contextuality.

full rationale

The central equivalence in Theorem 4.17 is loaded into the definitions: noncontextuality is defined as the image of Θ, and μ is introduced in Diagram (47) so that Equation (50) makes the two sides of the theorem equal. The proof then cites Equation (50) as its entire content, so the main characterization reduces to a commuting square built by definition. This is a genuine partial circularity of the central claim, though the paper also contains substantial independent categorical material: the event-scenario/bundle-scenario equivalence, monoidal structures, and mapping-scenario constructions are parameter-free and not preloaded with Theorem 4.17. The heavy reliance on [1,5,7,13] is standard mathematical reuse of stated definitions rather than circularity. The Appendix B naturality gap noted in the reader's take is a correctness issue, not a circularity issue, so it is not scored here. Because the main theorem reduces by construction but the surrounding framework has independent content, a score of 6 is appropriate.

Assumptions & free parameters 0 free parameters · 3 assumptions · 2 invented entities

The central claim rests on standard category-theoretic machinery plus one new domain assumption: the distribution monad has a gluing operation satisfying Definition 3.1. The proof of this for D is incomplete (see red flag). Event scenarios and stochastic simplicial scenarios are defined internally and carry no empirical parameters. No numbers are fitted. The axiom of choice is invoked in Proposition 4.3.

assumptions (3)

standard math Standard ZFC set theory and standard category theory: Grothendieck construction, simplicial sets and simplicial complexes, distribution monad, convex sets as algebras over the distribution monad.
Throughout: Definitions 2.5, 2.9, and 3.14 rely on these standard tools without proving them.
standard math Axiom of choice.
Invoked explicitly in Proposition 4.3 to choose preimages when constructing extensions of natural transformations; needed for local surjectivity of mapping scenarios.
domain assumption The distribution monad D has a gluing operation satisfying Definition 3.1, including its naturality axiom.
Proposition 3.2 is meant to prove this, but the proof of naturality in Appendix B uses unstated injectivity of γ. The stochastic category sScenD and the pullback functor π* rest on this assumption.

invented entities (2)

Event scenarios (Definition 2.3)
purpose: Functorial generalization of measurement scenarios as local, non-trivial, locally surjective presheaves on CΣ^op.
A new mathematical structure defined in the paper; no independent falsifiable prediction outside the formalism.
Stochastic simplicial scenarios (Definition 3.15)
purpose: Category sScenD whose morphisms are stochastic maps, used to define the simplicial distribution functor sDist.
Defined via the distribution monad; internal to the paper and without an external empirical handle.

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Cite this review

Pith. "Pith review of Simplicial methods in the resource theory of contextuality." pith.science (2026). https://pith.science/paper/WDZX4R36

@misc{pith2026250524010,
  author       = {Pith},
  title        = {Pith review of: Simplicial methods in the resource theory of contextuality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WDZX4R36}},
  note         = {Machine review of arXiv:2505.24010}
}

read the original abstract

We develop a resource theory of contextuality within the framework of symmetric monoidal categories, extending recent simplicial approaches to quantum contextuality. Building on the theory of simplicial distributions, which integrates homotopy-theoretic structures with probability, we introduce event scenarios as a functorial generalization of presheaf-theoretic measurement scenarios and prove their equivalence to bundle scenarios via the Grothendieck construction. We define symmetric monoidal structures on these categories and extend the distribution functor to a stochastic setting, yielding a resource theory that generalizes the presheaf-theoretic notion of simulations. Our main result characterizes convex maps between simplicial distributions in terms of non-contextual distributions on a corresponding mapping scenario, enhancing and extending prior results in categorical quantum foundations.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Possibilistic collapse and extremality of simplicial distributions
math.CT 2026-07 accept novelty 6.5 of 10

Strong connectivity of the support bundle (or a categorical analogue for event scenarios) is a sufficient condition for a simplicial distribution or empirical model to be extremal.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · cited by 1 Pith paper

[4]

Closing Bell: Boxing black box simulations in the resource theory of contextuality,

R. S. Barbosa, M. Karvonen, and S. Mansfield, “Closing Bell: Boxing black box simulations in the resource theory of contextuality,” inSamson Abramsky on Logic and Structure in Computer Science and Beyond(A. Palmigiano and M. Sadrzadeh, eds.), vol. 25 ofOutstanding Contributions to Logic, Springer, 2023

work page 2023
[1]

Simplicial quantum contextuality,

C. Okay, A. Kharoof, and S. Ipek, “Simplicial quantum contextuality,”Quantum, vol. 7, p. 1009, May 2023

work page 2023
[2]

The sheaf-theoretic structure of non-locality and contex- tuality,

S. Abramsky and A. Brandenburger, “The sheaf-theoretic structure of non-locality and contex- tuality,”New Journal of Physics, vol. 13, no. 11, p. 113036, 2011

work page 2011
[3]

A mathematical theory of resources,

B. Coecke, T. Fritz, and R. W. Spekkens, “A mathematical theory of resources,”Information and Computation, vol. 250, pp. 59–86, 2016

work page 2016
[5]

Simplicial distributions, convex categories and contextuality,

A. Kharoof and C. Okay, “Simplicial distributions, convex categories and contextuality,”Theory and Applications of Categories, vol. 44, no. 13, pp. 372–409, 2025

work page 2025
[6]

Topological proofs of contextuality in quantum mechanics,

C. Okay, S. Roberts, S. D. Bartlett, and R. Raussendorf, “Topological proofs of contextuality in quantum mechanics,”Quantum Information & Computation, vol. 17, no. 13-14, pp. 1135–1166, 2017

work page 2017
[7]

A bundle perspective on contextuality: Empirical models and simplicial distributions on bundle scenarios

R. S. Barbosa, A. Kharoof, and C. Okay, “A bundle perspective on contextuality: Empirical models and simplicial distributions on bundle scenarios,”arXiv preprint arXiv:2308.06336, 2023

work page Pith review arXiv 2023
[8]

The operadic theory of convexity,

R. Haderi, C. Okay, and W. H. Stern, “The operadic theory of convexity,”Applied Categorical Structures, vol. 33, no. 17, 2025

work page 2025

Show all 15 references

[9]

Mac Lane,Categories for the working mathematician, vol

S. Mac Lane,Categories for the working mathematician, vol. 5. Springer Science & Business Media, 2013

2013
[10]

Convexity, duality and effects,

B. Jacobs, “Convexity, duality and effects,” inIFIP International Conference on Theoretical Computer Science, pp. 1–19, Springer, 2010

2010
[11]

Riehl,Category theory in context

E. Riehl,Category theory in context. Courier Dover Publications, 2017

2017
[12]

P. G. Goerss and J. F. Jardine,Simplicial homotopy theory. Springer Science & Business Media, 2009

2009
[13]

Homotopical characterization of strongly contextual simplicial distributions on cone spaces,

A. Kharoof and C. Okay, “Homotopical characterization of strongly contextual simplicial distributions on cone spaces,”Topology and its Applications, vol. 352, p. 108956, 2024. A Grothendieck construction In this section, we introduce the Grothendieck construction [9] and a rel...

2024
[14]

The pullback(δ Σ)∗ :Bund(Σ)→Bund(Σ)along the unit map is the identity functor
[15]

Pullback is functorial with respect to Kleisli composition: (π1 ⋄π 2)∗ =π ∗ 2 ◦π ∗ 1. Proof. Part (1): For a bundle scenario f: Γ → Σ, by [7, Lemma A.9] we have ( δΣ)∗(f ) = f . On the other hand, for a morphism α:f→g in Bund(Σ), due to the naturality of δ, we obtain the follo...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [4]

Closing Bell: Boxing black box simulations in the resource theory of contextuality,

R. S. Barbosa, M. Karvonen, and S. Mansfield, “Closing Bell: Boxing black box simulations in the resource theory of contextuality,” inSamson Abramsky on Logic and Structure in Computer Science and Beyond(A. Palmigiano and M. Sadrzadeh, eds.), vol. 25 ofOutstanding Contributions to Logic, Springer, 2023

work page 2023

[2] [1]

Simplicial quantum contextuality,

C. Okay, A. Kharoof, and S. Ipek, “Simplicial quantum contextuality,”Quantum, vol. 7, p. 1009, May 2023

work page 2023

[3] [2]

The sheaf-theoretic structure of non-locality and contex- tuality,

S. Abramsky and A. Brandenburger, “The sheaf-theoretic structure of non-locality and contex- tuality,”New Journal of Physics, vol. 13, no. 11, p. 113036, 2011

work page 2011

[4] [3]

A mathematical theory of resources,

B. Coecke, T. Fritz, and R. W. Spekkens, “A mathematical theory of resources,”Information and Computation, vol. 250, pp. 59–86, 2016

work page 2016

[5] [5]

Simplicial distributions, convex categories and contextuality,

A. Kharoof and C. Okay, “Simplicial distributions, convex categories and contextuality,”Theory and Applications of Categories, vol. 44, no. 13, pp. 372–409, 2025

work page 2025

[6] [6]

Topological proofs of contextuality in quantum mechanics,

C. Okay, S. Roberts, S. D. Bartlett, and R. Raussendorf, “Topological proofs of contextuality in quantum mechanics,”Quantum Information & Computation, vol. 17, no. 13-14, pp. 1135–1166, 2017

work page 2017

[7] [7]

A bundle perspective on contextuality: Empirical models and simplicial distributions on bundle scenarios

R. S. Barbosa, A. Kharoof, and C. Okay, “A bundle perspective on contextuality: Empirical models and simplicial distributions on bundle scenarios,”arXiv preprint arXiv:2308.06336, 2023

work page Pith review arXiv 2023

[8] [8]

The operadic theory of convexity,

R. Haderi, C. Okay, and W. H. Stern, “The operadic theory of convexity,”Applied Categorical Structures, vol. 33, no. 17, 2025

work page 2025