{"id":"72cc952f-54ca-4e5d-aa95-2785a952ea69","arxiv_id":"2502.05958","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Introduces weak partial monoids and simplicial effects, and claims a new simplicial effect beyond effect algebroids whose states match all density operators.","lead":"This paper defines a new category of simplicial effects that generalizes effect algebras and effect algebroids by weakening associativity. It also constructs a claimed example of a simplicial effect that is not an effect algebroid, with a Gleason-type theorem identifying its states with all quantum states.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The non-effect-algebroid witness Psi in Construction 6.11 fails normalization and the claimed identity d2(Psi)=d1(Pi) is false, so the proof that Z is not 2-Segal loses its witness.","rationale":"The structural material in Sections 3-5, including the nerve equivalences and the hierarchy of associativity notions, appears carefully developed and is not the source of my objection. The load-bearing part of the paper is the explicit example in Construction 6.11: Definition 6.5 only becomes interesting if there is a simplicial effect that is not an effect algebroid, and Proposition 6.12 is the only displayed evidence for that. Its proof depends on a specific 2-simplex Psi. Direct computation from the displayed formulas shows Psi is not a normalized projective measurement, so it cannot be a simplex of PH(N Z/3); moreover, even treating the formula formally, the claimed face equality d2(Psi)=d1(Pi) is false. This is an internal computational inconsistency, not a matter of disagreeing with a background consensus, and it invalidates the non-2-Segal witness. The state theorem for Z is part of the same advertised package, and while Proposition 6.16 may in principle be independent of Psi, the overall claim that Z is a simplicial effect outside EffAlgd with a Gleason-type state space is no longer established by the displayed arguments. The reader's weakest assumption identifies the same normalization defect, and I concur. The appropriate verdict is REJECT unless the example can be repaired or replaced.","tokens_in":29265,"tokens_out":11880,"duration_ms":108970,"concrete_test":"Recompute the sum of the nine entries defining Psi and evaluate d2(Psi) and d1(Pi) using the fibre-sum formula in Definition 2.13. If sum(Psi) = I - Gamma_22, or if d2(Psi)_1 - d1(Pi)_1 = Gamma_10 + Gamma_22, then the boundary data in Proposition 6.12 do not exist. To make the repair explicit, also check whether any normalized 2-simplex in X2 satisfying the imposed zero conditions can have d2(Psi)=d1(Pi); if no such replacement exists, the non-2-Segal argument needs a different witness or a different example.","verdict_should_be":"REJECT","load_bearing_attack":"Construction 6.11 defines Psi with nonzero entries Psi_20=Gamma_+, Psi_10=Pi_01, Psi_01=Gamma_00, Psi_22=Gamma_-, and zeros elsewhere. Since Gamma_+ + Gamma_- = Gamma_02 + Gamma_20, the sum of the nine projectors is Gamma_00 + (Gamma_01+Gamma_11+Gamma_21+Gamma_12) + Gamma_02 + Gamma_20 = I - Gamma_22. Thus Psi is not a normalized projective measurement and is not an element of PH(N Z/3), hence not in Z. Independently of normalization, the stated face identity fails: using the fibre-sum face maps from Definition 2.13, d2(Psi)_0 = Gamma_00, d2(Psi)_1 = Pi_01 = Gamma_01+Gamma_11+Gamma_21+Gamma_12, d2(Psi)_2 = Gamma_02+Gamma_20, whereas d1(Pi)_0 = Gamma_00, d1(Pi)_1 = Pi_01 + Gamma_10 + Gamma_22, d1(Pi)_2 = Gamma_02+Gamma_20. The c=1 entries differ by Gamma_10+Gamma_22. The assertion 'a quick computation to see that d2(Psi)=d1(Pi)' is therefore false, and the claimed pair of 2-simplices used to test 2-Segality does not exist. Proposition 6.12(2) thus has no displayed witness, so the central claim that Z is a simplicial effect outside effect algebroids is unsupported as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces weak partial monoids and weakly associative partial groups as intermediate associativity notions between Segal's partial monoids and Chermak's partial groups, then defines a category SimpEff of simplicial effects as spiny, inverseless, weakly 2-Segal cyclic sets. The main structural results characterize nerves of partial unital magmas, weak partial monoids, and partial monoids in terms of spiny reduced 2-coskeletal sets, spiny reduced weakly 2-Segal sets, and spiny reduced 2-Segal sets, respectively. The paper's advertised application is a constructed object Z, a full simplicial subset of PH(N Z/3), claimed to be a simplicial effect that is not an effect algebroid, with a Gleason-type bijection between density operators and states on Z.","tokens_in":29584,"tokens_out":10987,"duration_ms":99556,"significance":"If the main example were correct, the paper would extend the theory of effect algebras and effect algebroids to a weakly associative simplicial setting, unifying work on simplicial distributions, commutative nerves, and partial groups. The categorical framework in Sections 3 and 4 is detailed and appears to contain substantial contributions: the nerve equivalences for WPM and Mag are argued carefully, and the relationship with Chermak's partial associativity structures is clarified. However, the central example in Construction 6.11 and Proposition 6.12 contains a load-bearing error: the purported 2-simplex Psi is not a normalized projective measurement and the claimed face identity fails. As a result, the paper's headline claim that Z lies outside effect algebroids is not established as written. The significance of the paper therefore depends on whether that example can be repaired.","major_comments":[{"comment":"The 2-simplex Psi defined in Construction 6.11 is not a normalized projective measurement. With the stated entries, the sum over its nine projectors is Gamma_00 + Pi_01 + Gamma_+ + Gamma_- = Gamma_00 + (Gamma_01 + Gamma_11 + Gamma_21 + Gamma_12) + (Gamma_02 + Gamma_20) = I - Gamma_10 - Gamma_22. Since PH(N Z/3)_2 consists precisely of projector-valued functions summing to the identity, Psi is not an element of PH(N Z/3) and hence not an element of Z. Consequently the map Delta^2 -> Z constructed from Psi does not exist, and the proof that Z is not 2-Segal in Proposition 6.12(2) loses its witness.","section":"§6.3, Construction 6.11 and Proposition 6.12"},{"comment":"Independently of normalization, the asserted identity d2(Psi) = d1(Pi) is false under the fibre-sum face maps determined by Definition 2.13. For the face maps d2(P)_k = sum_b P_{k,b} and d1(P)_k = sum_{a+b=k} P_{a,b}, one obtains d2(Psi)_0 = Gamma_00, d2(Psi)_1 = Pi_01, and d2(Psi)_2 = Gamma_02 + Gamma_20, whereas d1(Pi)_0 = Gamma_00, d1(Pi)_1 = Pi_01 + Gamma_10 + Gamma_22, and d1(Pi)_2 = Gamma_02 + Gamma_20. The c=1 entries differ by Gamma_10 + Gamma_22, so the 'quick computation' asserted in the text is contradicted by the definitions.","section":"§6.3, Construction 6.11"},{"comment":"Because the only displayed witness to non-2-Segality is invalid, the paper does not currently establish that Z is a simplicial effect outside the category of effect algebroids. The state-space theorem in Proposition 6.16 is a separate claim and may be repairable, but it does not supply the missing non-2-Segal witness. The authors should provide a valid pair of 2-simplices satisfying the required face identifications, verify normalization and the Z-defining conditions, and then check the non-commutation argument for the associated unitaries A, B, C.","section":"§6.3, Proposition 6.12 and §6.5, Proposition 6.16"}],"minor_comments":[{"comment":"There is a duplicate Definition 2.12: the paragraph beginning 'Definition 2.12 ([1])' ends with 'A more general version of this definition will appear in Definition 2.12,' which is circular and should be renumbered and corrected.","section":"§2.4"},{"comment":"In the proof of Proposition 4.14, 'multaplicable' should be 'multiplicable'.","section":"§4.1"},{"comment":"In the proof of Proposition 3.13, 'it is similarly easy to se that' should read 'it is similarly easy to see that'.","section":"§3.2"},{"comment":"In Example 2.19, 'On can verify all of the cyclic identities' should read 'One can verify all of the cyclic identities'.","section":"§2.5"},{"comment":"The proof of Lemma 6.9 invokes the statement that weak 2-Segality 'amounts to' the unique extension property against Delta^w_n -> Delta^n. This equivalence is not stated explicitly in Section 4; adding a short proof or precise reference would improve clarity, since the lemma is used to show that Z is weakly 2-Segal.","section":"§4.3, Lemma 6.9"}],"recommendation":"major_revision","confidential_remarks":"The general framework in Sections 3 and 4 appears to contain substantial and interesting material, but the key example advertised in the abstract and introduction is defective as written. The error is concrete and localized: the 2-simplex Psi is not normalized and does not satisfy the claimed face identity. If the authors can supply a corrected witness and verify the non-2-Segal and state claims, the paper may be suitable for publication. I would not recommend acceptance in the current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this paper has two halves. The categorical half—weak partial monoids, weak 2-Segal sets, nerve equivalences—is careful, original, and likely correct. The applicational half, centered on the simplicial effect Z built from PH(N Z/3), contains a load-bearing error in the displayed 2-simplex Psi, so the paper's headline example does not work as written.\n\nWhat's genuinely new: the intermediate associativity notion between Segal and Chermak is a useful idea, and the nerve characterizations in Sections 3–4 are argued in real detail. The definitions of weak partial monoids and weak 2-Segal simplicial sets are natural. The category SimpEff and the embeddings Eff -> EffAlgd -> SimpEff are interesting. If those structural results survive contact with a referee, they alone justify a paper.\n\nThe weak spot is Section 6. The stress-test note is right. The 2-simplex Psi in Construction 6.11 sums to I - Gamma_22, not I, so it is not in PH(N Z/3) and not in Z. And the claimed face identity d2(Psi)=d1(Pi) is false: the middle entries differ by Gamma_10+Gamma_22. Since Psi is the only displayed witness for non-2-Segality, Proposition 6.12(2) is unsupported. The proof of Proposition 6.16 (states on Z) also leans on Example 6.15(2) in a way that doesn't survive once the construction is corrected; at minimum it needs a fresh argument. This isn't a nitpick—the abstract promises a concrete simplicial effect outside effect algebroids, and that is exactly what the error removes.\n\nMy guess is the categorical framework can be repaired: either Z has a different witness, or a modified construction works. But as it stands, the central claim is unproven, and the paper shouldn't be accepted without serious revision.\n\nWho is this for: people working on higher Segal spaces, effectus theory, and simplicial contextuality. The structural sections deserve a serious referee; the example does not hold. I'd send it to review rather than desk-reject—the flaw is concrete and fixable, and the good parts are substantial enough that a referee report pointing at Psi would likely be productive. But I would not cite the example until it's corrected.","headline":"Solid categorical framework, but the paper's showcase example—the simplicial effect outside effect algebroids—has a concrete error in its defining 2-simplex, so the main application doesn't hold as written.","tokens_in":30150,"tokens_out":2681,"would_cite":false,"duration_ms":25114,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18N50","81P10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that by replacing full associativity with weak associativity, quantum measurements form a new category of simplicial effects that strictly extends effect algebras and effect algebroids, and it constructs a concrete member…","keywords":["simplicial effects","weak partial monoids","weakly 2-Segal spaces","effect algebroids","simplicial distributions","quantum measurements","cyclic sets","density operators"],"falsifier":"Recompute the projector sum of the 2-simplex $\\Psi$ in Construction 6.11. If the entries sum to $I - |10\\rangle\\langle 10| - |22\\rangle\\langle 22|$ rather than $I$, then $\\Psi$ is not a valid 2-simplex of PH(N Z/3), and the map used to show Z is not 2-Segal is not well-defined; a different witness would be needed for that claim and for the state theorem.","tokens_in":29027,"feed_emoji":"⚛️","tokens_out":9519,"duration_ms":88690,"temperature":0.7,"pith_summary":"The paper tries to establish that the algebraic scaffolding of quantum measurements can be weakened: when full associativity is relaxed to weak associativity, a new category of simplicial effects emerges that contains effect algebras and effect algebroids as a proper part. It constructs a concrete object Z from the projective-measurement space PH(N Z/3) that is a simplicial effect but not an effect algebroid, since it fails the 2-Segal condition. It then shows that the states of Z are in bijection with density operators, so the familiar correspondence between quantum states and effects survives in the new setting. A careful reader should care because this provides a strictly larger category of measurement-like structures that still carries a meaningful state theory.","feed_headline":"Simplicial effects strictly widen the algebra of quantum measurements","feed_subtitle":"A concrete object outside effect algebroids has states given exactly by density operators.","key_machinery":"The central device is the nerve functor from partial unital magmas with associativity data to simplicial sets, together with the weak 2-Segal condition that characterizes weak partial monoids. In degree 3, the weak 2-Segal condition says that if both $(a\\cdot b)\\cdot c$ and $a\\cdot(b\\cdot c)$ are defined then they agree, which is the exact associativity notion the paper uses. The construction Z is formed by taking the full simplicial subset of the projective-measurement simplicial set PH(N Z/3) cut out by zeros in certain projector entries; this removes inverses while preserving weak 2-Segalness, and the cyclic structure supplies the orthocomplement.","core_discovery":"On the paper's own terms, the central claim is that a simplicial effect is exactly a spiny, inverseless, weakly 2-Segal cyclic set: spiny means the 1-Segal maps are injective, inverseless means no non-identity morphism has a left or right inverse, weakly 2-Segal is the nerve-level shadow of weak associativity, and the cyclic structure encodes the orthocomplement. The paper proves that the nerve construction yields fully faithful embeddings Eff -> EffAlgd -> SimpEff, and that the full simplicial subset Z of PH(N Z/3) whose 2-simplices satisfy $\\Pi_{11}=\\Pi_{21}=\\Pi_{12}=0$ is a simplicial effect, is not an effect algebroid, and has state space in bijection with the density operators on $\\mathbb{C}^3\\otimes\\mathbb{C}^3$ via the formula $\\Pi \\mapsto \\mathrm{Tr}(\\rho(\\bar{\\Pi}_0 - \\tfrac12\\Pi_2))$.","pith_inferences":["Beyond the paper: the same full-subset cut-out that builds Z could be run on other d-torsion commutative nerves, yielding a family of simplicial effects with possibly different state spaces; the paper does not test this.","Beyond the paper: the weak 2-Segal condition may correspond to a concrete probabilistic constraint in the simplicial-distributions picture, such as non-signalling or contextuality; the paper does not draw this connection.","Beyond the paper: if the state bijection for Z is correct, it suggests quantum state spaces can be encoded by cyclic-simplicial structure alone, without an underlying effect algebra; the authors do not state this as a general slogan.","Testable extension: perturbing the zero conditions $\\Pi_{11}=\\Pi_{21}=\\Pi_{12}=0$ one at a time would show which parts of the state formula are load-bearing; the paper does not vary these conditions."],"forward_implications":["Every effect algebra and every effect algebroid embeds fully faithfully into SimpEff, so the new category is a common home for both classical and simplicial effect structures.","The commutative d-torsion nerve N(Z/d, U(H)) is a weakly associative partial group, placing projective-measurement spaces at the invertible extreme of the hierarchy, while simplicial effects form the inverseless extreme.","The object Z is a simplicial effect that is not an effect algebroid, so the weak 2-Segal condition captures measurement-like structures that full 2-Segalness misses.","States on Z are in bijection with density operators on $\\mathbb{C}^3\\otimes\\mathbb{C}^3$, giving a concrete state theorem in the new setting.","Under mild hypotheses, the first cyclic cohomology of a simplicial effect is the vector space spanned by its states."],"supporting_citations":[{"why":"Supplies the simplicial-distribution and projective-measurement framework, including the identification of PH(N Z/d) with a classifying space.","marker":"[3]"},{"why":"Defines effect algebroids and characterizes them as 2-Segal cyclic sets, the category being extended.","marker":"[4]"},{"why":"Provides the partial-monoid and nerve framework whose associativity condition is weakened to weak associativity.","marker":"[2]"},{"why":"Supplies the partial-group associativity data that forms the other extreme in the hierarchy of weak partial monoids.","marker":"[5]"},{"why":"Defines 2-Segal spaces and membrane sets, giving the simplicial conditions used throughout Section 4.","marker":"[20]"},{"why":"Provides the classical bijection between density operators and states on projectors that the state theorem for Z extends.","marker":"[11]"},{"why":"Introduces commutative nerves, the simplicial sets from which the example Z is carved out.","marker":"[15]"},{"why":"Introduces d-torsion commutative nerves, which appear in the projective-measurement example via the identification with PH(N Z/3).","marker":"[16]"},{"why":"Sets out the cyclic category and the cyclic-cohomology conventions used to define states on cyclic sets.","marker":"[18]"}],"fun_headline_variants":["Simplicial effects widen quantum measurements beyond effect algebroids","Concrete simplicial effect yields exactly density operator states","Weak associativity in partial monoids births simplicial effects and groups","Simplicial effect outside algebroids: states are density operators"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on a specific assignment of projectors being a valid measurement, meaning its projectors must sum to the identity; if that assignment is incorrect, the proof that Z is not an effect algebroid and the state bijection for Z both lose their stated support.","fun_headline_variants_meta":{"raw":{"variants":["Simplicial effects widen quantum measurements beyond effect algebroids","Concrete simplicial effect yields exactly density operator states","Weak associativity in partial monoids births simplicial effects and groups","Simplicial effect outside algebroids: states are density operators"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000849,"raw_usage":{"total_tokens":3632,"prompt_tokens":823,"completion_tokens":2809,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":439,"completion_tokens_details":{"reasoning_tokens":2743}},"tokens_in":439,"tokens_out":2809,"duration_ms":19133,"temperature":1.0,"reasoning_tokens":2743,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T17:14:07.990703+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the projector sum of the 2-simplex $\\Psi$ in Construction 6.11. If the entries sum to $I - |10\\rangle\\langle 10| - |22\\rangle\\langle 22|$ rather than $I$, then $\\Psi$ is not a valid 2-simplex of PH(N Z/3), and the map used to show Z is not 2-Segal is not well-defined; a different witness would be needed for that claim and for the state theorem.","supporting_citations":[{"cited_title":"Simplicial quantum contextuality,","cited_arxiv_id":null,"evidence_quote":"Supplies the simplicial-distribution and projective-measurement framework, including the identification of PH(N Z/d) with a classifying space."},{"cited_title":"Roumen,Effect Algebroids","cited_arxiv_id":null,"evidence_quote":"Defines effect algebroids and characterizes them as 2-Segal cyclic sets, the category being extended."},{"cited_title":"Configuration-spaces and iterated loop-spaces,","cited_arxiv_id":null,"evidence_quote":"Provides the partial-monoid and nerve framework whose associativity condition is weakened to weak associativity."},{"cited_title":"Fusion systems and localities,","cited_arxiv_id":null,"evidence_quote":"Supplies the partial-group associativity data that forms the other extreme in the hierarchy of weak partial monoids."},{"cited_title":"Higher Segal spaces","cited_arxiv_id":null,"evidence_quote":"Defines 2-Segal spaces and membrane sets, giving the simplicial conditions used throughout Section 4."},{"cited_title":"Measures on the closed subspaces of a hilbert space,","cited_arxiv_id":null,"evidence_quote":"Provides the classical bijection between density operators and states on projectors that the state theorem for Z extends."},{"cited_title":"Commuting elements, simplicial spaces and filtrations of classifying spaces,","cited_arxiv_id":null,"evidence_quote":"Introduces commutative nerves, the simplicial sets from which the example Z is carved out."},{"cited_title":"Commutative d-torsion K-theory and its applications,","cited_arxiv_id":null,"evidence_quote":"Introduces d-torsion commutative nerves, which appear in the projective-measurement example via the identification with PH(N Z/3)."},{"cited_title":"Loday,Cyclic homology, vol","cited_arxiv_id":null,"evidence_quote":"Sets out the cyclic category and the cyclic-cohomology conventions used to define states on cyclic sets."}],"review_version":1}