{"id":"9f623442-03fe-4c39-8f87-79aad333ebcd","arxiv_id":"2608.09262","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every partial Boolean algebra is isomorphic to the core of a consistent classical propositional theory, and Kochen-Specker non-colourability is reformulated as the absence of a dispersion-free model.","lead":"This paper shows that the non-Boolean logic of quantum propositions can be reproduced inside ordinary classical logic, by identifying any partial Boolean algebra with the 'core' of a consistent classical theory. Its Kochen-Specker obstruction becomes a statement about which classical models exist, illustrated by a new 140-vertex partial Boolean algebra built from the Peres configuration.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 16.5 asserts P6 of V140 in one sentence; transitivity of the global order is not proved or checked by the Appendix C script, and compatibility of consecutive pairs does not supply the required common side.","rationale":"The reader's conditional verdict is reasonable: the heavy finite verification is unshipped. My stress-test agrees that the finite side of V140 is the fragile part, but I identify a different, more immediate load-bearing gap within the written proof itself. Theorem 16.5 asserts P6 in one sentence, and the Appendix C certificate does not list any check of transitivity or antisymmetry for the global order. The proof of transitivity cannot be obtained simply by applying Lemma 16.2(7), because that lemma requires the family to be pairwise compatible already, and compatibility of consecutive pairs does not imply compatibility of the endpoints. This is not a disagreement with the likely truth of the construction; it is a precise missing argument in the verification chain. Since the issue is finite and independently checkable, the appropriate posture remains CONDITIONAL rather than rejection: if the authors add the order-transitivity check to the certificate, or supply a hand proof, the concern is resolved. The reader's weakest assumption and my concern are related but not identical, so agreement is partial: they focus on Lemma 16.2(7) and maximal-clique enumeration; I focus on the unverified P6 axiom, which the Appendix C script does not explicitly test. No objection is raised to the conceptual contributions: the model-theoretic realization of partial Boolean algebras, the cluster characterization, and the parity obstruction are internally coherent given the finite verification. The concrete test I propose would settle the P6 question with modest computation and without requiring the authors' code.","tokens_in":30649,"tokens_out":6875,"duration_ms":67591,"concrete_test":"From the side list (Σ) alone, compute the 140 congruence classes and the global relation ≤ defined before Proposition 16.4. Then exhaustively check, for all triples of classes x, y, z, that x ≤ y and y ≤ z imply x ≤ z, and for all pairs that x ≤ y and y ≤ x imply x = y. This is a finite check of roughly 140^3 ordered triples (or about 2.7 million comparisons) and can be implemented independently of the authors' script. The test should be added to Appendix C as an explicit verification step, or replaced by a hand proof from Lemma 16.2. If the check passes, the P6 gap is closed; if it fails, V140 is not a partial Boolean algebra and the main non-KS corollary is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central non-KS example V140 depends on satisfying all six partial-Boolean-algebra axioms. Theorem 16.5 dispatches P6 with: 'The partial order is the Boolean order on each component; well-definedness and transitivity follow from Proposition 16.4 and the common-component property.' That is not enough. To prove transitivity of the global order one must show: if [[u]] ≤ [[v]] via representatives u' ⊆ v' on side x, and [[v]] ≤ [[w]] via representatives v'' ⊆ w' on side y, then [[u]] ≤ [[w]]. The proof needs a single side containing representatives of [[u]] and [[w]]. Pairwise compatibility of u,v and of v,w does not imply compatibility of u,w; compatibility is not transitive in a partial Boolean algebra, as Remark 4.5 itself warns. Lemma 16.2(7) can only be invoked after the family {u,w} is known to be pairwise compatible, which is exactly the missing fact. Appendix C verifies congruence, representative-independence of operations, component intersections, maximal cliques, parity, exhaustive KS-search, and the Peres isomorphism, but its nine listed steps do not include a check of transitivity or antisymmetry of the global order ≤. Thus one load-bearing axiom of (V140, Π140) is supported by an assertion rather than by a proof or by a stated computation. If the global order is not transitive or antisymmetric, V140 is not a partial Boolean algebra, Theorem 6.12 cannot be applied to it, and Corollary 16.8 collapses. This is a finite, checkable issue, but it is distinct from the reader's highlighted Lemma 16.2(7): even with maximal-clique enumeration accepted, P6 remains unverified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a classical model-theoretic representation of partial Boolean algebras (pbas). For a surjection g from the set of initial formulas of a propositional language onto a pba (V,Π), the authors define a theory T_g via the axioms (g1)–(g4) encoding order and orthogonality of V, prove its consistency, characterize all models as upward-closed clusters (Theorems 6.6–6.9), and prove the central Theorem 6.12: the core of T_g — the ordered set of equivalence classes of initial formulas — carries a unique pba structure isomorphic to (V,Π). They then give a model-theoretic reformulation of KS-colourability (Theorem 10.6), introduce finite-spectrum observables with a possibilistic uncertainty measure based on pointer sets and locally admissible outcomes (Sections 11–13), a measurement-update rule yielding repeatability and a finite back-action example in the 12-vertex algebra V12 (Section 14), and a dispersion-free criterion equivalent to KS-colourability and to two-valued homomorphisms (Theorems 15.3–15.4). The main negative example is the 140-vertex algebra V140, built from a 3×3 array of modules by side-complement word congruence, claimed to be a four-dimensional pba (Theorem 16.5), shown not KS-colourable by a parity argument over nine loops (Theorem 16.7), with incidence structure isomorphic to the Peres 24-ray configuration.","tokens_in":30947,"tokens_out":26162,"duration_ms":221750,"significance":"Assuming the construction of V140 is fully validated, this is a substantial contribution. It shows that the structure usually associated with quantum contextuality, namely the Kochen–Specker obstruction, can be expressed as a property of the models of an ordinary classical propositional theory, with no modification of deduction, and the paper says precisely what it does and does not claim: no probabilities, no Leggett–Garg inequality, no resource monotone, and no computational speed-up. The proofs I checked in detail — Theorems 6.6, 6.12, 10.6, 15.3, and 16.7 — appear correct, and the cluster characterization of models is a clean and potentially reusable technique. The general framework rests on the stated axioms (P1)–(P6), and the measurement-update rule is honestly declared a postulate in Section 14. However, the flagship example currently rests on an incomplete verification of one load-bearing property, axiom (P6) for V140: the finite certificate described in Appendix C does not cover transitivity or antisymmetry of the global order, and the verification code is not shipped. This must be fixed before the central claims can be relied upon.","major_comments":[{"comment":"The proof of axiom (P6) for (V140, Π140) is not supplied. Theorem 16.5 says that 'well-definedness and transitivity follow from Proposition 16.4 and the common-component property', but the common-component property (Lemma 16.2(7)) can be invoked only for a family already known to be pairwise compatible. From [[u]] ≤ [[v]] and [[v]] ≤ [[w]] one obtains compatible pairs {u,v} and {v,w}, not the pair {u,w}; compatibility is not transitive, as Remark 4.5 itself warns. No argument in Section 16.2 produces a single side containing representatives of [[u]] and [[w]], so transitivity of the global order is asserted rather than proved. Antisymmetry is also not addressed explicitly, although it appears to follow from the fact that congruence preserves word size. The nine verification steps of Appendix C check congruence, representative-independence, component intersections, maximal cliques, parity, exhaustive KS-search, and the Peres isomorphism, but they do not include a check of transitivity or antisymmetry of ≤. This is load-bearing: Theorem 6.12, Theorem 15.4, Theorem 15.3, and Corollary 16.8 all presuppose that V140 is a pba; if the global order is not a partial order, Corollary 16.8 collapses. Because the condition is finite and checkable on 140 vertices, I request either a structural proof of transitivity or an explicit, reported verification step (with the certificate), to be added to Appendix C.","section":"16.2, Theorem 16.5"},{"comment":"The certificate for the paper's central example is not independently reproducible from the manuscript. Appendix C describes nine verification steps and states that a short Python program confirms them, but also that the code is 'available from the authors on request'; the Declarations repeat this. Since Lemma 16.2(7) (maximal-clique enumeration of the 140-vertex compatibility graph) and the missing order-theoretic checks of Theorem 16.5 are finite computational facts that the paper does not prove by hand, the validity of the non-KS example currently depends on an unshipped program. The accompanying assertion that every step 'can equally be carried out by hand from the side list (Σ)' is not credible at this scale. I recommend shipping the code as supplementary material or providing a machine-checkable certificate, and adding the transitivity and antisymmetry checks for ≤ to the enumerated list in Appendix C.","section":"Appendix C; Declarations (Code availability)"}],"minor_comments":[{"comment":"The sentence 'g(p2) ⊓ g(p8) = g(p7)' uses the partial operation ⊓ for vertices 2 and 8, which are incompatible in (V12, Π◦) by Proposition 5.1; by Definition 4.3 and Proposition 4.2, the meet is defined only for compatible pairs. The computation is correct as a meet in the underlying orthomodular lattice of Figure 1, but the text does not say so, and the example conflates the ambient lattice meet with the partial pba operation.","section":"7.1"},{"comment":"Definition 13.1 defines the pointer set ΔM(P) but only Remark 13.2 notes that (P, ΔM(P)) need not itself be true in M; since this is an easy inference to draw incorrectly, the warning belongs in or immediately after the definition, not in a later remark.","section":"13, Definition 13.1"},{"comment":"The counts for T12 (53 models, 5 dispersion-free, 18 making both P0 and Q0 sharp, 11 leaving one observable non-actualizable) are asserted in Remark 13.2 and after Proposition 15.2 and are said to be reproduced by the unshipped program; adding a short table or counting derivation would make these numeric claims verifiable independently of the code.","section":"13.2; 15.1"}],"recommendation":"major_revision","confidential_remarks":"The concern from the stress-test about Theorem 16.5 lands: the one-sentence justification of (P6) for V140 does not establish transitivity of the global order, and the gap is load-bearing for Corollary 16.8. I recommend major revision rather than rejection because the missing item is a finite, checkable property squarely within the manuscript's scope, and the general framework (Sections 6–15) is, as far as I checked, correct. If the authors supply the proof or the verification of P6 and ship the certificate, the paper would in my view be close to publishable. The paper is transparent about building on the late O. Q. Malhas's own earlier programme, and I see no citation or novelty problem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper up front. First, the central representation theorem is real: every partial Boolean algebra is the core of a consistent classical propositional theory, and the proof (Theorem 6.12) is careful and self-contained. The cluster characterization of models and the model-theoretic reformulation of KS-colourability are genuinely useful and are not in Malhas's earlier papers. Second, the new 140-vertex non-KS example has a load-bearing gap that the paper does not close.\n\nWhat is actually new: the systematic cluster semantics, the model-theoretic KS criterion, the finite-spectrum uncertainty measure, the measurement-update rule, and the V140 construction with its parity argument. The paper is honest about scope—no probabilities, no temporal inequalities, no resource claims—and that restraint earns it credit.\n\nNow the soft spot. The reader's flagged concern about Lemma 16.2(7) (unshipped enumeration) is real but not the deepest issue. The stress-test note is the sharper one: Theorem 16.5 asserts P6, the partial order condition, in a single sentence. Transitivity of the global order on V140 is not proved, and the appendix's nine verification steps do not include a transitivity check. This is not a minor omission: without transitivity (or antisymmetry), V140 is not a partial Boolean algebra, so Theorem 6.12 cannot be applied, Corollary 16.8 collapses, and the non-KS claim loses its foundation. The issue is finite and checkable, so the paper can likely be fixed, but as written the flagship example is unsupported at a load-bearing point.\n\nThe rest of the mathematics appears sound. The quotient construction and the parity argument are clear, and the connection to the Peres configuration is credible. If the transitivity gap is closed (by a hand proof or a stated exhaustive check), this becomes a valuable contribution to quantum logic.\n\nWho is this for? People working on Kochen-Specker, quantum logic, and model-theoretic approaches to contextuality. A serious referee should engage with it, but the paper needs revision before it should be published. My recommendation: send it to peer review, but with the explicit requirement that the authors either prove P6 directly or include a verifiable script that checks transitivity and antisymmetry of the global order on V140, not just maximal cliques.","headline":"Solid core theorem, but the flagship V140 example has an unverified transitivity claim that needs fixing before I'd trust it.","tokens_in":31517,"tokens_out":1662,"would_cite":false,"duration_ms":16920,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03G12","81P10","03B05","06C15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every partial Boolean algebra is the core of a consistent classical propositional theory.","keywords":["partial Boolean algebra","Kochen-Specker theorem","classical propositional logic","dispersion-free model","contextuality","uncertainty","Lindenbaum algebra","Peres configuration"],"falsifier":"An independent exhaustive check of the side list (Σ) would settle the central claim: build the compatibility graph on the 140 congruence classes and enumerate its maximal cliques; if any maximal clique is not exactly one of the 24 sixteen-element side components, or if an exhaustive search over all $2^{24}$ assignments finds a function that assigns exactly one 1 to every side, then Theorem 16.7 and Corollary 16.8 fail.","tokens_in":30388,"feed_emoji":"🧩","tokens_out":6393,"duration_ms":62130,"temperature":0.7,"pith_summary":"The paper aims to show that quantum-like uncertainty and Kochen-Specker contextuality can be reproduced inside ordinary classical propositional logic, without changing truth tables, deduction, or the consequence relation. The method is constructive: given any partial Boolean algebra and a surjection of atomic formulas onto its vertices, the axioms of the induced theory are chosen so that the theory's core—the equivalence classes of those atomic formulas—is isomorphic to the original partial Boolean algebra. The paper then proves that models of this theory are exactly upward-closed clusters, that KS-colourability is equivalent to the existence of a dispersion-free model, and that a constructed 140-vertex example has no such model, by a parity argument. A concrete measurement-update postulate shows that measuring an incompatible observable can destroy sharpness. The conclusion is that nonclassicality is located in the organization of propositions and models, not in nonclassical deduction.","feed_headline":"140-vertex algebra defeats every dispersion-free model","feed_subtitle":"Inside classical propositional logic, the Kochen-Specker obstruction survives as a structural fact about models.","key_machinery":"The core of a theory is the central object: the subset of the Lindenbaum algebra consisting of equivalence classes of initial formulas. A cluster is a set of initial formulas containing no orthogonal pair; every cluster extends by upward closure to a model, and every model is such an extension. Pre-frames are preimages of frames, and a dispersion-free model is one that meets every pre-frame in exactly one primitive formula. The carrying identity is the isomorphism f([p])=g(p), which transfers KS-colourability into the existence of a dispersion-free model. The non-colourable example V_140 is built from a 3×3 array of modules whose quotient by side-complement congruence has 140 vertices and 24 maximal Boolean components; the parity count over nine loops forbids a KS-colouring.","core_discovery":"The central discovery is Theorem 6.12: for any partial Boolean algebra (V,Π) and any surjection g from the initial formulas of a propositional language onto V, the induced theory T_g is consistent and its core C(T_g)—the set of equivalence classes of initial formulas modulo provable equivalence—carries a unique partial Boolean algebra structure on which the map f([p])=g(p) is an isomorphism onto (V,Π). Equivalently, every partial Boolean algebra, including the propositional structure of a finite-dimensional Hilbert space, is the core of an ordinary classical propositional theory. Building on this, the paper characterizes models as upward-closed clusters, proves that a nontrivial n-dimensional partial Boolean algebra is KS-colourable exactly when the induced theory has a model meeting every pre-frame in exactly one primitive formula, and exhibits a 140-vertex four-dimensional partial Boolean algebra with no such model; the obstruction is a parity argument over nine loops, and the incidence structure is isomorphic to the Peres 24-ray configuration in $R^{4}$. Consequently, every model of the corresponding theory fails to make every finite-spectrum observable sharp (Corollary 16.8).","pith_inferences":["The model space of T_g is finite for finite V, so the update rule defines a finite transition system on models; the paper does not explore its reachable states, fixed points, or recurrent classes, but those could be analyzed without adding probabilities.","The definition of uncertainty as a count of locally admissible outcomes is possibilistic; pairing it with a probability law on globally admissible outcomes would be the natural next step toward meaningful temporal inequalities, which the paper explicitly leaves open.","The same logical transport should apply to any parity-based Kochen-Specker proof, not only the Peres/Cabello one: any finite ray configuration with a parity argument yields a classical propositional theory with no dispersion-free model.","Many models of T_g are neither pure-state nor support-state certainty models, so the framework may encode more than state certainty; what, if anything, those extra models represent operationally is a question the paper does not settle."],"forward_implications":["Every finite-dimensional partial Boolean algebra, hence the quantum propositional structure of any finite-dimensional Hilbert space, is the core of a consistent classical propositional theory.","A partial Boolean algebra is KS-colourable if and only if its induced theory has a dispersion-free model, so the Kochen-Specker obstruction appears as the nonexistence of a certain kind of classical model rather than a breakdown of classical logic.","If every finite-spectrum observable is sharp in a model, the underlying algebra must be KS-colourable; consequently the V_140 theory has no model in which every observable is sharp.","With the measurement-update postulate, measuring an incompatible observable can turn a sharp observable into a nonsharp one, giving a finite, logic-level description of back-action.","Because V_140's incidence structure is isomorphic to the Peres configuration, the same core is realized by genuine quantum propositions in dimension four."],"supporting_citations":[{"why":"Supplies the Kochen-Specker theorem and the partial-Boolean-algebra framing of hidden-variable obstructions that the paper re-expresses model-theoretically.","marker":"[8]"},{"why":"Provides the component definition of partial Boolean algebras and the quantum propositional structure used throughout.","marker":"[16]"},{"why":"Launches the programme of treating quantum logic as classical propositional calculus, which the paper continues.","marker":"[10]"},{"why":"Earlier construction of the lattice of quantum propositions as the poset of a theory; the concept of a core grows out of it.","marker":"[13]"},{"why":"Supplies the Peres 24-ray configuration to which V_140's incidence structure is isomorphic, giving the Hilbert-space realization.","marker":"[15]"},{"why":"Provides the 18-vector, 9-basis parity proof that the nine loops of V_140 reproduce, serving as the template for the parity obstruction.","marker":"[1]"},{"why":"Sets out the elementary propositional-logic syntax, valuations, and Lindenbaum algebras underlying the construction of T_g.","marker":"[5]"},{"why":"Supplies the orthocomplemented lattice example V_12 that the paper uses for its colourable and back-action illustrations.","marker":"[17]"}],"fun_headline_variants":["140-vertex algebra blocks all dispersion-free models","Classical logic reproduces quantum uncertainty exactly","Every partial Boolean algebra is a classical theory's core","Kochen-Specker obstruction survives in classical logic","Model theory proves quantum uncertainty without leaving classical logic"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The V_140 example depends on the finite combinatorial claim that every pairwise compatible family of vertices lies in one of the 24 side components; the paper says an exhaustive maximal-clique enumeration certifies this, but the verification code is not included.","fun_headline_variants_meta":{"raw":{"variants":["140-vertex algebra blocks all dispersion-free models","Classical logic reproduces quantum uncertainty exactly","Every partial Boolean algebra is a classical theory's core","Kochen-Specker obstruction survives in classical logic","Model theory proves quantum uncertainty without leaving classical logic"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000274,"raw_usage":{"total_tokens":1702,"prompt_tokens":1068,"completion_tokens":634,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":684,"completion_tokens_details":{"reasoning_tokens":562}},"tokens_in":684,"tokens_out":634,"duration_ms":6917,"temperature":1.0,"reasoning_tokens":562,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T20:36:03.042254+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"An independent exhaustive check of the side list (Σ) would settle the central claim: build the compatibility graph on the 140 congruence classes and enumerate its maximal cliques; if any maximal clique is not exactly one of the 24 sixteen-element side components, or if an exhaustive search over all $2^{24}$ assignments finds a function that assigns exactly one 1 to every side, then Theorem 16.7 and Corollary 16.8 fail.","supporting_citations":[{"cited_title":", title =","cited_arxiv_id":null,"evidence_quote":"Supplies the Kochen-Specker theorem and the partial-Boolean-algebra framing of hidden-variable obstructions that the paper re-expresses model-theoretically."},{"cited_title":"and Garc\\'ia-Alcaine, Guillermo , title =","cited_arxiv_id":null,"evidence_quote":"Provides the component definition of partial Boolean algebras and the quantum propositional structure used throughout."},{"cited_title":"Reviews of Modern Physics , volume =","cited_arxiv_id":null,"evidence_quote":"Launches the programme of treating quantum logic as classical propositional calculus, which the paper continues."},{"cited_title":", title =","cited_arxiv_id":null,"evidence_quote":"Earlier construction of the lattice of quantum propositions as the poset of a theory; the concept of a core grows out of it."},{"cited_title":"Journal of Physics A: Mathematical and General , volume =","cited_arxiv_id":null,"evidence_quote":"Supplies the Peres 24-ray configuration to which V_140's incidence structure is isomorphic, giving the Hilbert-space realization."},{"cited_title":"Duke Mathematical Journal , volume =","cited_arxiv_id":null,"evidence_quote":"Sets out the elementary propositional-logic syntax, valuations, and Lindenbaum algebras underlying the construction of T_g."},{"cited_title":"Nature , volume =","cited_arxiv_id":null,"evidence_quote":"Supplies the orthocomplemented lattice example V_12 that the paper uses for its colourable and back-action illustrations."}],"review_version":1}