Finite quantum-logical constructions can appear classical or nonclassical depending on which structure is retained. We distinguish incidence tests based on valuations, colorings, and partition representations; simplex-embedding tests for specified prepare-and-measure fragments; and operator-functional tests imposing spectral and product rules. We show that the collective GHZ joint measurement is a single Boolean context and becomes nonclassical only when a common assignment of local factors, together with product preservation, is required. For selected labelled ray fragments, we compute state-depolarizing thresholds for a restricted projector-cone factorization and for a specified vector-generated operational closure, separating exact primal--dual certificates from numerical estimates. These values are properties of the stated fragments and noise model, not invariants of the underlying hypergraphs or a universal ordering of contextuality.
Finite quantum-logical constructions can appear classical or nonclassical depending on which structure is retained. We distinguish incidence tests based on valuations, colorings, and partition representations; simplex-embedding tests for specified prepare-and-measure fragments; and operator-functional tests imposing spectral and product rules. We show that the collective GHZ joint measurement is a single Boolean context and becomes nonclassical only when a common assignment of local factors, together with product preservation, is required. For selected labelled ray fragments, we compute state-depolarizing thresholds for a restricted projector-cone factorization and for a specified vector-generated operational closure, separating exact primal--dual certificates from numerical estimates. These values are properties of the stated fragments and noise model, not invariants of the underlying hypergraphs or a universal ordering of contextuality.
At one frozen setting, the probabilities observed in a sharp quantum context are indistinguishable, as detector-click statistics, from ordinary probabilities on the atoms of a classical partition. But an actual analyzer usually comes with a calibrated knob: a tangible handle on the apparatus. If the measurement configuration is co-varied continuously through this physical parameter, the operational object is no longer one point of a simplex but a response curve. Classical linear responses, Malus-type Hilbert-space responses, softmax links, non-homomorphic parameter transcriptions, and discontinuous threshold limits are different maps from settings to probabilities. Continuity, calibration, and preservation of the physical composition law are then part of the experimental meaning of the knob. Such comparisons distinguish specified, calibrated response models; by themselves they do not constitute a classical-versus-quantum impossibility theorem. The static operational coincidence can also persist for two intertwined contexts: if the common outcomes receive the same probabilities, the remaining masses can always be coupled by a classical joint distribution. Genuine multi-context nonclassicality begins when a family of local shadows cannot be glued into one nonnegative global distribution or one simplex factorization. Farkas' lemma gives the exact alternative: either the classical extension exists, or a separating linear inequality certifies its impossibility.
Quantum logic is usually presented as a non-classical departure from ordinary reasoning forced on us by quantum mechanics, with classical logic kept as the secure starting point. We argue for the opposite order of explanation in a finite and fully computable setting. The free orthomodular lattice on two generators has ninety-six elements, the direct product of a six-element non-distributive factor and a sixteen-element Boolean factor. Reading the first factor as a register of contexts and the second as Boolean content, we obtain a calculus whose elements are context--bit-vector pairs and whose operations act component by component. With this calculus we establish three results. First, we classify the six layers by commutativity, identifying the central kernel of context-neutral propositions together with a dual central layer in which all complementary contexts are present. Second, we show that orthocomplementation rearranges the layers exactly as the complementation of the small factor rearranges its elements, which makes the duality among the layers rigid rather than accidental. Third, we prove that the operation forgetting the context is a surjective homomorphism of orthocomplemented lattices whose quotient is the classical Boolean algebra, so that classical logic is a six-to-one, information-losing image of the contextual calculus.
This work examines a foundational question that precedes many debates about quantum superposition: what mathematical map is required to translate quantum support into definite classical actuality? Quantum states, Born probabilities for a fixed measurement, and Boolean truth valuations of classical events are treated as distinct structures. For a finite outcome space, the positive-support indicator Tₚ(E) = 1 iff p(E) > 0 preserves unions, but whenever two exclusive outcomes have nonzero probability it fails to preserve intersections. Hence it is not a Boolean truth valuation unless the support is a singleton. In particular, a superposition α|U⟩ + β|D⟩ with αβ ≠ 0 does not become the classical conjunction U ∧ D through the Born map alone. A deterministic single-outcome selector is also unavailable if it is required to respect the exchange symmetry of a balanced binary distribution. The analysis then grants the strongest charitable version of the disputed literal reading by moving to four-valued, non-explosive Belnap–Dunn semantics. In a correlated trigger–carrier–response model, a local glut propagates only when both positive and negative semantic support are explicitly transferred along the relevant causal links. This yields a precise signed-reachability criterion: the downstream contradiction is not a consequence of quantum dynamics alone, but of additional semantic transfer rules. Explicit countermodels show where the inference fails when one of these rules is removed. The result separates the physical channel from the semantic bridge and prevents classical explosion from hiding the actual logical structure of the argument. A complementary unitary dilation demonstrates that the required trigger–response correlations can be generated by ordinary quantum evolution without producing contradictory classical outcomes. Joint support is kept distinct from its marginals, coherent superpositions are distinguished from incoherent mixtures, and labelled-history models separate coarse causal order from microscopic history. The central diagnosis is therefore structural: any contradiction obtained under the tested assumptions originates in added translation and compositional postulates, not in an inconsistency of quantum mechanics itself. The framework provides a compact audit tool for claims that move from Hilbert-space superposition to classical co-actuality, and then from local semantic contradiction to causal propagation.
Bell and contextuality scenarios are unified within the Acín-Fritz-Leverrier-Sainz hypergraph formalism, in which measurements are hyperedges. Multipartite measurements are obtained from product rules that combine independent contextuality scenarios into a single hypergraph. Several such rules exist: they are structurally distinct, yet yield the same classical, quantum, and boxworld correlations. Here, we show that this difference is not incidental, unveiling an unexpected connection. The measurements of composite Bell scenarios are exactly the process functions, the deterministic classical communication protocols that may be non-causal, i.e., lack a definite causal order without generating paradoxes. Validity of a hyperedge and logical consistency of a process function are thus one and the same condition. This allows us to characterize process functions by normalization and no-signaling alone, providing transformations that generate non-causal process functions from no-communication ones. The product rules, in turn, order process functions by causal structure, from no-communication to non-causality, which yields a graphical criterion for the latter: a process function is non-causal precisely when its hyperedge is generated by the disjunctive but not the maximal product. We thereby obtain the first joint hypergraph representation of (non-causal) process functions and (nonlocal-without-entanglement) measurements, recasting classical non-causality as a non-standard way of composing black boxes.
Intertwined complete projective experiments share outcomes, motivating the search for a common spectral labeling across contexts. Strong coloring tests whether complete contexts admit one context-independent nondegenerate spectrum; when that labeling exists, chromatic separability tests which event distinctions it preserves. We distinguish these questions from ordinary Kochen--Specker statelessness, geometric realizability, event-order representation, and Born-probability models. A state extends to a strong d-coloring exactly when deletion of its support leaves a (d-1)-colorable graph. Faithful real 17- and 21-ray restrictions of completed Yu--Oh have separating ordinary states but forced color identities; one quantum preparation requires exactly 1/9 nonextendable weight despite admitting an ordinary-state model. Independent Hermitian projection arguments correct earlier inferences from a chromatic gap to statelessness or nonrealizability. Hesse's complex 37-ray example separates real and complex geometry and has exact state-family minima 6/7/18 for three representation requirements. A rational two-plane construction gives a linear real 65-ray example with 393,216 states and explicit event-order and probability failures. General and specified ancestry results distinguish geometric KS extensions from abstract chromatic-to-stateless constructions. Detailed enumerations, regenerated state certificates, and extended proofs are supplementary.
Abstract Quantum contextuality represents a fundamental form of nonclassicality in quantum mechanics. To provide a more complete characterization of nonclassical properties in quantum systems, we adopt a logical perspective and propose a mathematical framework based on exclusive partial Boolean algebras. This framework enables a unified description of contextuality and nonclassicality across finite general, quantum, and classical systems. We establish a unified and minimal classical counterpart for any finite general system. Within this framework, we formalize major categories of quantum contextuality, demonstrating that: 12 projectors suffice to generate Kochen–Specker scenarios; 10 projectors suffice to witness state-independent contextuality; and 3 observables suffice to witness quantum contextuality. Finally, we prove that contextuality is a sufficient but not necessary condition for nonclassicality.