Sidiney B. Montanhano

Contextuality has long been associated with topological properties. In this work, such a relationship is elevated to identification in the broader framework of generalized contextuality. We employ the usual identification of states, effects, and transformations as vectors in a vector space and encode them into a tangent space, rendering the noncontextual conditions the generic condition that discrete closed paths imply null phases in valuations, which are immediately extended to the continuous case. Contextual behavior admits two equivalent interpretations in this formalism. In the geometric or intrinsic-realistic view, termed "Schrödinger", flat space is imposed, leading to contextual behavior being expressed as non-trivial holonomy of probabilistic functions, analogous to the electromagnetic tensor. As a modification of the valuation function, we use the equivalent curvature to connect contextuality with interference, noncommutativity, and signed measures. In the topological or participatory-realistic view, termed "Heisenberg", valuation functions must satisfy classical measure axioms, resulting in contextual behavior needing to be expressed in topological defects in events, resulting in non-trivial monodromy. We utilize such defects to connect contextuality with non-embeddability and to construct a generalized Vorob’ev theorem, a result regarding the inevitability of noncontextuality. We identify in this formalism the contextual fraction for models with outcome-determinism, and a pathway to address disturbance in ontological models as non-trivial transition maps. We also discuss how the two views for encoding contextuality relate to interpretations of quantum theory.

Multi-agent scenarios, like Wigner’s friend and Frauchiger–Renner scenarios, can show contradictory results when a non-classical formalism must deal with the knowledge between agents. Such paradoxes are described with multi-modal logic as violations of the structure in classical logic. Even if knowledge is treated in a relational way with the concept of trust, contradictory results can still be found in multi-agent scenarios. Contextuality deals with global inconsistencies in empirical models defined on measurement scenarios even when there is local consistency. In the present work, we take a step further to treat the scenarios in full relational language by using knowledge operators, thus showing that trust is equivalent to the Truth Axiom in these cases. A translation of measurement scenarios into multi-agent scenarios by using the topological semantics of multi-modal logic is constructed, demonstrating that logical contextuality can be understood as the violation of soundness by supposing mutual knowledge. To address the contradictions, assuming distributed knowledge is considered, which eliminates such violations but at the cost of lambda-dependence. We conclude by translating the main examples of multi-agent scenarios to their empirical model representation, contextuality is identified as the cause of their contradictory results.