Strong $O$-valued contextuality: ruling out discrete nondeterministic alternatives to quantum theory

• URL: http://arxiv.org/abs/2607.14931v1
• Date: Thu, 16 Jul 2026 12:46:16 GMT
• Title: Strong $O$-valued contextuality: ruling out discrete nondeterministic alternatives to quantum theory
• Abstract summary: Gleason's theorem identifies the Born rule via non-contextuality over an infinite continuous lattice of projectors.<n>The Kochen-Specker (KS) theorem rules out the specific class of deterministic ($0,1$-valued) noncontextual models using finite sets of projectors.<n>We extend this framework by introducing strong $O$-valued contextuality, showing that quantum theory evades global sections of all finite-valued presheaves.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Gleason's theorem identifies the Born rule via non-contextuality over an infinite continuous lattice of projectors, while its corollary the Kochen-Specker (KS) theorem rules out the specific class of deterministic ($\{0,1\}$-valued) noncontextual models using finite sets of projectors. Gleason's theorem also indicates the existence of KS-type finite vector constructions to rule out other discrete nondeterministic alternatives to quantum theory beyond the $\{0, 1\}$ case. Here, we construct finite measurement configurations to rule out noncontextual empirical models with outcome probabilities drawn from an arbitrary finite subset $O \subset [0,1]$. We do this by two means: (i) constructing a family of experimentally feasible state-dependent Hardy-type tests, and (ii) proving a generalized KS theorem for a broad class of $O$ that includes prior results as special cases. In the sheaf-theoretic framework of Abramsky and Brandenburger, a hierarchy of probabilistic-possibilistic-strong contextuality has been established quantifying contextuality as a resource. We extend this framework by introducing strong $O$-valued contextuality, showing that quantum theory evades global sections of all finite-valued presheaves. We also discuss the implications of the result on finite many-valued logics as viable ontological models for quantum theory and for contextuality-based (semi)-device-independent protocols.

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