arXiv:2611.05656v1 Announce Type: new Abstract: For nearly sixty years, refinements of the Bell-Kochen-Specker theorem have concentrated on finite configurations, combinatorics and robustness; once a configuration is fixed, contextuality is witnessed by the inconsistency of a finite family of Boolean constraints. We study the theorem instead with infinitary real-valued logic and a model existence game. We axiomatize, in real-valued $L_{\omega_1\omega}$, the lattice of closed subspaces of a separable infinite-dimensional complex Hilbert space together with a quantum probability measure. In the standard models the theory recovers Gleason's theorem. Once the axiom that the measure is $\{0,1\}$-valued is adjoined, the theory has no models, standard or nonstandard, although every finite set of its axioms has one; a model existence game converts this infinitary inconsistency into finite inconsistent sets of interval commitments on the values of sentences. Over the standard models, the subsets of $[0,1]$ that formulas define through the values of the measure are exactly the $\sigma$-projective sets, the members of the least class containing the Borel sets and closed under complements, countable unions and projections along $\mathbb{N}^\mathbb{N}$; the classification is proved in ZFC. For a specified strategy in the game, the plays that complete the parity argument for the eighteen-ray configuration of Cabello, Estebaranz and Garc\'ia-Alcaine record the geometric identities between the occurrences of a ray in different contexts, and the use that the measure makes of them. Probabilities are tracked up to rational tolerances, and $\sigma$-additivity along a named basis selects the four-dimensional block in which the argument takes place. The refinement is thus not a smaller Kochen-Specker set but a finer analysis of the logical form of the obstruction.
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