Contextual Probability and Operational Minimality in Tug-of-War Decision Making

What connects minimal decision-making to contextual probability? We address this question through the Tug-of-War (TOW) principle, a resource-coupled mechanism for multi-armed bandit decision-making. Starting from classical probability, we show that when reward-learning functionality and specified information requirements are fixed, minimizing the necessary relaxation of common event semantics can lead to nonclassical probability. We derive the exact minimum semantic repair and identify the conditions under which information efficiency, calibration, and compositional consistency select contextual statistics. We then construct a single quantum Tug-of-War (QTOW) instrument in which the same decision--reward process both improves stationary decision performance and drives the unconditional learned state beyond the Klyachko--Can--Binicioğlu--Shumovsky (KCBS) noncontextuality bound. A finite diagnostic based on the instrument's recovery operations also yields a corrected contextuality witness. These results connect minimal decision-making, operational economy, and the emergence of contextual probability.