Operational Shadows of Hilbert-Space Probabilities

Abstract

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.