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The formalism of generalized probabilistic theories (GPTs) was originallydeveloped as a way to characterize the landscape of conceivable physicaltheories. Thus, the GPT describing a given physical theory necessarily includesall physically possible processes. We here consider the question of how toprovide a GPT-like characterization of a particular experimental setup within agiven physical theory. We show that the resulting characterization is notgenerally a GPT in and of itself-rather, it is described by a more generalmathematical object that we introduce and term an accessible GPT fragment. Wethen introduce an equivalence relation, termed cone equivalence, betweenaccessible GPT fragments (and, as a special case, between standard GPTs). Wegive a number of examples of experimental scenarios that are best describedusing accessible GPT fragments, and where moreover cone-equivalence arisesnaturally. We then prove that an accessible GPT fragment admits of a classicalexplanation if and only if every other fragment that is cone-equivalent to italso admits of a classical explanation. Finally, we leverage this result toprove several fundamental results regarding the experimental requirements forwitnessing the failure of generalized noncontextuality. In particular, we provethat neither incompatibility among measurements nor the assumption of freedomof choice is necessary for witnessing failures of generalized noncontextuality,and, moreover, that such failures can be witnessed even using arbitrarilyinefficient detectors.

Accessible fragments of generalized probabilistic theories, cone equivalence, and applications to witnessing nonclassicality