Since the enlightening proofs of quantum contextuality first established by Kochen and Specker, and also by Bell, various simplified proofs have been constructed to exclude the non-contextual hidden variable theory of our nature at the microscopic scale. The conflict between the non-contextual hidden variable theory and quantum mechanics is commonly revealed by Kochen-Specker (KS) sets of yes-no tests, represented by projectors (or rays), via either logical contradictions or noncontextuality

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... qualities in a state-(in)dependent manner. Here we first propose a systematic and programmable construction of a state-independent proof from a given set of nonspecific rays in ^3 according to their Gram matrix. This approach brings us a greater convenience in the experimental arrangements. Besides, our proofs in ^3 can also be generalized to any higher dimensional systems by a recursive method.