  The Bell-Kochen-Specker conditions (BKS) for a deterministic noncontextual
hidden-variable model are wonderfully simple to state, deal with just
one-dimensional projectors on a Hilbert space H and make no reference to a
probabilistic phase space or quantum system. They only ask for an assignment of
zero or one to every projector such that the assignment respects orthogonal
resolutions of the identity. Various no-go results in the literature show that
the pair of statements {BKS is valid; dim H greater than or equal to 3} are
inconsistent. Here we show, more radically, that the pair actually contradicts
the dimensionality of the space itself, by implying that there can exist at
most a single one-dimensional projector acting on H. Our derivation involves
only elementary inner product spaces. It is non-probabilistic, inequality-free,
state independent, does not use entanglement, and is simultaneously valid in
all dimensions three or greater.
