  Kochen-Specker theorems assure the breakdown of certain types of
non-contextual hidden variable theories through the non-existence of global,
holistic frame functions; alas they do not allow us to identify where this
breakdown occurs, nor the extent of it. It was recently shown [Phys. Rev. A 86,
062109 (2012)] that this breakdown does not occur everywhere; here we show that
it is maximal in that it occurs almost everywhere, and thus prove that quantum
indeterminacy--often referred to as contextuality or value indefiniteness--is a
global property as is often assumed. In contrast to the Kochen-Specker theorem,
we only assume the weaker non-contextuality condition that any potential value
assignments that may exist are locally non-contextual. Under this assumption,
we prove that once a single arbitrary observable is fixed to occur with
certainty, almost (i.e. with Lebesgue measure one) all remaining observables
are indeterminate.
