  Previous experimental tests of quantum contextuality based on the
Bell-Kochen-Specker (BKS) theorem have demonstrated that not all observables
among a given set can be assigned noncontextual eigenvalue predictions, but
have never identified which specific observables must fail such assignment. We
now remedy this shortcoming by showing that BKS contextuality can be confined
to particular observables by pre- and postselection, resulting in anomalous
weak values that we measure using modern neutron interferometry. We construct a
confined contextuality witness from weak values, which we measure
experimentally to obtain a $5\sigma$ average violation of the noncontextual
bound, with one contributing term violating an independent bound by more than
$99\sigma$. This weakly measured confined BKS contextuality also confirms the
quantum pigeonhole effect, wherein eigenvalue assignments to contextual
observables apparently violate the classical pigeonhole principle.
