  Quantum effects inside the chiral bag induce a color anomaly which requires a
compensating surface term to prevent breakdown of color gauge invariance. We
show that the presence of this surface term first discovered several years ago
allows one to derive in a gauge-invariant way a chiral-bag version of the
Shore-Veneziano two-component formula for the flavor-singlet axial charge of
the proton. This has relevance to what is referred to as the ``proton spin
problem" on the one hand and to the Cheshire-Cat phenomenon in hadron structure
on the other. We show that when calculated to the leading order in the color
gauge coupling and for a specific color electric monopole configuration in the
bag, one can obtain a striking Cheshire-Cat phenomenon with a negligibly small
singlet axial charge.
