  The flavor singlet axial charge has been a source of study in the last years
due to its relation to the so called {\it Proton Spin Problem}. The relevant
flavor singlet axial current is anomalous, i.e., its divergence contains a
piece which is the celebrated $U_A(1)$ anomaly. This anomaly is intimately
associated with the $\eta^\prime$ meson, which gets its mass from it. When the
gauge degrees of freedom of QCD are confined within a volume as is presently
understood, the $U_A(1)$ anomaly is known to induce color anomaly leading to
"leakage" of the color out of the confined volume (or bag). For consistency of
the theory, this anomaly should be canceled by a boundary term. This ``color
boundary term" inherits part or most of the dynamics of the volume (i.e., QCD).
In this paper, we exploit this mapping of the volume to the surafce via the
color boundary condition to perform a complete analysis of the flavor singlet
axial charge in the chiral bag model using the Cheshire Cat Principle. This
enables us to obtain the hitherto missing piece in the axial charge associated
with the gluon Casimir energies. The result is that the flavor singlet axial
charge is small independent of the confinement (bag) size ranging from the
skyrmion picture to the MIT bag picture, thereby confirming the (albeit
approximate) Cheshire Cat phenomenon.
