  A measure-preserving formalism (MPF) is constructed and applied to spin/band
models, which yield observations about pumping. It occurs at topological phase
transition (TPT), i.e., a $Z_2$-flip, suggesting that $Z_2$ can imply bulk
effects. The model's asymptotic behavior is analytically solved via MPF. The
pumping probability is geometric, fractional, and has a ceiling of
$\frac{1}{2}$. Intriguingly, theorems are proved about occurrence conditions,
which are linked to the system's dimension and the distinction between rational
and irrational numbers. Experimental detection is discussed.
