  We present extensive X-ray Absorption Fine Structure (XAFS) measurements on
La_{1-x}Ca_xMnO_3 as a function of B-field (to 11T) and Ca concentration, x
(21-45%). These results reveal local structure changes (associated with polaron
formation) that depend only on the magnetization for a given sample,
irrespective of whether the magnetization is achieved through a decrease in
temperature or an applied magnetic field. Furthermore, the relationship between
local structure and magnetization depends on the hole doping. A model is
proposed in which a filamentary magnetization initially develops via the
aggregation of pairs of Mn atoms involving a hole and an electron site. These
pairs have little distortion and it is likely that they pre-form at
temperatures above T_c.
