The most powerful unfalsified model (MPUM), i.e., the least complex exact model for the given data, is well established for linear time-invariant (LTI) systems. It has not been generalized for linear parameter-varying (LPV) systems. In order to do this, we define the notions of complexity for LPV systems with shifted-affine scheduling dependence. The MPUM leads to identifiability conditions and a method for exact LPV system identification. The method is based on lifting the LPV system to a higher dimensional space and LTI embedding in the lifted space. It is made rigorous by proving a formal connection between the parameters of the LTI embedding and the original LPV system.