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This paper concerns the variational description of prestrained materials, inthe context of dimension reduction for thin films $\Omega^h=\omega\times(-\frac{h}{2}, \frac{h}{2})$. Given a Riemann metric $G$ on $\Omega^1$, westudy the question of what is the infimum of the averaged pointwise deficit ofa given immersion from being an orientation-preserving isometric immersion of$G_{\mid \Omega^h}$ on $\Omega^h,$ over all weakly regular immersions. Thisdeficit is measured by the non-Euclidean energies $\mathcal{E}^h$, which can beseen as modifications of the classical nonlinear three-dimensional elasticity.  Building on our previous results, we complete the scaling analysis of$\mathcal{E}^h$ and the derivation of $\Gamma$-limits of the scaled energies$h^{-2n}\mathcal{E}^h$, for all $n\geq 1$. We show the energy quantisation inthe sense that the even powers $2n$ of $h$ are indeed the only possible ones(all of them are also attained). For each $n$, we identify the equivalentconditions for the validity of the corresponding scaling, in terms of thevanishing of appropriate Riemann curvatures of $G$ to certain orders, and interms of the matched isometry expansions. We also establish the asymptoticbehaviour of the minimizing immersions as $h\to 0$.

Quantitative immersability of Riemann metrics and the infinite hierarchy of prestrained shell models