Towards coarse moduli of augmentations

A variety $X$ over a suitable base $Z$ gives rise to a highly structured algebra $C^*(X)$ in motives over $Z$; a $Z$-point gives rise to an augmentation $C^*(X) \to \mathbf{1}$; this assignment $X(Z) \to \operatorname{Aug}\big(C^*(X)\big)$ factors the unipotent Kummer map. In one direction, this suggests the possibility of performing Chabauty-Kim theory motivically without waiting for a motivic t-structure. In a different (largely independent) direction, this may allow us to extract arithmetic information from the full rational homotopy type going beyond $\pi_1$. In both directions, $ \operatorname{Aug}\big(C^*(X)\big)$ would benefit from a structure of finite type $\mathbb{Q}$-variety. I'll show preliminary results and examples in this direction.