Special varieties package conjecture for local-system moduli

Let $X$ be a smooth complex algebraic variety and $r$ a natural number. Let $\mathcal{M}_B(X,r)$ be the moduli space of semisimple complex local systems of rank $r$. An absolute $\overline{\mathbb{Q}}$-point means a point satisfying the paper's absolute arithmetic conditions, and geometric origin is understood in the sense used for local systems. Special varieties package conjecture. (1) The absolute $\overline{\mathbb{Q}}$-constructible subsets of $\mathcal{M}_B(X,r)(\mathbb{C})$ are generated from the absolute $\overline{\mathbb{Q}}$-closed subsets by taking irreducible components, intersections, finite unions, and complements; in particular, the Euclidean, equivalently Zariski, closure of an absolute $\overline{\mathbb{Q}}$-constructible set is absolute $\overline{\mathbb{Q}}$-closed. (2) An absolute $\overline{\mathbb{Q}}$-closed set is the Zariski closure of its absolute $\overline{\mathbb{Q}}$-points. (3) A point is an absolute $\overline{\mathbb{Q}}$-point if and only if it is of geometric origin. (4) An irreducible component of the Zariski closure of an infinite set of absolute $\overline{\mathbb{Q}}$-points is an absolute $\overline{\mathbb{Q}}$-closed subset in $\mathcal{M}_B(X',r')(\mathbb{C})$ for possibly a different smooth complex algebraic variety $X'$ and natural number $r'$, where $\mathcal{M}_B(X',r')$ is isomorphic to $\mathcal{M}_B(X,r)$ over $\mathbb{Q}$. The package is presented as a conjectural analogue of the Manin–Mumford, Mordell–Lang, and André–Oort principles; the final part is explicitly motivated only by analogy and the assertions remain open.