Abstract
<jats:p>Active learning for machine-learned force fields usually selects from an existing pool. We study the harder generative question: among all admissible periodic structures, allowing cell, atom count, and composition to change, which one carries the most information not already spanned by the labelled set, per unit labelling cost? Posing this precisely forces a representational constraint — an invariant kernel sees only the image of a configuration under a feature map that has quotiented out translation, rotation, lattice-basis change, and permutation, so everything it can certify is a function on that quotient, whatever the size of the training set.</jats:p> <jats:p>We build that quotient and, when it is nonempty and the stated descriptor and cost hypotheses hold, prove that the cost-normalized maximum exists and map what the resulting design controls. We organize the new material around three headline contributions: a finite-shell construction that supplies cell-deformation continuity for the AMD structural component used by the existence theorem; an obstruction–restoration pair showing that no fixed state-independent anisotropic strain direction descends uniformly even to the rotation quotient while the isotropic one does, carrying native-space error control to hydrostatic pressure; and a cycle defect showing that the practical mixed acquisition considered here need not be the marginal of any set function, so standard submodular guarantees do not transfer. Cartesian forces and the deviatoric stress tensor lie outside the scalar theory by construction. A companion paper [27] executes and audits a practical approximation of this loop on lithium.</jats:p>