Abstract
<jats:p>Active learning for machine-learned force fields is well developed for local atomic environments, yet global variation in cell, atom count, composition, and thermodynamic conditions is often handled by incompatible generators and informal guarantees. We formulate kernel experimental design on a compact quotient of periodic structures, with continuous isometry invariants (AMD/PDD) and positive-definite factors for configuration, cell, count, and composition, and prove an explicit guarantee ledger: under the hypotheses attached to each object, regularized power-function selection attains a maximizer, equals a log-determinant gain, and bounds native-space error for invariant scalar targets; its integral against a fixed thermodynamic measure is a GP Bayes risk; and pure information-gain batching retains the (1− 1/e) greedy factor. The boundaries are equally explicit and proved by counterexample. Force and deviatoric-stress components need derivative or equivariant kernels—a quotient-canonical isotropic-derivative kernel, supplied here, transfers energy native-space control to hydrostatic pressure; the finite-width neural-tangent-kernel perturbation bound is fixed-design and can be numerically vacuous; and the exact mixed power–MMD–barrier score generally has no set-function potential, its natural surrogate is non-submodular, and partition-constrained information gain or a facility-coverage objective provably restores the guarantee. Where the source condition or feature injectivity fails, a misspecification oracle bound and finite-pool certificates replace the assumption with a measurable quantity. A companion paper [10] executes these constructions on lithium and audits the theory-to-implementation gap. The contribution is a quotient-safe construction and a reproducible map of which kernel-design guarantees survive, which need extra hypotheses, and which fail.</jats:p>