The Geometry of Transfer

A new target task may have little labeled data but access to a library of fine-tuned models whose source datasets are unavailable. Existing transferability scores rank source representations, while model-merging methods combine task vectors, but neither directly quantifies which part of target adaptation is already contained in the source library. We formulate transfer as projection in a target-conditioned functional tangent space. Each source checkpoint induces a tangent field on target inputs, while the target loss induces a curvature-normalized descent field. Their weighted $L_2$ geometry defines a principal transfer angle for each source subspace and a transfer completeness index measuring the fraction of target adaptation captured by a sparse source span. We introduce GEOTRANSFER, which estimates this geometry from source checkpoints and a small target sample, performs group-sparse source projection, learns an orthogonality-regularized innovation adapter for residual target structure, and uses a held-out risk gate for target-only fallback. We prove a local projection identity, first-order invariance to smooth reparameterization, a sparse-span oracle inequality, and a finite-sample guarantee that gated deployment does not exceed the target-only baseline under a prespecified bounded target risk. Simulations and real-data experiments show that the geometry predicts realized source utility, is stable under parameter rescaling, and improves few-shot transfer when useful and harmful source directions coexist.