Functional Tangent Geometry Selects Adapter Placement in Equivariant Interatomic Potentials

The best adapter is determined more by placement than parameter count because different locations expose different energy-force correction spaces. We formalize these spaces with functional tangent geometry, using a Jacobian to map bounded adapter coefficients to reachable output corrections. A held-out transfer score then tests whether coefficients fitted on one configuration split represent the same correction on another. The score is invariant to adapter reparameterization when the coefficient metric transforms with it. Controlled experiments recover all four generating placements, and local molecular experiments show that tangent rankings predict constrained nonlinear adaptation. Across six rMD17 molecules, native-coordinate geometry selects the readout adapter. It remains best on untouched 1,000-frame holdouts, reducing normalized residual by 20.71% relative to the first equivariant product block. A local Ti-Al task instead selects that product block, whereas column normalization reverses the molecular ranking. Functional tangent geometry chooses adapter locations from the corrections they can express, avoiding a nonlinear sweep over placements.