What Your Model Threw Away and Why You'll Want It Back: Masking, Fingerprinting, and Privacy from Discarded Geometry

We develop a framework for the information discarded by machine learning models whose inputs carry a Lie group action. Given a representation π of a Lie group G on a space V and a learned function fcolon V → ℝ, we define two objects measuring the symmetry invisible to f. The null fiber at a point x ∈ V is the set NG(f,x) = g ∈ G : f(π(g-1) · x) = f(x) of group elements whose inverse action on x is undetectable by f. When NG(f,x) is independent of x, it coincides with the stabilizer StabG(f), the largest subgroup of G under which f is invariant. For smooth maps to ℝ, the preimage theorem guarantees that null fibers have dimension at least dim G - 1 at generic inputs, regardless of architecture. For compact groups acting on themselves, the Peter--Weyl theorem yields a spectral characterization of both objects in terms of the Fourier coefficient matrices of f. We show that null fiber elements can be computed efficiently via Newton iteration on the orbit map, at a cost comparable to a few gradient evaluations. Applications to data masking, model fingerprinting, and privacy-preserving computation are developed and tested experimentally on molecular property prediction under SO(3) and spherical image classification under the M"obius group PSL(2, mathbbC). The framework applies uniformly to classical neural networks and variational quantum circuits.