A recognizer that compares rotation-invariant descriptors sees a surface only up to the fiber of the descriptor. We measure this fiber by its radius in the orbit distance from the enrolled surface. A large radius admits decoys, that is, distant shapes that pass the matcher. A small radius discloses the enrolled shape to anyone who captures the stored value. For star-shaped surfaces truncated to spherical harmonics of degree at most
L, with
n coefficients, a descriptor of generic rank
r has generic fibers of dimension
n−3−r modulo rotations. The standard pool of band powers, even bispectra, and three invariants of the degree-three band therefore admits decoy families of dimension
5,
13,
20 at
L=4,6,8. Its rank first reaches
n−3 at
L=16, and a mirror decoy remains at every
L. The odd bispectra remove the mirror decoy generically for
L≥4. Yet at fixed mean radius the same pool determines the enclosed volume exactly, and it does not determine whether a surface meets a clearance requirement. We certify two cases by exact and interval arithmetic. At
L=6 a decoy matches all
32 invariants to relative precision
2⋅10−18 at orbit distance at least
0.87 times the norm of the enrolled tuple. For the radar shape model of asteroid (101955) Bennu, the pool recovers the modeled volume, misses the handedness, and leaves the keep-out radius uncertain by more than
7m.