We introduce the Geometric Observability Index (GOI), a per-feature sensitivity measure for pose estimation on SE(3). For a Gauss-Newton curvature matrix
H=E[J⊤WJ] and a Riemannian metric
G on the Lie algebra, the index is the
G-norm of the influence a single measurement exerts on the estimated pose:
GOI(z)=∥AOO−1POφ(z)∥G, where
ψ(z)=J⊤Wr(z) is the score,
φ=G−1ψ its gradient representative,
A=G−1H the curvature operator (self-adjoint in the
G-inner product),
O=range(A) the observable subspace, and
AOO its restriction. This single object (i) equals the norm of the M-estimator influence function, (ii) is governed by the Fisher information, which coincides with the curvature, (iii) exposes weak observability through the smallest eigenvalue
λmin, which (iv) also governs finite-sample stability. Operationally the theory cuts both ways. The index is the exact per-measurement attribution: it predicts the true leave-one-out pose shift with log-correlation
r=1.00. But we also prove that the influence standardized by its inlier null covariance collapses exactly to the classical chi-square residual statistic: residual gating is the leverage-corrected influence test, explaining its robustness from first principles, while raw-influence gating conflates a measurement's information with its harm and over-rejects high-leverage inliers in weakly observable geometry. Experiments on synthetic problems, five TUM RGB-D dynamic sequences, and two KITTI odometry sequences confirm the picture: the two criteria coincide under well-conditioned geometry, and raw-influence gating degrades significantly at
cond(H)≈104, as the leverage analysis predicts for noise-dominated weak directions. All quantitative claims are validated; code is released.