  The Conley index of an isolated invariant set is a fundamental object in the
study of dynamical systems. Here we consider smooth functions on closed
submanifolds of Euclidean space and describe a framework for inferring the
Conley index of any compact, connected isolated critical set of such a function
with high confidence from a sufficiently large finite point sample. The main
construction of this paper is a specific index pair which is local to the
critical set in question. We establish that these index pairs have positive
reach and hence admit a sampling theory for robust homology inference. This
allows us to estimate the Conley index, and as a direct consequence, we are
also able to estimate the Morse index of any critical point of a Morse function
using finitely many local evaluations.
