  Remote detectability is often taken as a physical assumption in the study of
topologically ordered systems, and it is a central axiom of mathematical
frameworks of topological quantum field theories. We show under the
entanglement bootstrap approach that remote detectability is a necessary
property; that is, we derive it as a theorem. Starting from a single wave
function on a topologically-trivial region satisfying the entanglement
bootstrap axioms, we can construct states on closed manifolds. The crucial
technique is to immerse the punctured manifold into the topologically trivial
region and then heal the puncture. This is analogous to Kirby's torus trick. We
then analyze a special class of such manifolds, which we call pairing
manifolds. For each pairing manifold, which pairs two classes of excitations,
we identify an analog of the topological $S$-matrix. This pairing matrix is
unitary, which implies remote detectability between two classes of excitations.
These matrices are in general not associated with the mapping class group of
the manifold. As a by-product, we can count excitation types (e.g., graph
excitations in 3+1d). The pairing phenomenon occurs in many physical contexts,
including systems in different dimensions, with or without gapped boundaries.
We provide a variety of examples to illustrate its scope.
