  We define a trace map for every cohomological correspondence in the motivic
stable homotopy category over a general base scheme, which takes values in the
twisted bivariant groups. Local contributions to the trace map give rise to
quadratic refinements of the classical local terms, and some
$\mathbb{A}^1$-enumerative invariants, such as the local $\mathbb{A}^1$-Brouwer
degree and the Euler class with support, can be interpreted as local terms. We
prove an analogue of a theorem of Varshavsky, which states that for a
contracting correspondence, the local terms agree with the naive local terms.
