We use constructive bounded Kasparov [Formula: see text]-theory to investigate the numerical invariants stemming from the internal Kasparov products [Formula: see text], [Formula: see text], where the last morphism is provided by a tracial state. For the class of properly defined finitely-summable Kasparov [Formula: see text]-cycles, the invariants are given by the pairing of [Formula: see text]-theory of [Formula: see text] with an element of the periodic cyclic cohomology of [Formula: see text], which we call the generalized Connes–Chern character. When [Formula: see text] is a twisted crossed product of [Formula: see text] by [Formula: see text], [Formula: see text], we derive a local formula for the character corresponding to the fundamental class of a properly defined Dirac cycle. Furthermore, when [Formula: see text], with [Formula: see text] the algebra of continuous functions over a disorder configuration space, we show that the numerical invariants are connected to the weak topological invariants of the complex classes of topological insulators, defined in the physics literature. The end products are generalized index theorems for these weak invariants, which enable us to predict the range of the invariants and to identify regimes of strong disorder in which the invariants remain stable. The latter will be reported in a subsequent publication.