  We use constructive bounded Kasparov K-theory to investigate the numerical
invariants stemming from the internal Kasparov products $K_i(\mathcal A) \times
KK^i(\mathcal A, \mathcal B) \rightarrow K_0(\mathcal B) \rightarrow \mathbb
R$, $i=0,1$, where the last morphism is provided by a tracial state. For the
class of properly defined finitely-summable Kasparov $(\mathcal A,\mathcal
B)$-cycles, the invariants are given by the pairing of K-theory of $\mathcal B$
with an element of the periodic cyclic cohomology of $\mathcal B$, which we
call the generalized Connes-Chern character. When $\mathcal A$ is a twisted
crossed product of $\mathcal B$ by $\mathbb Z^k$, $\mathcal A = \mathcal B
\rtimes_\xi^\theta \mathbb Z^k$, we derive a local formula for the character
corresponding to the fundamental class of a properly defined Dirac cycle.
Furthermore, when $\mathcal B = C(\Omega) \rtimes_{\xi'}^{\phi} \mathbb Z^j$,
with $C(\Omega)$ 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.
