Let $M$ be a smooth manifold, let $G$ be a Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$, and let $\pi:P\to M$ be a smooth principal $G$-bundle. Let $k\ge 1$, let $\Lambda\subset \mathbb R$ be a discrete additive subgroup, and let
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\begin{align*}
P_0:\mathfrak g^k\to \mathbb R
\end{align*}
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be a symmetric $\operatorname{Ad}(G)$-invariant $k$-[linear map](/page/Linear%20Map). Assume that $P_0$ is normalized so that, for every principal connection $B$ on $P$, the de Rham cohomology class of the Chern-Weil form $P_0(F_B)$ is the image of a fixed class
where $I_\omega\in C^j(M;\mathbb R)$ denotes the singular integration cochain of a smooth $j$-form $\omega\in\Omega^j(M)$ on smooth singular $j$-chains, and a $\Lambda$-valued singular cochain