be an invariant symmetric $k$-linear polynomial whose Chern-Weil de Rham class is the image, under the de Rham comparison map, of an integral universal characteristic class
Fix a natural Cheeger-Simons differential refinement $\widehat c_P$ of $c_P$. This means that for every smooth manifold $X$ and every smooth principal $G$-bundle with connection $(Q,B)$ over $X$, the refinement assigns a degree-$2k$ differential character
whose curvature is the Chern-Weil form $P(F_B)$, whose values are taken on smooth singular $(2k-1)$-cycles in $X$ with values in $\mathbb R/\mathbb Z$, and whose boundary formula is as follows: for every smooth singular $2k$-chain $C=\sum_{j=1}^N n_j\sigma_j$ in $X$, with $n_j\in\mathbb Z$ and smooth singular simplices $\sigma_j:\Delta^{2k}\to X$,
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\begin{align*}
\widehat c_P(Q,B)(\partial C)=\sum_{j=1}^N n_j\int_{\Delta^{2k}}\sigma_j^*P(F_B)\,d\mathcal L^{2k}\mod \mathbb Z.
\end{align*}
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The assignment is natural under isomorphisms of principal bundles with connection: if $\Phi:Q'\to Q$ is a smooth principal $G$-bundle isomorphism covering a smooth map $f:X'\to X$ and $B'=\Phi^*B$, then
Let $M$ be a closed oriented smooth manifold of dimension $2k-1$, let $\pi:E\to M$ be a smooth principal $G$-bundle, and let $A$ be a smooth connection on $E$. If $z_M$ is any smooth singular $(2k-1)$-cycle representing the fundamental class $[M]\in H_{2k-1}(M;\mathbb Z)$, define the integral Chern-Simons invariant of $A$ by
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\begin{align*}
\operatorname{CS}_{\widehat c_P}(A):=\widehat c_P(E,A)(z_M)\in \mathbb R/\mathbb Z.
\end{align*}
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This value is independent of the chosen fundamental cycle representative. If $u:E\to E$ is a smooth gauge transformation, meaning a smooth principal $G$-bundle automorphism covering $\operatorname{id}_M$, and the gauge-transformed connection is defined by