Ambiguity of a differential refinement with fixed characteristic class and curvature

Let $x=x_1-x_0$. Then $I(x)=0$, so [F1] gives $x=a(\eta )$ for some $(k-1)$-form $\eta$, modulo a closed integral-period form. Since $R(x)=0$ and $R(a(\eta ))=d\eta$, the chosen representative is closed. Thus its de Rham class determines a class in $H_{\mathrm{d}\mathrm{R}}^{k-1}(M;\mathbb{R})$.

For every $j\ge 0$, a closed $j$-form $\theta$ has integral periods if and only if its real cohomology class realifies an integral class. Indeed integration gives a real cocycle $w$ on the free integral smooth chains. If its periods are integral, evaluation defines $\lambda :H_j^{\mathrm{\infty }}(M;\mathbb{Z})\to \mathbb{Z}$. The integer evaluation surjection in [F3] supplies an integral cocycle $c$ with that evaluation. Real evaluation is injective by [F3], hence $[w]=[c_{\mathbb{R}}]$. The natural smooth-to-ordinary cohomology comparison transports $[c]$ to an ordinary integral class, whose realification is $[\theta ]$ by [F2]. Conversely, such a realification evaluates integrally on every smooth integral cycle. The argument includes $j=0$, where integration evaluates functions on point chains and there are no negative-degree boundaries.

Two closed representatives give the same differential character exactly when their difference lies in ${\mathrm{\Omega }}_{\mathbb{Z}}^{k-1}(M)$ by [F1]. By step 1.2 and [F2], this subgroup maps onto exactly the image of $H^{k-1}(M;\mathbb{Z})$, so the quotient displayed in the Statement maps bijectively onto $\ker I\cap \ker R$. Conversely every closed form gives an element of that common kernel by [F1]; taking $x_0=0$ realizes every quotient class as a difference.