Hopkins–Singer connective differential K-model on smooth stacks

Let $k$ be the connective cover of $KU$ and let $c:k\to H\mathbb{C}[u]$ be the fixed complexified Chern character, with the source grading. For each integer $q$, define the Hopkins–Singer degree-$q$ object as the homotopy pullback of sheaves of spectra where $Z$ is the smart heart: its only nonzero group is the sheaf of closed total-degree-$q$ forms in chain degree zero. The bottom map is the shifted Chern character followed by the locally constant inclusion, and the other map includes closed forms. The pullback projections induce the forgetful map and curvature map. Closed forms are taken before derived global sections; their discrete sheaf has good-cover descent. Define ${\widehat{k}}^q(X)={\pi }_0\mathbf{R}\mathrm{\Gamma }(X;\widehat{k}(q))$. These degree-dependent truncations, together with the product induced by the coefficient ring and wedge multiplication, are the source's graded Hopkins–Singer connective model. Taking all degrees from one unshifted smart-heart pullback is not this convention. No equivalence with Freed–Lott cycle groups is included.

Smooth moduli stack of G-connections and its forms

Connective complex K-theory spectrum

Complex K-theory is a two-periodic generalized cohomology theory

The degree index is included explicitly to retain the source's shift and smart-heart truncation. The definition does not compute curvature kernels or suppress flat classes.