Moduli of $G$-bundles on rigid gerbes over affine curves

We geometrize the basic cohomology set $H^{1}(\text{Kal}_{F}, G)_{\text{basic}}$ for a global function field $F$. We do this by constructing a v-stack $\text{Bun}_{G,F}^{e}$ which has localization maps to Fargues' analogous stack $\text{Bun}_{G,F_{v}}^{e}$ for all places $v$ of $F$ and whose semistable locus is the disjoint union of $\text{Bun}_{G_{b},F}$ for all $b \in H^{1}(\text{Kott}_{F} \times_{F} \text{Kal}_{F},G)_{\text{basic}}$. We also prove a version of Tate-Nakayama duality for $H^{1}(\text{Kott}_{F} \times_{F} \text{Kal}_{F},G)_{\text{basic}}$, which lets us state a conjectural multiplicity formula for discrete automorphic representations of $G(\mathbb{A}_{F})$ adapted to this new cohomology set.