@EdoardoLanari Do you mean the construction of k-invariants in VI.5? That seems fundamentally inexplicit: they don't really give a formula for the k-invariant. In any case what they do doesn't apply to the generalization I want to consider.

Suppose R is a nonconnective E_∞-ring, and take its associated nonconnective spectral DM stack Spét(R). Is it possible that the structure sheaf ends up being connective?