Derived Picard reconstruction for graded surfaces with stops

We study reconstruction of graded surfaces with stops $(S,M,\eta)$ from their mapping class groups and the derived Picard groups of their partially wrapped Fukaya categories. In genus at least two, we prove that the ungraded mapping class group determines the surface with stops $(S,M)$, while the derived Picard group with the shift distinguished determines the underlying graded surface $(S,\eta)$. Nevertheless, the derived Picard group with both the shift and the Serre functor distinguished has countably infinite fibres. We also construct arbitrarily large finite families of pairwise non-equivalent categories with the same invariant and Grothendieck rank. For line fields induced by nowhere-vanishing vector fields, we show that the rational Serre representation determines the surface with stops $(S,M)$ in every genus. For arbitrary line fields in genus at least two, the full integral derived Picard representation, with the shift distinguished, has fibres of cardinality at most four. It completely determines the graded surface with stops $(S,M,\eta)$ when $S$ has one or at least five boundary components, while non-singleton fibres occur for two, three, and four boundary components.