  Given a noncommutative partial resolution $A=\mathrm{End}_R(R\oplus M)$ of a
Gorenstein singularity $R$, we show that the relative singularity category
$\Delta_R(A)$ of Kalck-Yang is controlled by a certain connective dga
$A/^{\mathbb{L}}\kern -2pt AeA$, the derived quotient of Braun-Chuang-Lazarev.
We think of $A/^{\mathbb{L}}\kern -2pt AeA$ as a kind of `derived exceptional
locus' of the partial resolution $A$, as we show that it can be thought of as
the universal dga fitting into a suitable recollement. This theoretical result
has geometric consequences. When $R$ is an isolated hypersurface singularity,
it follows that the singularity category $D_\mathrm{sg}(R)$ is determined
completely by $A/^{\mathbb{L}}\kern -2pt AeA$, even when $A$ has infinite
global dimension. Thus our derived contraction algebra classifies threefold
flops, even those $X \to \mathrm{Spec} (R)$ where $X$ has only terminal
singularities. This gives a solution to the strongest form of the derived
Donovan-Wemyss conjecture, which we further show is the best possible
classification result in this singular setting.
