  We define and study refined Gopakumar-Vafa invariants of contractible curves
in complex algebraic 3-folds, alongside the cohomological Donaldson--Thomas
theory of finite-dimensional Jacobi algebras. These Gopakumar-Vafa invariants
can be constructed one of two ways: as cohomological BPS invariants of
contraction algebras controlling the deformation theory of these curves, as
defined by Donovan and Wemyss, or by feeding the moduli spaces that Katz used
to define genus zero Gopakumar-Vafa invariants into the machinery developed by
Joyce et al. The conjecture that the two definitions give isomorphic results is
a special case of a kind of categorified version of the strong rationality
conjecture due to Pandharipande and Thomas, that we discuss and propose a means
of proving. We prove the positivity of the cohomological/refined BPS invariants
of all finite-dimensional Jacobi algebras. This result supports this
strengthening of the strong rationality conjecture, as well as the conjecture
of Brown and Wemyss stating that all finite-dimensional Jacobi algebras for
appropriate symmetric quivers are isomorphic to contraction algebras.
