  The aim of this paper is two-fold: Firstly, we prove Toda's
$\chi$-independence conjecture for Gopakumar--Vafa invariants of arbitrary
local curves. Secondly, following Davison's work, we introduce the BPS
cohomology for moduli spaces of Higgs bundles of rank $r$ and Euler
characteristic $\chi$ which are not necessary coprime, and show that it does
not depend on $\chi$. This result extends the Hausel--Thaddeus conjecture on
the $\chi$-independence of E-polynomials proved by Mellit,
Groechenig--Wyss--Ziegler and Yu in two ways: we obtain an isomorphism of mixed
Hodge modules on the Hitchin base rather than an equality of E-polynomials, and
we do not need the coprime assumption.
  The proof of these results is based on a description of the moduli stack of
one-dimensional coherent sheaves on a local curve as a global critical locus
which is obtained in the companion paper by the first author and Naruki Masuda.

Algebraic Geometry;High Energy Physics - Theory