  A complex algebraic surface $S$ is a $\mathbb{Q}$-homology plane if
$H_{i}(S,\mathbb{Q})=0$ for $i>0$. The Negativity Conjecture of Palka asserts
that $\kappa(K_{X}+\tfrac{1}{2}D)=-\infty$, where $(X,D)$ is a log smooth
completion of $S$.
  We give a complete description of smooth $\mathbb{Q}$-homology planes
satisfying the Negativity Conjecture. We restrict our attention to those of log
general type, as otherwise their geometry is well understood. We show that, as
conjectured by tom Dieck and Petrie, they can be arranged in finitely many
discrete series, each obtained in a uniform way from an arrangement of lines
and conics on $\mathbb{P}^{2}$. We infer that these surfaces satisfy the
Rigidity Conjecture of Flenner and Zaidenberg; and a conjecture of Koras, which
asserts that $\#\operatorname{Aut}(S)\leq 6$.
