Half-Spin Tautological Relations and Fabers Proportionalities of Kappa Classes

We employ the $1/2$-spin tautological relations to provide a particular combinatorial identity. We show that this identity is a statement equivalent to Fabers formula for proportionalities of kappa-classes on $mathcal{M}_g$, $ggeq 2$. We then prove several cases of the combinatorial identity, providing a new proof of Fabers formula for those cases.