Explicit separation of quadratic irrationals from the middle-third Cantor set

Assuming a mild non-degeneracy condition excluding very low-level Cantor endpoints, and assuming a counting/input hypothesis for the contribution of non-deep orbit indices, we show that for the quadratic field $K=\mathbb{Q}(\alpha )$ there exist constants $A_K,B_K>0$ such that $$\mathrm{e}\mathrm{x}\mathrm{i}\mathrm{t}(\alpha )\le A_K({\log }_{3}H{)}^2+B_K\mathrm{.}$$ Consequently, $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(\alpha ,\mathcal{C})\ge H^{-{\kappa }_K\log H}$ for some ${\kappa }_K>0$.