Totally positive integers with 8 partitions in real quadratic fields

We construct an explicit totally positive integer $ \alpha_D\in\mathcal O_K^+ $ and prove that $ p_K(\alpha_D) = 8 $ whenever $ K = \mathbb Q(\sqrt D) $ is a real quadratic field satisfying $ \{\xi_D\} < 1/3 $. The set of such radicands $ D $ has positive density. Hence our construction provides an explicit positive density family of real quadratic fields in which the value $ 8 $ occurs. Independently, Zindulka's general theorem implies that the exceptional set $ \mathcal D(8) $ has natural density zero, so $ 8 $ is attained in almost all real quadratic fields. The construction applies uniformly to both congruence classes $ D\equiv 2, 3\pmod 4 $ and $ D\equiv 1\pmod 4 $. We also exhibit the complete list of 8 partitions for several square-free radicands and show that infinitely many fields satisfy the required condition.