In order to study bounded t-structures, B. Keller and D. Vossieck [Bull. Soc. Math. Belg., Sér. A 40, No. 2, 239–253 (1988; Zbl 0671.18003)] introduced silting objects. They established a one-to-one correspondence between equivalence classes of silting objects and algebraic t-structures for path algebras of Dynkin quivers. Later, this correspondence was studied in many settings, such as: homologically smooth non-positive dg algebras [B. Keller and P. Nicolás, Int. Math. Res. Not. 2013, No. 5, 1028–1078 (2013; Zbl 1312.18007)], finite-dimensional algebras [A. B. Buan et al., Math. Z. 271, No. 3–4, 1117–1139 (2012; Zbl 1246.05171); S. Koenig and D. Yang, Doc. Math. 19, 403–438 (2014; Zbl 1350.16010); J. Rickard and R. Rouquier, J. Algebra 475, 287–307 (2017; Zbl 1369.16004)] and proper non-positive dg algebras over an algebraically closed field \(k\) [H. Su and D. Yang, Algebr. Represent. Theory 22, No. 1, 219–238 (2019; Zbl 1428.18024)]. In this paper, the author establishes this correspondence for proper non-positive dg algebra over arbitrary field \(k\). It should be noted that, in [Zbl 1428.18024], Su and Yang used Koszul duality between dg algebras and \(A_{\infty}\)-algebras. One main reason for Su and Yang to use \(A_{\infty}\)-algebras is that the simple modules over the \(A_{\infty}\)-Koszul dual are easily constructed and characterized. In this paper, the author used the results of Keller and Nicolás [loc. cit.] to construct and characterize the simple modules over the dg-Koszul dual. In this way, the constraint in [Zbl 1428.18024] that the given simple-minded collection is elementary is removed.