This paper explores Koszul duality patterns in representation theory, focusing on the relationship between parabolic-singular duality and Koszul duality. The authors introduce the concept of Koszul rings, which are positively graded rings with semisimple degree zero components and specific projective resolutions. They show that the endomorphism ring of a projective module in a category of modules over a Koszul ring is itself a Koszul ring. The paper discusses the parabolic-singular duality, where certain categories of modules over universal enveloping algebras of Lie algebras are shown to be governed by Koszul rings. It also introduces the notion of Koszul duality, which relates two Koszul rings through their duals. The authors prove that the Koszul dual of a Koszul ring is also Koszul, and that this duality can be understood in terms of derived categories. The paper further explores how Koszul rings arise naturally in the context of mixed geometry, which includes theories of mixed complexes and mixed Hodge modules. It shows that certain categories of perverse sheaves and intersection cohomology complexes can be understood as categories of modules over Koszul rings. The authors also provide a detailed analysis of Koszul rings, including their definitions, properties, and relationships with other algebraic structures. They prove that any Koszul ring is quadratic and that the quadratic dual of a Koszul ring is also Koszul. The paper concludes with a discussion of the implications of these results for representation theory and the broader context of Koszul duality in mathematics.This paper explores Koszul duality patterns in representation theory, focusing on the relationship between parabolic-singular duality and Koszul duality. The authors introduce the concept of Koszul rings, which are positively graded rings with semisimple degree zero components and specific projective resolutions. They show that the endomorphism ring of a projective module in a category of modules over a Koszul ring is itself a Koszul ring. The paper discusses the parabolic-singular duality, where certain categories of modules over universal enveloping algebras of Lie algebras are shown to be governed by Koszul rings. It also introduces the notion of Koszul duality, which relates two Koszul rings through their duals. The authors prove that the Koszul dual of a Koszul ring is also Koszul, and that this duality can be understood in terms of derived categories. The paper further explores how Koszul rings arise naturally in the context of mixed geometry, which includes theories of mixed complexes and mixed Hodge modules. It shows that certain categories of perverse sheaves and intersection cohomology complexes can be understood as categories of modules over Koszul rings. The authors also provide a detailed analysis of Koszul rings, including their definitions, properties, and relationships with other algebraic structures. They prove that any Koszul ring is quadratic and that the quadratic dual of a Koszul ring is also Koszul. The paper concludes with a discussion of the implications of these results for representation theory and the broader context of Koszul duality in mathematics.