We characterize the $k$-Schur functions as the graded characters of simple objects in an additive module category. This confirms a set of conjectures formulated in the Ph.D. thesis of Chen, written under the direction of Mark Haiman, and thereby establishes the algebraic framework proposed therein. As a consequence, we deduce that the modified Macdonald polynomials are $k$-Schur positive, thus realizing the original motivation behind the definition of the $k$-Schur functions by Lapointe, Lascoux, and Morse. Our approach builds on our previous work on the algebraic and geometric realization of Catalan symmetric functions, which encompasses both the $k$-Schur functions and the Hall--Littlewood functions.