This article aims to present a new mathematical theory around the concept we call "recursive sum." The notion of recursive sum, in our view, is defined as the process of reducing an integer to a single digit between 0 and 9 by repeatedly adding its digits. Once defined, we explore the fundamental arithmetic properties of this operation, including its periodicity, its relationship to congruence modulo 9 and congruence modulo 3, and the determination of a prime number. In addition, we define, for an integer a between 0 and 9, another concept related to recursive sum called "inverse recursion," defined as the set of all integers whose recursive sum is equal to a. We discuss the theoretical implications of these concepts and their relationship to numerical arithmetic. Finally, we present practical examples illustrating the application of this theory in solving mathematical problems.