A Modular Reduction Process Defined by the Digital Root

This work introduces an arithmetic function defined on the set of positive natural numbers using the concept of the digital root. For every integer n, the function F(n) is defined as the remainder of the division of n by its digital root dr(n), that is F(n) = n mod dr(n).

The process can also be interpreted operationally as the repeated subtraction of the digital root from the initial number until no further subtraction is possible; the remaining value represents the output of the function.

The study investigates the structural properties of this transformation, its behavior on digital classes, and the asymptotic distribution of its values. In particular, it is shown that the function only assumes the values 0,1,2,3,4,5,6,7, while the value 8 never occurs. Furthermore, the iterative application of the process always reaches zero in at most two steps.

The work provides a simple but non-trivial modular structure emerging from the interaction between natural numbers and their digital roots.

Note: The core idea and underlying mechanism presented in this work were developed independently by the author. The mathematical formalization was refined with the support of assistive tools.