Formal Mathematical Framework of a Modular Reduction Process via Digital Root: A Technical Revision (V2)

This technical revision (V2) formalizes an arithmetic function defined on the set of positive natural numbers based on the concept of the digital root. The function F(n) = n (mod dr(n)) is rigorously analyzed, providing formal proofs of its structural properties. In particular, the work demonstrates the restriction of the range, including the strict exclusion of the value 8 in base 10, and proves that the function is nilpotent of index 2, meaning that F(F(n)) = 0 for all n. The asymptotic distribution of the function values is also derived, showing that the set of numbers satisfying F(n) = 0 (referred to as 9-Harshad numbers) has a natural density of approximately 52.42%. Finally, the framework is generalized to arbitrary positional numeral systems, proving that the exclusion of the value b − 2 is a universal structural property.