Digit-Cycle Recurrence (DCR): n-Digit Generalization and Maximum Period Structure

This paper presents an n-digit generalization of the Digit-Cycle Recurrence (DCR), a digit-based iterative transformation in base b.We formalize DCR as a linear congruence over (Z/bZ)ⁿ, derive the maximum period formula using matrix order and the Chinese Remainder Theorem, and prove that all irregular cycles divide the maximum period.This work extends previous fixed-digit studies and provides a unified theoretical framework applicable to arbitrary digit lengths and bases.