A General Framework for Kaprekar Dynamics on n-Digit Integers

This preprint presents a general digit-gap framework for Kaprekar dynamics on fixed-width integers. The paper defines the Kaprekar transformation using fixed digit length with leading zeros, proves basic invariants, and shows that every orbit eventually becomes periodic because the system is finite. It also derives a symmetric digit-gap interpretation, discusses the classical decimal cases such as 495 and 6174, and includes reproducible verification of representative base-10 cycles up to 9 digits.