Formal Introduction of Irregular Sequences and Functions

Classical sequencetheory typically assumes regular indexing and one-to-one correspondence between indices and values. However, many naturally occurring sequences-- arising in discrete geometry, combinatorics, and step-wise dynamical models-- exhibit systematic repetition, grouping, or non-uniform progression of values. Such behavior is not adequately captured by existing frameworks and isoften treated as exceptional rather than structural.

In this paper, we develop a formal theory of irregular sequences and functions, in which irregularity is characterized by the cardinality of repeated values and theirdistribution along the index set. We introduce precise definitions of irregular sequences based on repetition cardinality, classify different types of irregularity, and establish a structural decomposition separating regular and irregular components.

Functions generated by irregular sequences are then studied, leading to a natural notion of generalised difference and derivative operators that reflect the underlying repetition structure. Several theorems are then proved to demonstrate fundamental properties of irregular sequences (functions), including invariance under translation and conditions for bounded repetition. Examplesdrawnfrom discrete geometric counting problems illustrate the applicabilityof the framework. The results provide a foundation for treating irregularity as an intrinsic and mathematically rigorous feature of discrete systems.