A Layer-Based Framework for Differences of Powers

This research note presents a layer-based framework for studying differences of powers. It defines the layer operator as L_n(k) = k^n - (k-1)^n and develops its telescoping, polynomial, finite-difference, recurrence, combinatorial, and arithmetic interpretations. Classical identities and standard finite-difference results are identified explicitly, while arithmetic divisibility, greatest-common-divisor, residue-class, and higher-difference questions are presented as directions for further research. The note is intended as a structured framework for theorem development rather than a claim of a completed new foundational classification theorem.