Isolated Gaps in Almost Symmetric Arf Numerical Semigroups: Characterization Modulo m Classes and Enumeration

In this study, the isolated gaps of almost symmetric Arf numerical semigroups are systematically investigated. Taking the multiplicity m of the numerical semigroup as a reference, the isolated gaps are classified modulo m. For each isolated gap set Ii, an explicit description is given, its characterization is established, and its cardinality is computed. Consequently, new structural results concerning the gap structure of almost symmetric Arf numerical semigroups are obtained.