This study explores the cognitive architectures supporting abstract algebraic reasoning, a foundational component of higher mathematics. Despite its centrality, little is known about the mental structures and processes that facilitate such reasoning. Through a mixed-methods approach integrating neurocognitive assessments, eye-tracking analyses, and qualitative problem-solving interviews, We identify key cognitive patterns, including the use of symbolic schemas, hierarchical abstraction, and procedural-structural duality. The findings provide evidence for domain-specific cognitive modules in advanced algebraic reasoning, with implications for educational practices and cognitive theory development.