A Temporal Neutrosophic Nano–Interior Ideal Framework for Performance Boundary Optimization in University Volleyball Courses

University volleyball courses demand continuous assessment and adaptation to address the varying skill levels, uncertainties, and development rates of students. Traditional evaluation systems often fail to account for indeterminacy and transitional performance states. This paper proposes a novel Temporal Neutrosophic Nano–Interior Ideal Framework that integrates the precision of neutrosophic nano semi-frontier theory with the structural rigor of ℵ–interior ideals. The model represents each volleyball skill as a dynamic neutrosophic triplet over time, capturing truth, indeterminacy, and falsity components for multiple skill dimensions. The nano semi-frontier component identifies players whose performance lies near the boundary between mastery and non-mastery, while the ℵ–interior ideal structure ensures the stability of performance progression within the training system. A temporal extension tracks weekly changes and optimizes training schedules to maximize overall skill advancement by the end of the semester. The framework is demonstrated through a case study using a complete semester of hypothetical volleyball course data. The proposed approach offers a mathematically rigorous, uncertainty-aware methodology for performance optimization in sports education.