Neutrosophic Network Flow with Truth, Indeterminacy, and Falsity Capacities: An Innovative Mathematical Framework for Efficiency Evaluation of Achievement Transformation of College Student Innovation and Entrepreneurship Training Programs

This paper introduces a new mathematical framework that combines neutrosophic logic and network flow theory to improve the design of student innovation and entrepreneurship training programs. In the proposed approach, each connection in the training process is represented as a directed edge with three separate capacities: truth, indeterminacy, and falsity. These capacities describe how much confirmed learning, uncertainty, and failure can pass between stages of the program. By applying conservation rules for each component and adapting classical flow optimization methods, we can identify training paths that increase confirmed learning while reducing uncertainty and limiting the spread of failure. The framework also extends the max flow/min-cut theorem to the neutrosophic setting, allowing rigorous analysis under incomplete or contradictory data. A fully worked case study with synthetic student data shows how the model can guide curriculum adjustments, demonstrating measurable improvements in training effectiveness. This work opens a new research direction by linking neutrosophic probability with flow networks, offering a versatile tool for decision making in complex educational systems.