Complex uncertain variables are measurable functions from an uncertainty space to the set of complex numbers and are used to model complex uncertain quantities. This paper introduces the convergence concepts of complex uncertain sequences: convergenc

Background Teachers' professional identity plays a critical role in shaping educational environments, particularly through its influence on teacher-student relationships and students' psychological resilience. Although the connection between professional identity and these factors is established, the indirect effect of professional identity on student resilience—mediated by teacher-student relationships and moderated by teachers' life satisfaction—has not been fully explored. This study investigates the impact of novice teachers' professional identity on seventh-grade students' psychological resilience, focusing on the mediating role of teacher-student relationships and the moderating effect of teachers' life satisfaction. Methods Data were collected from 69 novice teachers and 2,147 students across 10 middle schools in China. The study employed Hierarchical Linear Modeling (HLM) to analyze the direct and indirect effects of teachers' professional identity on students' psychological resilience, accounting for the nested data structure. Results The analysis revealed that a strong professional identity among novice teachers significantly enhances students' psychological resilience, primarily through the improvement of teacher-student relationships. However, the expected moderating effect of teachers' life satisfaction was not supported, possibly due to the uniformly high levels of life satisfaction among the participants. Conclusion These results underscore the importance of fostering a strong professional identity in novice teachers to cultivate positive teacher-student relationships, which are crucial for enhancing students' psychological resilience. The study offers valuable insights for educational policies and interventions aimed at supporting teacher development and improving student outcomes during the critical transition to middle school.