Objective: This study explores the associations between parent–child exercise and children’s mental health from the perspective of family physical education. Methods: In total, 527 valid questionnaires were collected from students in grades four to six of three primary schools in Yuhang District, Hangzhou City, including a survey of the status of children’s exercise and family sports and the SCL-90 symptom self-measurement scale. Based on an analysis of practical challenges in family sports engagement and children’s mental health status, the data were analyzed and modeled using structural equation modeling to obtain a model of children’s mental health promotion, with individual children’s exercise as the primary factor and parent–child exercise as the mediator. Results: Both individual children’s exercise and parent–child exercise were significant predictors of children’s mental health promotion. Parent–child activities show a more significant negative correlation with symptoms of anxiety and depression than individual exercise alone. They also partially mediated the relationship between individual exercise and depression/anxiety symptoms. The indirect effects had confidence intervals of [−0.008, −0.001] for depression and [−0.007, −0.001] for anxiety. The direct effects of individual exercise on mental health (depression: β = −0.115; anxiety: β = −0.127) were stronger than the indirect effects and significantly positively correlated with parent–child exercise (β = 0.444, p < 0.05), suggesting that individual exercise may encourage more parent–child exercise. Conclusions: We propose a relational pathways model incorporating parent–child exercise as a mediating variable and individual exercise as the primary activity. This model is more closely aligned with real-life conditions and practical feasibility than approaches lacking such a family-based component.

Achieving “dual carbon” targets by containing carbon emissions while sustaining economic growth is challenging. This study examines the varying carbon dependency levels among China’s 30 provincial-level administrative units, considering spatial correlations in emissions. Using a semi-parametric varying coefficient spatial autoregressive panel model on 2004–2019 panel data, this study shows the following: (i) The relationship between economic growth and carbon emissions forms an “S”-shaped curve, with the contribution decreasing as tertiary industry grows, defining three stages of carbon dependency. (ii) There is significant heterogeneity in carbon dependency across provinces, with some advancing to “weak dependency” or an “economic carbon peak” due to advantages and policies. (iii) Dependency levels shift over time, with “weak dependency” being the predominant stage, though transitions occur. (iv) A positive spatial spillover effect in emissions was noted. This study recommends tailored policies for each provincial-level administrative unit based on their carbon dependency and development stage.