Climate change and sustainable development drive transformation in economic development models. Carbon emission reduction and the circular economy propel climate change and sustainable development, yet it's unclear if they synergize or counteract each other. This study examines the question from theoretical and practical perspectives. Using a theory-practice framework, bibliometric and big data analyses were conducted on the Web of Science and Chinese case data, totaling 2.29GB, to explore synergies between carbon emission reduction and the circular economy. The study finds predominantly synergistic interactions between the circular economy and carbon emission reduction, with minimal offsetting effects. That is, the circular economy markedly enhances carbon emissions reduction. At the theoretical level, the two fields are gradually evolving towards in-depth research, while at the practical level, collaboration is coalescing around four areas: hot fields, potential fields, auxiliary fields and common goals. A noteworthy contribution of this study is the development of a framework that synergizes theory and practice, providing a structured approach for future research in this domain. By quantifying the synergistic and offsetting relationship between the circular economy and carbon emissions reduction through systematic big data analysis, this research offers insights essential for achieving the UN's Sustainable Development Goals. We also stress the need for diverse case studies and multi-dimensional analyses in ongoing research.