## 1 Introduction
These results contribute to the broader discourse on the interactions between trade, energy policy, and environmental sustainability, providing valuable guidance for policymakers seeking to balance economic growth with climate change mitigation efforts. This study offers four significant contributions to the literature, particularly in the realm of environmental sustainability within South Korea. First, while existing research has explored the relationship between trade liberalization and CO2 emissions, this study innovates by quantitatively analyzing the short- and long-term effects of trade policy, energy efficiency, and foreign direct investment using the Fourier autoregressive distributed lag model. This methodological advancement addresses limitations of traditional models, offering a more detailed understanding of the dynamic interactions between these variables, and accounting for structural breaks and non-linear patterns often overlooked in previous studies. Second, while much of the existing scholarship focuses on economic growth and industrialization as primary drivers of emissions, this study introduces energy efficiency as a central factor. The findings reveal that energy efficiency plays a dual role: in the short term, it mitigates the negative externalities associated with trade expansion, while over the long term, it decouples economic growth from carbon emissions. This contribution stands out as earlier studies, such as [PERSON] et al. [7] and [PERSON] et al. [8], have not adequately considered the cumulative impact of energy efficiency on emissions reduction over time. Third, this study broadens the conversation on FDI by highlighting its divergent short- and long-term impacts on emissions in South Korea. Initially, FDI is associated with increased emissions due to higher industrial activity but, over time, it supports the adoption of cleaner, more energy-efficient technologies. This analysis deepens the insights provided by [PERSON] and [PERSON] [9] and [PERSON] et al. [10], who primarily focus on the economic advantages of FDI without fully addressing its environmental consequences. Lastly, the research offers a comprehensive policy framework that aligns trade policy with environmental sustainability objectives, recommending the integration of energy efficiency and green technology innovation into South Korea's trade agreements. This forward-thinking policy perspective fills a gap in the literature, as prior research has tended to focus on technological adoption within national borders without sufficiently exploring the global dimensions of trade and sustainability. By underscoring the role of international collaboration, this study sets a new standard for future research in environmental economics and policy development. The structure of this paper is outlined as follows: Section 2 provides a thorough review of the relevant literature that underpins the research. Section 3 introduces the methodology, including the formulation of variables and model specification. Section 4 presents an in-depth analysis of the empirical results, offering a detailed interpretation of the findings. Finally, Section 5 summarizes the study's conclusions, highlighting key insights and their broader implications for policy and future research directions.