## 4 Conclusions  
This study explores the impact of C-class solar flares (XRA) on the ionosphere and low-frequency signal propagation by analyzing the lag time, correlation coefficients, and Granger causality between low-frequency signal strength and XRA data. The main conclusions are as follows:  
1. Correlation Analysis: The results indicate that within the lag intervals of \((-5,0)\) and \((0,\) 5) minutes, the low-frequency signal strength and XRA show the highest correlation, suggesting a rapid ionospheric response to XRA disturbances. Significant changes in signal strength are observed even before the peak of the flare. The negative lag intervals \((-10,\)\(-5)\) and earlier periods support the theory of ionospheric pre-response, indicating that changes in signal strength may precede XRA events, which serves as an early warning of impending solar flares. In the positive lag interval \((0,\) 10), the signal strength closely follows the increase in XRA, reflecting the sustained impact of XRA on the ionosphere. 2. Correlation Coefficient Analysis: As the intensity of C-class solar flares increases (from C1-C3 to C7-C9), the Pearson correlation coefficient increases from \(0.38\) to \(0.58\), demonstrating a stronger linear effect of higher intensity flares on signal propagation. Both the Spearman and Kendall Tau coefficients also rise with increasing flare intensity, indicating a stronger relationship between signal strength and XRA in terms of non-linear trends and consistency. These findings highlight that low-frequency signal strength is highly sensitive to XRA disturbances, supporting the hypothesis that C-class solar flares have a direct impact on the ionosphere. 3. Granger Causality Test: The Granger causality test further confirms the causal relationship between signal strength and XRA data. The significance levels in the lag intervals \((-5,\) 0) and \((0,\) 5) are the highest, reaching \(60\%\) and \(66.67\%\), respectively, indicating that XRA has the most significant causal impact on signal strength during these periods. In the negative lag interval \((-10,\)\(-5)\), the significance level is \(41.25\%\), supporting the idea that low-frequency signal strength responds predictively to impending XRA events. In the positive lag interval \((0,\) 10), the significance reaches \(70\%\), indicating that the impact of XRA on signal strength persists for several minutes after the flare. Through a multi-faceted analysis of lag time, correlation coefficients, and Granger causality, this study confirms the significant influence of C-class solar flare events on low-frequency signal propagation in the ionosphere and the Earth-ionosphere wave interference region. This impact is not only characterized by the ionosphere's rapid response to XRA events and its early warning capabilities but also by the lasting disturbance effects following the event. Despite the relatively low energy of C-class solar flares, their impact on the ionosphere and low-frequency signals is immediate, sustained, and predictive. These findings reveal the correlation between C-class solar flares and low-frequency time-code signal strength, providing theoretical support for the application of low-frequency signals in space weather monitoring and early warning systems. However, it is crucial to emphasize that these results do not establish a strong causal relationship between variations in low-frequency signal strength and C-class solar flare events, as statistical causality, such as that identified through Granger causality, does not confirm the underlying physical causality. The practical application of these results requires further validation in real-world scenarios, including the incorporation of long-term observational data from additional fixed stations to enhance the predictive accuracy and applicability of the proposed methods.