## 1 Introduction  
Karst landscapes, defined by features such as sinkholes and caves, cover about 15% of the Earth's surface and are globally distributed [1]. China hosts some of the largest karst systems, particularly in provinces like Guizhou, Guangxi, and Yunnan, where these terrains offer both opportunities and challenges for infrastructure, notably tunnel construction [2]. Current research and field observations indicate that the construction of karst tunnels is notably influenced by factors such as karst caves, the presence of filling materials, and the risks of water and mud inrushes. A common engineering issue encountered in karst areas pertains to the instability and potential collapse of rock formations located between the tunnel and the karst cave. This poses substantial threats to the safety and integrity of tunnel construction and subsequent operations. Consequently, investigating the optimal safe thickness of rock strata between tunnels and karst caves is of considerable practical importance. This research aims to ensure the stability of these rock strata, thereby safeguarding the construction process and operational safety of tunnels in karst areas [3, 4]. The stability of tunnels and the safety thickness of dikes in karst areas is a critical issue attracting the attention of numerous researchers. Approaches to address these challenges include empirical methods, numerical simulations, and theoretical calculations. In empirical research, [PERSON] et al. [5] reviewed construction experiences across several major Chinese cities, suggesting methods for karst cave pretreatment. [PERSON] et al. [6] assessed rainfall impacts on tunnel water inflow, proposing safety measures for tunnel structures. [PERSON] et al. [7] introduced a novel grouting method for underwater tunnel construction, while [PERSON] et al. [8] developed a semi-empirical model predicting the safety thickness of karst cave roofs. For numerical simulations, [PERSON] et al. [9] developed a method to predict hidden karst cave tops via displacement monitoring, while [PERSON] et al. [10] optimized shield velocity to minimize cave disturbance. [PERSON] et al. [11] and [PERSON] et al. [12] explored discrete element models and lattice spring models, analyzing rock stability and failure modes during tunnel excavation. Theoretical research has furthered the understanding of tunnel collapse mechanisms. [PERSON] et al. [13; 14; 15; 16] and [PERSON] and [PERSON] [17] used advanced criteria and theorems to analyze collapse forms and derive analytical solutions. [PERSON] et al. [18] and [PERSON] and [PERSON] [19] focused on calculating minimum safe thicknesses, while [PERSON] et al. [20] and [PERSON] et al. [21] examined geological disaster impacts and backfill strength deterioration. [PERSON] et al. [22] proposed novel stability analysis methods for filled karst caves. Overall, empirical methods, while conservative, often lack precision due to varying geological conditions. Numerical simulations incorporate multiple factors but require practical validation. Theoretical calculations offer a robust foundation, yet their complexity and parameter sensitivity limit practical applicability. In recent years, the catastrophe theory has been increasingly applied to address the stability issues in geotechnical engineering. This theory determines the critical failure conditions through the principle of total potential energy, facilitating the assessment of structural stability. [PERSON] et al. [23] explored slope sliding instability using the cusp abrupt change theory, developing an instability criterion. [PERSON] et al. [24] integrated landslide case models with abrupt change theory to derive a peak abrupt change model and its inversion method for analyzing the medium and long-term evolution of landslides. [PERSON] et al. [25] employed the cusp mutation model to study the calculation of safe roof thickness in karst areas, outlining necessary instability conditions for subgrade caves.