## 1 Introduction  
Karst landforms are widely distributed globally, each with unique characteristics in terms of morphology, genesis, and distribution range [1]. When extrapolated to un -derground mining, this results in significant variations in the burial depth of local coal seams, pronounced differentiation in the distribution of virgin rock stress, and complex stress concentration conditions in the roof. Consequently, the longwall mining face under such geological features faces challenges in controlling the stability of the surrounding rock in gob-side entry retention (GSER) [2,3]. Based on this, ensuring the stability of the surrounding rock in gob-side entries is a severe issue confronting mining engineering operations worldwide. Tracing the root of the problem, the retained roadway space, it overburdens rocks in the mining area, and the stress distribution of the large-scale structure directly affects the deformation and instability of the small-scale structure [4, 5, 6]. Thus, the extent of stress variation in the rock mass and the concentrated roof stress in karst mountainous areas are the key factors affecting the stability control of GSER surrounding rock. As a commonly used technical means for GSER [7, 8, 9], the prerequisite for roof cutting and pre-splitting is to select reasonable cutting parameters [10, 11, 12]. To this end, exploring the selection of reasonable roof-cutting parameters under the special geological conditions of karst mountainous areas will serve as the foundation for the widespread application of the roof-cutting and pre-splitting technology for gob-side entry retention in global mining engineering operations [13, 14, 15]. The structural characteristics of the roof in GSER are primarily influenced by the location of the main roof fracture [16, 17]. As the working face advances, the immediate roof undergoes irregular or regular collapse and subsidence, resulting in separation from the upper main roof. After the collapse of the immediate roof, the main roof undergoes fracture, rotation, and subsidence, eventually forming a \"masonry beam\" structure [18]. This fracture pattern is influenced by multiple parameters; the properties of the surrounding rock, mining depth, roadway layout, support methods, and mining techniques all play crucial roles, and the location of the fracture line can be on the side of the goat, directly above the roadway, or on the side of the solid coal wall [19]. Currently, computer numerical simulation is the mainstream method adopted by scholars worldwide to analyze the location and shape of the fracture of the main roof in gob-side entry retention [20, 21, 22]. From the perspective of active control, inducing the roof fracture along the goat-side cut-off line by adopting roof-cutting measures is crucial for achieving stability control and stress optimization of the GSER roof [23, 24]. In addition, scholars at home and abroad have conducted research on the relationship between roof-cutting parameters and the stability of the roof structure in gob-side entry retention [25, 26, 27, 28]. [PERSON], [PERSON] et al. used the Hoek-Brown rock failure criterion to determine the safety factor of barrier pillars [29]. [PERSON] discussed the stability index of long-wall faces [30]. [PERSON] analyzed the stability of mining roadways based on field monitoring results and numerical model analysis [31]. [PERSON] et al. analyzed the stress distribution and displacement characteristics of the surrounding rock in GSER when roof cutting is insufficient or sufficient based on the theory of short cantilever beam with roof cutting and determined key parameters such as the cutting angle and the support resistance of the constant resistance anchor cable [32]. [PERSON] et al. established a mechanical model of short cantilever beams and solved it using energy theory and a displacement variational method, further analyzing the equilibrium characteristics of the short cantilever structure [33, 34]. [PERSON] et al.