## 1 Introduction
Initially, micro-fracture emergence is dominant, and as the stress continues to be loaded, the fracture expands along the direction of the maximum principal stress, and they penetrate into each other. Experiments show that when fracture density reaches the critical threshold, the risk of local coal wall instability rises sharply [16]. In a mine in Yulin, Shaanxi Province, for example, the fracture expansion rate of the coal wall in a working face with a mining height of 7.2 m was 32 mm per hour, and a penetrating rupture surface with a depth of 2.4 m was formed within 48 h, which ultimately triggered large-scale ganging and led to an increase in the deviation coefficient of the top beam load of the stent from 0.18 to 0.47. Furthermore, the deflection rate of the column was up to 28% [17]. The sustained development of gangs will trigger multiple disaster chain reactions. (1) Roof instability: gangs lead to an increase in the distance between empty roofs at the end face (for every increase of 0.5 m, the amount of roof subsidence rises by 12-15%), and the depth of gangs in a working face in the Shendong Mining Area is 3.1 m, which shortens the distance between the top plate cycle and the pressure step by 30% and increases the frequency of opening the safety valve of the stent by two times [18]. (2) Equipment damage: the impact load generated by the collapse of the coal wall (the peak value can be up to 1.5 times the working resistance of the bracket) can easily trigger column cylinder cracks, connecting pin fractures and other structural damage. In a mine in Huainan, the annual maintenance cost of this type of accident increased by CNY 15 million. (3) Production capacity constraints: the accumulation of broken coal in the rib spalling increases the blockage rate of the transportation system by 40%. The Tashan Coal Mine of Jinneng Holdings has reduced the daily average production capacity by 25% due to the rib spalling, and the economic loss per month exceeds 8 million yuan [19]. This study aims to solve the key problem of roof instability induced by coal wall ganging in shallow buried large mining height working faces, and the specific objectives include the following. 1. Establishing the classification model of roof structures after coal wall ganging. Aiming at the geological characteristics of single and double key seams in shallow buried coal seams and combining the theories of step rock beams and masonry beams [20], we put forward four types of structural models of roof slab-coal wall collaborative damage (single key layer Categories I and II and double key layer Categories I and II). 2. Derivation of the theory of dynamic calculation of support resistance. Based on the roof structure model, the coupling relationship between the depth \(lp\) of the sheet gang and the roof load is quantified, and the calculation formula of the support resistance considering the breakage angle of the rock layer (\(\alpha\)), the rotation angle of the key block (\(\theta_{1}\)), and the coefficient of fracture expansion (\(kp\)) is constructed. 3. Validate the applicability of the model and propose optimization strategies, verify the accuracy of the model through the field data of typical mining areas in western China (Xialianta and Daliuta), analyze the threshold value of support resistance under different geological conditions, and propose an optimization scheme to dynamically regulate the matching of support resistance and mining height. The specific realization methods are as follows. 1. Roof structure model construction (theoretical analysis): based on the key layer theory and rock beam articulation structure [21], a single/double key layer roof fracture mechanics model is established, and the concept of \"step rock beam\" is introduced to describe the discontinuous bearing behavior of the roof after fracture in shallow buried coal seams. 2.