## 6 Conclusions  
The study's findings are listed below:  
1. The numerical simulation method simulated the penetration and extension law of fracture at the boundary of coal-rock after fracturing in the roof stratum. It was proven that the fracture in the roof stratum can extend to the coal seam. The fracture within the coal can spread rapidly and the fracture width of the coal seam was wider than that of the roof stratum. 2. With the increase in the false mudstone roof thickness and under the condition of constant fracturing volume, the width of fractures in the coal seam decreased. The reason was that the fracturing process consumed certain kinetic energy in the fracture of the mudstone pseudo roof, leading to a reduction in pore pressure when the fractures extended into the coal seam, and consequently, a decrease in fracture width. The width of fractures in the coal seam roof initially increased rapidly and then slowed down, with the final fracturing width showing a negative correlation with the thickness of the false mudstone roof. The reason was that the initiation time of the fracture at the interface between the immediate roof and the pseudo roof (sandstone-mudstone interface) increased during the fracturing process, resulting in initially larger widths of roof fractures. After the fracture of the mudstone pseudo roof, the energy loss caused a rapid decrease in the growth rate of fracture width in the later stage. 3. Considering the development of the false roof in the mudstone, physical similarity simulation was employed to simulate the expansion fracture. The results indicated that with horizontal boreholes placed within the sandy mudstone of the fractured soft coal seam roof, under reasonable vertical distance and high-volume fracturing fluid construction, and if the coal seam had a relatively thin false roof of mudstone, the fracture can penetrate through the direct roof-false roof interface (sandy mudstone-mudstone interface) and the false roof-coal seam interface (mudstone-coal seam interface), extending into the underlying coal seam. The fracture then extended to form a curved and irregular stepped pattern, demonstrating that hydraulic fracturing can achieve the goal of fracturing and increasing production in both soft and hard coal seams. However, when the coal seam developed a thicker false roof of mudstone, the mudstone created a blocking effect on hydraulic fracturing, making it difficult for the fracture to open the underlying coal seam. It was evident that the thick false mudstone roof was a detrimental factor for hydraulic fracturing and production enhancement of the coal seam roof. Therefore, it is recommended to increase the construction volume in areas where the false mudstone roof is well developed. 4. The engineering verification conducted at the Panxie coal mine and the Luling coal mine showed that by utilizing a construction drainage rate of 7.5 cubic meters per minute at Panxie coal mine, the maximum fracture length reached 218.3 m, with a maximum fracture height of 36.8 m. The maximum daily gas production of a single well reached 1450 cubic meters per day, with a total gas extraction volume of 43.62 \(\times\) 10\({}^{4}\) cubic meters over two years. At Luling coal mine, utilizing a construction -drainage rate of 10 cubic meters per minute, the maximum fracture length reached 169.1 m, with a maximum fracture height of 20.5 m. The maximum daily gas production of a single well reached 10,775 cubic meters per day, with a total gas extraction volume of 590 \(\times\) 10\({}^{4}\) cubic meters for 1090 days. This indicated that the fracture within the roof can penetrate the composite roof of coal seams and extended to the interior of the coal seams, achieving the purpose of transforming fractured and low-permeability coal seams and providing an effective mode of gas extraction. Methodology, B.W. and Q.L. ; Software, B.W., Z.L., T.F. and Y.G. ; Validation, B.W. and Z.L. ; Formal analysis, E.H. and W.D. ; Investigation, T.F. and B.M. ; Data curation, B.W., L.M. and Z.G.