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In order to quantitatively calculate the critical depth and critical load of mines affected by rock burst, and to achieve effective prevention and control of rock burst in coal mines, this paper proposes a mechanical model for predicting the occurrence of rock burst in coal mine roadways based on catastrophe theory. Additionally, a theoretical calculation formula for initiating rock burst is derived. The first step was to establish a mechanical analysis model, which directly correlated with the in-situ stress, physical and mechanical characteristics of the coal-rock mass, and engineering structural parameters. Following this, a mechanical instability criterion was derived for the key load-bearing circle within the surrounding rock of the roadway. In the final step, the critical depth and load for rock burst initiation were verified for 25 distinct coal mines in China that were prone to rock burst hazards. The research results demonstrate that the discrepancy between the theoretically calculated critical depth and the actual measured statistical values was less than 35%. In addition, the difference between the theoretically determined critical depth and the value calculated by Pan Yishan was less than 32%. Notably, the ratio of the theoretically calculated critical load to the uniaxial compressive strength of the coal-rock mass ranged from 0.38 to 1.93. This aligns with empirical data on rock burst occurrences, as set out in the engineering classification standards for rock masses. These research outcomes substantiated the practical utility of the proposed theory, thereby laying a robust theoretical groundwork for the quantitative control of rock burst. [ABSTRACT FROM AUTHOR]