<p>Clarifying critical stress threshold ranges during rock brittle failure under uniaxial compression is crucial for engineering stability analysis and instability mitigation in rock masses. Current investigations into these thresholds primarily rely on laboratory mechanical testing or statistical regression methods, often yielding conclusions with limited generalizability and ambiguous physical significance, highlighting the need for further research. This study integrates Renormalization Group theory with a rock damage constitutive model to derive mathematical expressions for the ratios of crack initiation stress to peak stress and damage stress to peak stress under uniaxial compression, thereby establishing explicit theoretical mechanical relationships among crack initiation stress, damage stress, and peak stress. The results indicate that for Class I–V rocks, the ratios of crack initiation stress to peak stress range within [0.43, 0.60], [0.43, 0.62], [0.47, 0.62], [0.43, 0.79], and [0.58, 0.73], while the ratios of damage stress to peak stress fall within [0.63, 0.88], [0.63, 0.92], [0.78, 0.92], [0.63, 0.96] and [0.84, 0.95]. These theoretical ranges were validated by 298 uniaxial compression datasets of rock specimens from varied locations and lithologies. Furthermore, based on the derived relationships, this study calculated and experimentally validated the ratio range of damage stress to peak stress for gas hydrate-bearing sediments, and proposed a corresponding range for crack initiation stress to peak stress. These findings provide critical theoretical support for the safe and efficient extraction of natural gas hydrates._</p>