## 3 Verification of the HTP-MC Criterion  
To verify the reliability of the HTP-MC criterion, this study collected high-temperature and high-pressure triaxial compression test data for various types of rocks. As shown in the Table 1, the rock types include granite, shale, and carbonate rocks. The collected dataset includes two sets of triaxial compression tests. One set was conducted under room temperature conditions, and the other was conducted after high-temperature treatment. Considering those ultra-deep formations, with burial depths greater than 6500 m, typically reach formation temperatures of 160-220 \({}^{\circ}\)C, we selected triaxial compression test data for rocks treated at approximately 200 \({}^{\circ}\)C as far as the available data allowed. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & **Granite \({}^{\text{a}}\)** & & & **Mudstone \({}^{\text{b}}\)** & & & **Carbonate Rock \({}^{\text{c}}\)** \\ \hline
**T (\({}^{\text{c}}\)C)** & \(\sigma_{3}\) **(MPa)** & \(\sigma_{1}\) **(MPa)** & **T (\({}^{\text{c}}\)C)** & \(\sigma_{3}\) **(MPa)** & \(\sigma_{1}\) **(MPa)** & **T (\({}^{\text{c}}\)C)** & \(\sigma_{3}\) **(MPa)** & \(\sigma_{1}\) **(MPa)** \\ \hline  & 0 & 178.5 & & 0 & 39.7 & & 0 & 73.1 \\  & 5 & 226.5 & & 5 & 58 & & 10 & 126.5 \\
25 & 10 & 246.5 & & 10 & 76.8 & & 20 & 169.2 \\  & 15 & 273.6 & & 20 & 91.1 & 25 & 30 &  
Currently, in addition to the MC criterion, the Hoek-Brown failure criterion (HB criterion) is also widely applied in rock engineering. Furthermore, [PERSON] et al. [45] and others, through quantitative comparative analysis of five failure criteria, concluded that the Bieniawski failure criterion has distinct advantages in describing the nonlinear strength characteristics of rocks. Therefore, in this study, the proposed HTP-MC criterion, the HB criterion, and the Bieniawski criterion are evaluated by comparing their accuracy in predicting rock triaxial strength. The HB criterion relates rock strength to geological parameters (such as the degree of joint development and disturbance) and rock mechanical parameters (such as uniaxial compressive strength) through empirical formulas, enabling a nonlinear strength representation of rock masses. Its mathematical expression is as follows:  
\[(\sigma_{1}-\sigma_{3})=\sigma_{\rm c}(\frac{m\sigma_{3}}{\sigma_{\rm c}}+1)^{ 0.5} \tag{10}\]  
where \(m\) is a constant related to the material, dimensionless; \(\sigma_{\rm c}\) is the uniaxial compressive strength of the rock, MPa. According to [PERSON], the peak \(\sigma_{1}\)-\(\sigma_{3}\) envelope based on rock triaxial test data typically exhibits a concave shape. It is proposed that the relationship between \(\sigma_{1}\) and \(\sigma_{3}\) should be expressed in an exponential form. Additionally, all stress components should be divided by the uniaxial compressive strength of the rock to ensure dimensional consistency in the mathematical relationship between \(\sigma_{1}\) and \(\sigma_{3}\) in exponential form. The mathematical expression is as follows:  
\[\frac{\sigma_{1}}{\sigma_{\rm c}}=1+A(\frac{\sigma_{3}}{\sigma_{\rm c}})^{k} \tag{11}\]  
where \(A\) and \(k\) are constants related to the rock material, which can be fitted using experimental data, dimensionless; \(\sigma_{\rm c}\) is the uniaxial compressive strength of the rock, MPa. To quantify the deviation, the evaluation criteria used in this study are given by Equations (12) and (13), which are the coefficient of determination (\(R^{2}\)) and the mean relative error (\(MRE\)), respectively.