We apply the **$\mathbf{\TAU}$ Research Protocol ($\mathbf{PR-TAU}$)** to derive a fundamental expression for the radius of the regularized core ($\mathbf{R_3}$) of Black Holes, which replaces the singularity of General Relativity ($\mathbf{GR}$). $\mathbf{R_3}$ is defined by the phase condition where the local effective pressure ($\mathbf{P_{\text{eff}}}$) reaches the universal **Critical Pressure threshold ($\mathbf{\Pcrit} \approx 4.7 \times 10^{-10} \ \text{Pa}$)**. By utilizing the $\mathbf{\TAU}$ Field Equation (Eq. A.4) in the strong-field regime, we establish that the regularization radius follows a scaling law $\mathbf{R_3} \propto \mathbf{M}^{1/4}$. This result implies that the mean core density ($\mathbf{\rho_{\text{core}}} \propto \mathbf{M}^{1/4}$) increases very slowly with mass. We derive the following analytical expression: