\[
d_t=\sqrt{(x_t-\mu_{g_t})^{\mathsf T}
(\Sigma_{g_t}+\rho I)^{+}(x_t-\mu_{g_t})}.
\]
\[
L_{a,s}(\lambda)=\frac{1}{|T_{a,s,\lambda}|}
\sum_{t\in T_{a,s,\lambda}}d_t.
\]
\[
\beta_{a,s}(\lambda)=
\frac{\left|L_{a,s}(\lambda)-L_{a,s}(0)\right|}
{\lambda_{\mathrm{realized}}+\varepsilon},
\qquad \beta_{a,s}(0)=0.
\]
\[
\beta_a(\lambda)=\max_{s\in P_a}\beta_{a,s}(\lambda),
\qquad
\beta_f^{\mathrm{shared}}=
\max_{a\in A_f^{\mathrm{shared}}}\max_{\lambda>0}\beta_a(\lambda).
\]
Here, \(\rho=10^{-3}\), \((\cdot)^{+}\) is the Moore–Penrose pseudoinverse, and \(\varepsilon=10^{-6}\). \(L\) is a rollout-level mean transition distance; \(\beta_{a,s}\) is an episode–seed clean–stress pair; \(\beta_a\) is an episode anchor; and \(\beta_f^{\mathrm{shared}}\) is a task-family shared-support summary. Lower \(\beta\) indicates greater Stability.
