SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \frac {1}{\frac {1}{(1 + \frac {\epsilon}{2}) \mathcal {V} (\tau_ {1})} + \left(\frac {1 + \epsilon}{1 + \frac {\epsilon}{2}} - \cos \Delta\right) (t - \tau_ {1})} <   \mathcal {U} (t) \leq \frac {1}{\frac {1}{\mathcal {U} (\tau_ {1})} + \left(1 - \frac {\cos \Delta}{1 + \frac {\epsilon}{2}}\right) (t - \tau_ {1})}, \forall t \in (\tau_ {1}, + \infty); \\ \frac {1}{\frac {1}{\mathcal {V} (\tau_ {1})} + \left((1 + \epsilon) - (1 + \frac {\epsilon}{2}) \cos \Delta\right) (t - \tau_ {1})} \leq \mathcal {V} (t) <   \frac {1}{\frac {1 + \frac {\epsilon}{2}}{\mathcal {U} (\tau_ {1})} + \left(1 + \frac {\epsilon}{2} - \cos \Delta\right) (t - \tau_ {1})}, \forall t \in (\tau_ {1}, + \infty). \\ \end{array}
$$
It means
$$
\mathcal {U} (t) = \Theta \left(\frac {1}{\frac {\frac {1}{p ^ {\frac {1}{1 + \cos \Delta}} + \Delta^ {2}} (t - \tau_ {1})}{\kappa_ {2} ^ {2}}}\right), \forall t \geq \tau_ {1} = \mathcal {O} \left(\frac {p ^ {\frac {1}{1 + \cos \Delta}} \log (1 / \Delta)}{\kappa_ {2} ^ {2}}\right);
$$
$$
\mathcal {V} (t) = \Theta \left(\frac {1}{\frac {\frac {1}{p ^ {\overline {{1 + \cos \Delta}}}} + \Delta^ {2} (t - \tau_ {1})}{\kappa_ {2} ^ {2}}}\right), \forall t \geq \tau_ {1} = \mathcal {O} \left(\frac {p ^ {\frac {1}{1 + \cos \Delta}} \log (1 / \Delta)}{\kappa_ {2} ^ {2}}\right).
$$
Step VI. The tight bound for  $\mathcal{U}(t) - \mathcal{V}(t)\cos \Delta$
From  $1 + \frac{\epsilon}{2} < \frac{\mathcal{U}(t)}{\mathcal{V}(t)} < \frac{1 + 2\epsilon}{\cos\Delta}$  proved in Step V, the two-side bound is straightforward: for any  $t \geq \tau_1$
$$
\mathcal {U} (t) - \mathcal {V} (t) \cos \Delta > \left(1 + \frac {\epsilon}{2} - \cos \Delta\right) \mathcal {V} (t) = \Theta \left(\Delta^ {2} \mathcal {V} (t)\right),
$$
$$
\mathcal {U} (t) - \mathcal {V} (t) \cos \Delta <   \left(\frac {1 + 2 \epsilon}{\cos \Delta} - \cos \Delta\right) \mathcal {V} (t) = \Theta \left(\Delta^ {2} \mathcal {V} (t)\right).
$$
$$
\mathcal {U} (t) - \mathcal {V} (t) \cos \Delta = \Theta \left(\Delta^ {2} \mathcal {V} (t)\right) = \Theta \left(\frac {1}{\frac {1}{p ^ {\frac {1}{1 + \cos \Delta}} + (\tau_ {1})}}\right), \quad \forall t \geq \tau_ {1}.
$$
Step VII. The tight bound of  $\mathcal{U}(t)\cos \Delta -\mathcal{V}(t)$
If we follow the proof in Step VI,  $1 + \frac{\epsilon}{2} < \frac{\mathcal{U}(t)}{\mathcal{V}(t)} < \frac{1 + 2\epsilon}{\cos\Delta}$  can only gives us a loose two-side bound for  $\mathcal{U}(t)\cos \Delta -\mathcal{V}(t)$ :
$$
\mathcal {U} (t) \cos \Delta - \mathcal {V} (t) > \Big (\cos \Delta + \frac {m _ {-}}{2 m _ {+}} \cos \Delta - 1 \Big) \mathcal {V} (t) ^ {(2 3)} > - \Theta \left(\Delta^ {2} \mathcal {V} (t)\right),
$$
$$
\mathcal {U} (t) \cos \Delta - \mathcal {V} (t) \cos \Delta <   \Big (1 + 2 \epsilon - 1 \Big) \mathcal {V} (t) = \Theta \left(\Delta^ {2} \mathcal {V} (t)\right).
$$
Hence, we need more fine-grained analysis to derive its sharper bounds.
We first focus on its sharper upper bound. From the dynamics (22), for any  $t \geq T_{\mathrm{I}}$ , we have
$$
\begin{array}{l} \frac {\mathrm {d}}{\mathrm {d} t} \left(\mathcal {U} (t) \cos \Delta - \mathcal {V} (t)\right) \\ = \left(- \Big (\mathcal {U} (t) \cos \Delta - \mathcal {V} (t) \Big) \Big (\mathcal {U} (t) + (\frac {1}{\cos \Delta} + 1 - \cos \Delta) \mathcal {V} (t) \Big) - \Big (\frac {1}{\cos \Delta} - \cos \Delta - \epsilon \Big) \mathcal {V} ^ {2} (t)\right) \\ = \left(- \left(\mathcal {U} (t) \cos \Delta - \mathcal {V} (t)\right) \left(\mathcal {U} (t) + (\frac {1}{\cos \Delta} + 1 - \cos \Delta) \mathcal {V} (t)\right) - \mathcal {V} ^ {2} (t) \left(\frac {1}{\cos \Delta} - \alpha\right) \sin^ {2} \Delta\right). \\ \end{array}
$$
We define the hitting time
$$
\tau_ {\mathcal {U} / \mathcal {V}} ^ {+} := \inf \left\{t > \tau_ {1}: \mathcal {U} (t) \cos \Delta - \mathcal {V} (t) \leq 0 \right\}.
$$
From  $\frac{\mathcal{U}(\tau_1)}{\mathcal{V}(\tau_1)} = \frac{1 + \epsilon}{\cos\Delta}$ , we know  $\tau_{\mathcal{U} / \mathcal{V}}^{+}$  exists and  $\tau_{\mathcal{U} / \mathcal{V}}^{+} > \tau_{1}$ .
Then for any  $t \in (\tau_1, \tau_{\mathcal{U} / \mathcal{V}}^+)$ , we have  $\mathcal{U}(t) \cos \Delta - \mathcal{V}(t) > 0$ , so
$$
\begin{array}{l} \frac {\mathrm {d}}{\mathrm {d} t} \left(\mathcal {U} (t) \cos \Delta - \mathcal {V} (t)\right) \\ = \left(- \left(\mathcal {U} (t) \cos \Delta - \mathcal {V} (t)\right) \left(\mathcal {U} (t) + \left(\frac {1}{\cos \Delta} + 1 - \cos \Delta\right) \mathcal {V} (t)\right) - \mathcal {V} ^ {2} (t) \left(\frac {1}{\cos \Delta} - \alpha\right) \sin^ {2} \Delta\right) \\ \end{array}
$$
$$
\stackrel {\mathrm {S t e p I V}} {<  } \left(- \Big (\mathcal {U} (t) \cos \Delta - \mathcal {V} (t) \Big) \Big (\frac {1}{\cos \Delta} + 2 + \frac {\epsilon}{2} - \cos \Delta \Big) \mathcal {V} (t) - \mathcal {V} ^ {2} (t) \Big (\frac {1}{\cos \Delta} - \alpha \Big) \sin^ {2} \Delta\right)
$$
$$
\overset {\mathrm {S t e p I V}} {\leq} - \frac {\left(\frac {\sin^ {2} \Delta}{\cos \Delta} + 2 + \frac {\epsilon}{2}\right) \left(\mathcal {U} (t) \cos \Delta - \mathcal {V} (t)\right)}{\frac {1}{\mathcal {V} (\tau_ {1})} + \left((1 + \epsilon) - (1 + \frac {\epsilon}{2}) \cos \Delta\right) (t - \tau_ {1})} - \frac {\left(\frac {1}{\cos \Delta} - \alpha\right) \sin^ {2} \Delta}{\left(\frac {1}{\mathcal {V} (\tau_ {1})} + \left((1 + \epsilon) - (1 + \frac {\epsilon}{2}) \cos \Delta\right) (t - \tau_ {1})\right) ^ {2}}.
$$
For simplicity, we denote  $A = \frac{1}{\mathcal{V}(\tau_1)}$ ,  $B = (1 + \epsilon) - (1 + \frac{\epsilon}{2})\cos \Delta$ ,  $C_1 = \frac{\sin^2\Delta}{\cos\Delta} + 2 + \frac{\epsilon}{2}$ ,  $C_2 = \left(\frac{1}{\cos\Delta} - \alpha\right)\sin^2\Delta$ . And we consider the auxiliary ODE:
$$
\frac {\mathrm {d} \mathcal {E} (t)}{\mathrm {d} t} = - \frac {C _ {1} \mathcal {E} (t)}{A + B (t - \tau_ {1})} - \frac {C _ {2}}{(A + B (t - \tau_ {1})) ^ {2}},
$$
where  $\mathcal{E}(\tau_1) = \mathcal{U}(\tau_1)\cos \Delta -\mathcal{V}(\tau_1) = \epsilon \mathcal{V}(\tau_1)$ .
Its solution is
$$
\mathcal {E} (t) = \left(1 + \frac {B}{A} (t - \tau_ {1})\right) ^ {- \frac {C _ {1}}{B}} \left(\mathcal {E} (\tau_ {1}) - \frac {C _ {2}}{A (C _ {1} - B)} \left(\left(1 + \frac {B}{A} (t - \tau_ {1})\right) ^ {\frac {C _ {1}}{B} - 1} - 1\right)\right).
$$
From the Comparison Principle of ODEs, for any  $t \in (\tau_1, \tau_{\mathcal{U} / \mathcal{V}}^+)$ , we have
$$
\mathcal {U} (t) \cos \Delta - \mathcal {V} (t) \leq \mathcal {E} (t).
$$
Let  $T_{\mathcal{E}} - \tau_1 = \frac{A}{B}\left(\left(1 + \frac{(C_1 - B)\epsilon}{C_2}\right)^{\frac{1}{B} - 1} - 1\right)$ , it is easy to verify  $\mathcal{E}(T_{\mathcal{E}}) = 0$ . Moreover,
$$
\begin{array}{l} \frac {A}{B} = \Theta \left(\frac {1}{\mathcal {V} (\tau_ {1}) \left((1 + \epsilon) - (1 + \frac {\epsilon}{2}) \cos \Delta\right)}\right) = \Theta \left(\frac {1}{\kappa_ {2} ^ {2} p ^ {- \frac {1}{1 + \cos \Delta}} \Delta^ {2}}\right) = \Theta \left(\frac {p ^ {\frac {1}{1 + \cos \Delta}}}{\kappa_ {2} ^ {2} \Delta^ {2}}\right); \\ \frac {1}{\frac {C _ {1}}{B} - 1} \log \left(1 + \frac {(C _ {1} - B) \epsilon}{C _ {2}}\right) = \Theta \left(\Delta^ {2} \log \left(1 + \frac {\Theta (\Delta^ {2})}{\Theta (\Delta^ {2})}\right)\right) = \Theta (\Delta^ {2}); \\ \left(1 + \frac {(C _ {1} - B) \epsilon}{C _ {2}}\right) ^ {\frac {1}{\frac {C _ {1}}{B} - 1}} - 1 = \exp \left(\frac {1}{\frac {C _ {1}}{B} - 1} \log \left(1 + \frac {(C _ {1} - B) \epsilon}{C _ {2}}\right)\right) - 1 = \Theta (\Delta^ {2}). \\ \end{array}
$$
$$
\begin{array}{l} \tau_ {\mathcal {U} / \mathcal {V}} ^ {+} \leq T _ {\mathcal {E}} = \tau_ {1} + \Theta \left(\frac {p ^ {\frac {1}{1 + \cos \Delta}}}{\kappa_ {2} ^ {4} \Delta^ {2}} \Delta^ {2}\right) \\ = \mathcal {O} \left(\frac {p ^ {\frac {1}{1 + \cos \Delta}} \log (1 / \Delta)}{\kappa_ {2} ^ {2}}\right) + \Theta \left(\frac {p ^ {\frac {1}{1 + \cos \Delta}}}{\kappa_ {2} ^ {2}}\right) = \mathcal {O} \left(\frac {p ^ {\frac {1}{1 + \cos \Delta}} \log (1 / \Delta)}{\kappa_ {2} ^ {2}}\right). \\ \end{array}
$$
Then we define the next hitting time
$$
\tau_ {\mathcal {U} / \mathcal {V}} ^ {-} := \inf  \left\{t \geq \tau_ {\mathcal {U} / \mathcal {V}} ^ {+}: \mathcal {U} (t) \cos \Delta - \mathcal {V} (t) \geq 0 \right\}.
$$
From  $\mathcal{U}(\tau_{\mathcal{U} / \mathcal{V}}^{+})\cos \Delta -\mathcal{V}(\tau_{\mathcal{U} / \mathcal{V}}^{+}) = 0$  and  $\left.\frac{\mathrm{d}}{\mathrm{d}t}\Bigl (\mathcal{U}(t)\cos \Delta -\mathcal{V}(t)\Bigr)\right|_{t = \tau_{\mathcal{U} / \mathcal{V}}^{+}} < 0$ , we know  $\tau_{\mathcal{U} / \mathcal{V}}^{-}$  exists and  $\tau_{\mathcal{U} / \mathcal{V}}^{-} > \tau_{\mathcal{U} / \mathcal{V}}^{+}$ .
For any  $t \in (\tau_{\mathcal{U} / \mathcal{V}}^{+}, \tau_{\mathcal{U} / \mathcal{V}}^{-})$ , we have  $\mathcal{U}(t) \cos \Delta - \mathcal{V}(t) < 0$ , so
$$
\begin{array}{l} \frac {\mathrm {d}}{\mathrm {d} t} \left(\mathcal {U} (t) \cos \Delta - \mathcal {V} (t)\right) \\ \begin{array}{l} = \left(- \left(\mathcal {U} (t) \cos \Delta - \mathcal {V} (t)\right) \left(\mathcal {U} (t) + (\frac {1}{\cos \Delta} + 1 - \cos \Delta) \mathcal {V} (t)\right) - \mathcal {V} ^ {2} (t) \left(\frac {1}{\cos \Delta} - \alpha\right) \sin^ {2} \Delta\right) \\ <   \frac {\text {S t e p I V}}{} \left(- \left(\mathcal {U} (t) \cos \Delta - \mathcal {V} (t)\right) \left(1 + \frac {\frac {1}{\cos \Delta} + 1 - \cos \Delta}{1 + \frac {\epsilon}{2}}\right) \mathcal {U} (t) - \mathcal {V} ^ {2} (t) \left(\frac {1}{\cos \Delta} - \alpha\right) \sin^ {2} \Delta\right) \end{array} \\ \stackrel {\text {S t e p I V}} {\leq} - \frac {\left(1 + \frac {\frac {1}{\cos \Delta} + 1 - \cos \Delta}{1 + \frac {\epsilon}{2}}\right) \left(\mathcal {U} (t) \cos \Delta - \mathcal {V} (t)\right)}{\frac {1}{\mathcal {U} (\tau_ {1})} + \left(1 - \frac {\cos \Delta}{1 + \frac {\epsilon}{2}}\right) (t - \tau_ {1})} - \frac {\left(\frac {1}{\cos \Delta} - \alpha\right) \sin^ {2} \Delta}{\left(\frac {1}{\mathcal {V} (\tau_ {1})} + \left((1 + \epsilon) - \left(1 + \frac {\epsilon}{2}\right) \cos \Delta\right) (t - \tau_ {1})\right) ^ {2}} \\ \end{array}
$$
$$
\begin{array}{l} \leq - \frac {\left(1 + \frac {\frac {1}{\cos \Delta} + 1 - \cos \Delta}{1 + \frac {\epsilon}{2}}\right) \left(\mathcal {U} (t) \cos \Delta - \mathcal {V} (t)\right)}{\frac {\cos \Delta}{1 + \epsilon} \frac {1}{\mathcal {V} (\tau_ {1})} + \frac {1}{3} \left((1 + \epsilon) - (1 + \frac {\epsilon}{2}) \cos \Delta\right) (t - \tau_ {1})} - \frac {\left(\frac {1}{\cos \Delta} - \alpha\right) \sin^ {2} \Delta}{\left(\frac {1}{\mathcal {V} (\tau_ {1})} + \left((1 + \epsilon) - (1 + \frac {\epsilon}{2}) \cos \Delta\right) (t - \tau_ {1})\right) ^ {2}} \\ \leq - \frac {3 \left(1 + \frac {\frac {1}{\cos \Delta} + 1 - \cos \Delta}{1 + \frac {\epsilon}{2}}\right) \left(\mathcal {U} (t) \cos \Delta - \mathcal {V} (t)\right)}{\frac {1}{\mathcal {V} (\tau_ {1})} + \left((1 + \epsilon) - (1 + \frac {\epsilon}{2}) \cos \Delta\right) (t - \tau_ {1})} - \frac {\left(\frac {1}{\cos \Delta} - \alpha\right) \sin^ {2} \Delta}{\left(\frac {1}{\mathcal {V} (\tau_ {1})} + \left((1 + \epsilon) - (1 + \frac {\epsilon}{2}) \cos \Delta\right) (t - \tau_ {1})\right) ^ {2}}. \\ \end{array}
$$
For simplicity, we denote  $C_3 = 3\left(1 + \frac{\frac{1}{\cos\Delta} + 1 - \cos\Delta}{1 + \frac{\epsilon}{2}}\right)$ . And we consider the auxiliary ODE:
$$
\frac {\mathrm {d} \mathcal {F} (t)}{\mathrm {d} t} = - \frac {C _ {3} \mathcal {F} (t)}{A + B (t - \tau_ {1})} - \frac {C _ {2}}{(A + B (t - \tau_ {1})) ^ {2}},
$$
where  $\mathcal{F}(\tau_{\mathcal{U} / \mathcal{V}}^{+}) = \mathcal{U}(\tau_{\mathcal{U} / \mathcal{V}}^{+})\cos \Delta -\mathcal{V}(\tau_{\mathcal{U} / \mathcal{V}}^{+}) = 0$
$$
\begin{array}{l} \mathcal {F} (t) = - \frac {C _ {2}}{A \left(C _ {3} - B\right) \left(1 + \frac {B}{A} \left(t - \tau_ {1}\right)\right) ^ {\frac {C _ {3}}{B}}} \left(\left(1 + \frac {B}{A} \left(t - \tau_ {1}\right)\right) ^ {\frac {C _ {3}}{B} - 1} - 1\right) \\ \leq - \frac {C _ {2}}{A C _ {3} \left(1 + \frac {B}{A} (t - \tau_ {1})\right)} \left(1 - \frac {1}{\left(1 + \frac {B}{A} (t - \tau_ {1})\right) ^ {\frac {C _ {3}}{B} - 1}}\right). \\ \end{array}
$$
Let  $\tau_1' = \tau_1 + \Theta \left(\frac{p^{\frac{1}{1 + \cos\Delta}}\log(1 / \Delta)}{\kappa_2^2}\right) \geq 2\tau_1$ . Then for any  $t \geq \tau_1'$ , it holds
$$
\begin{array}{l} \frac {1}{\left(1 + \frac {B}{A} (t - \tau_ {1})\right) ^ {\frac {C _ {3}}{B} - 1}} \leq \frac {1}{\left(1 + \frac {B}{A} (\tau_ {1} ^ {\prime} - \tau_ {1})\right) ^ {\frac {C _ {3}}{B} - 1}} = \exp \left(- \left(\frac {C _ {3}}{B} - 1\right) \log \left(1 + \frac {B}{A} (\tau_ {1} ^ {\prime} - \tau_ {1})\right)\right) \\ = \exp \left(- \Theta \left(\frac {1}{\Delta^ {2}}\right) \log \left(1 + \Theta \left(\frac {\kappa_ {2} ^ {4} \Delta^ {2}}{p ^ {\frac {1}{1 + \cos \Delta}}} \frac {p ^ {\frac {1}{1 + \cos \Delta}} \log (1 / \Delta)}{\kappa_ {2} ^ {2}}\right)\right)\right) \\ = \exp \left(- \Theta \left(\frac {1}{\Delta^ {2}}\right) \log \left(1 + \Theta \left(\Delta^ {2} \log (1 / \Delta)\right)\right)\right) = \exp \left(- \Theta \left(\frac {1}{\Delta^ {2}}\right) \Theta \left(\Delta^ {2} \log (1 / \Delta)\right)\right) \\ = \exp \left(- \Theta \left(\log (1 / \Delta)\right)\right) \leq \frac {1}{2}, \\ \end{array}
$$
$$
\mathcal {F} (t) \leq - \frac {C _ {2}}{A C _ {3} \left(1 + \frac {B}{A} (t - \tau_ {1})\right)} \left(1 - \frac {1}{2}\right) = - \Theta \left(\frac {\Delta^ {2}}{\frac {\frac {1}{p ^ {1 + \cos \Delta}}}{\kappa_ {2} ^ {2}} + \Delta^ {2} (t - \tau_ {1})}\right),
$$
$$
\forall t \geq \tau_ {1} ^ {\prime} = \tau_ {1} + \Theta \left(\frac {p ^ {\frac {1}{1 + \cos \Delta}} \log (1 / \Delta)}{\kappa_ {2} ^ {2}}\right) \geq 2 \tau_ {1}.
$$
If we let  $\tau_{2} = \Theta \left(\frac{p^{\frac{1}{1 + \cos\Delta}}\log(1 / \Delta)}{\kappa_{2}^{2}}\right) \geq 2\tau_{1}^{\prime}$ , then we have:
$$
\mathcal {F} (t) \leq - \Theta \left(\frac {\Delta^ {2}}{\frac {1}{p ^ {\frac {1 + \cos \Delta}{\kappa_ {2} ^ {2}}} + \Delta^ {2} (t - \tau_ {1})}\right),   \forall t \geq \tau_ {2}.
$$
From the Comparison Principle of ODEs, for any  $t \in (\tau_{\mathcal{U} / \mathcal{V}}^{+}, \tau_{\mathcal{U} / \mathcal{V}}^{-})$ , we have
$$
\mathcal {U} (t) \cos \Delta - \mathcal {V} (t) \leq \mathcal {F} (t) <   0.
$$
From the definition of  $\tau_{\mathcal{U} / \mathcal{V}}^-$ , we obtain
$$
\tau_ {\mathcal {U} / \mathcal {V}} ^ {-} = + \infty .
$$
$$
\mathcal {U} (t) \cos \Delta - \mathcal {V} (t) \leq - \Theta \left(\frac {\Delta^ {2}}{\frac {1}{p ^ {\frac {1}{1 + \cos \Delta}}} + \Delta^ {2} (t - \tau_ {1})}\right), \forall t \geq \tau_ {2} = \Theta \left(\frac {p ^ {\frac {1}{1 + \cos \Delta}} \log (1 / \Delta)}{\kappa_ {2} ^ {2}}\right).
$$
Recalling the lower bound at the beginning of Step VII, we obtain the tight bound:
$$
\mathcal {U} (t) \cos \Delta - \mathcal {V} (t) = - \Theta \left(\frac {\Delta^ {2}}{\frac {1}{p ^ {\frac {1 + \cos \Delta}{\kappa_ {2} ^ {2}}} + \Delta^ {2} (t - \tau_ {1})}}\right) = - \Theta \left(\Delta^ {2} \mathcal {V} (t)\right), \forall t \geq \tau_ {2} = \Theta \left(\frac {p ^ {\frac {1}{1 + \cos \Delta}} \log (1 / \Delta)}{\kappa_ {2} ^ {2}}\right).
$$
Lemma D.5 (Hitting time relationship).
$$
\begin{array}{l} T _ {\mathrm {I I}} ^ {+} = T _ {\mathrm {I I}} ^ {*} = \inf  \left\{t \geq T _ {\mathrm {I}}: \exists k \in \mathcal {K} _ {+}, \text {s . t .} \langle \boldsymbol {b} _ {k} (t), \boldsymbol {x} _ {-} \rangle \leq 0 \right\} \\ = \inf  \left\{t \geq T _ {\mathrm {I}}: \exists k \in \mathcal {K} _ {+}, \text {s . t .} \langle \boldsymbol {b} _ {k} (T _ {\mathrm {I}}), \boldsymbol {x} _ {-} \rangle + \int_ {T _ {\mathrm {I}}} ^ {t} \frac {\sqrt {m}}{\kappa_ {2} m _ {+}} \left(\mathcal {U} (s) \cos \Delta - \mathcal {V} (s)\right) \mathrm {d} s \leq 0 \right\}, \\ \end{array}
$$
where  $T_{\mathrm{II}}^{+}$  and  $T_{\mathrm{II}}^{*}$  are defined in (20)(21), and  $\mathcal{U}(t),\mathcal{V}(t)$  satisfy (22).
Proof of Lemma D.5.
Recall the definitions of  $T_{\mathrm{II}}^{+}$  and  $T_{\mathrm{II}}^{*}$ :
$$
\begin{array}{l} T _ {\mathrm {I I}} ^ {+} = \inf  \left\{t > T _ {\mathrm {I}}: \exists k \in \mathcal {K} _ {+}, \text {s . t .} \langle \boldsymbol {w} _ {k} (t), \boldsymbol {x} _ {+} \rangle \leq 0 \text {o r} \langle \boldsymbol {w} _ {k} (t), \boldsymbol {x} _ {-} \rangle \leq 0 \right\} \\ = \inf  \left\{t > T _ {\mathrm {I}}: \exists k \in \mathcal {K} _ {+}, \text {s . t .} \langle \boldsymbol {b} _ {k} (t), \boldsymbol {x} _ {+} \rangle \leq 0 \text {o r} \langle \boldsymbol {b} _ {k} (t), \boldsymbol {x} _ {-} \rangle \leq 0 \right\}, \\ \end{array}
$$
$$
T _ {\mathrm {I I}} ^ {*} = \inf  \left\{t > T _ {\mathrm {I}}: \langle \boldsymbol {F} _ {+} (t), \boldsymbol {x} _ {+} \rangle \leq 0 \text {o r} \exists k \in \mathcal {K} _ {+}, \text {s . t .} \langle \boldsymbol {b} _ {k} (t), \boldsymbol {x} _ {+} \rangle \leq 0 \text {o r} \langle \boldsymbol {b} _ {k} (t), \boldsymbol {x} _ {-} \rangle \leq 0 \right\},
$$
where  $\pmb{F}_{+}(t) = \frac{p}{1 + p} e^{-f_{+}(t)}\pmb{x}_{+} - \frac{1}{1 + p} e^{f_{-}(t)}\pmb{x}_{-}$ .
$$
\langle \pmb {F} _ {+} (t), \pmb {x} _ {+} \rangle = \frac {p}{1 + p} e ^ {- f _ {+} (t)} - \frac {1}{1 + p} e ^ {f _ {-} (t)} \cos \Delta = \frac {m}{\kappa_ {2} ^ {2} m _ {+}} \Big (\mathcal {U} (t) - \mathcal {V} (t) \cos \Delta \Big).
$$
And for any  $k\in \mathcal{K}_{+}$
$$
\begin{array}{l} \left\langle \boldsymbol {b} _ {k} (t), \boldsymbol {x} _ {+} \right\rangle = \left\langle \boldsymbol {b} _ {k} \left(T _ {\mathrm {I}}\right), \boldsymbol {x} _ {+} \right\rangle + \int_ {T _ {\mathrm {I}}} ^ {t} \left\langle \frac {\mathrm {d} \boldsymbol {b} _ {k} (s)}{\mathrm {d} s}, \boldsymbol {x} _ {+} \right\rangle \mathrm {d} s \\ = \langle \pmb {b} _ {k} (T _ {\mathrm {I}}), \pmb {x} _ {+} \rangle + \int_ {T _ {\mathrm {I}}} ^ {t} \frac {\sqrt {m}}{\kappa_ {2} m _ {+}} \Big (\mathcal {U} (s) - \mathcal {V} (s) \cos \Delta \Big) \mathrm {d} s; \\ \end{array}
$$
$$
\begin{array}{l} \left\langle \boldsymbol {b} _ {k} (t), \boldsymbol {x} _ {-} \right\rangle = \left\langle \boldsymbol {b} _ {k} (T _ {\mathrm {I}}), \boldsymbol {x} _ {-} \right\rangle + \int_ {T _ {\mathrm {I}}} ^ {t} \left\langle \frac {\mathrm {d} \boldsymbol {b} _ {k} (s)}{\mathrm {d} s}, \boldsymbol {x} _ {-} \right\rangle \mathrm {d} s \\ = \langle \pmb {b} _ {k} (T _ {\mathrm {I}}), \pmb {x} _ {-} \rangle + \int_ {T _ {\mathrm {I}}} ^ {t} \frac {\sqrt {m}}{\kappa_ {2} m _ {+}} \Big (\mathcal {U} (s) \cos \Delta - \mathcal {V} (s) \Big) \mathrm {d} s. \\ \end{array}
$$
$$
\begin{array}{l} T _ {\mathrm {I I}} ^ {*} = \sup  \left\{t > T _ {\mathrm {I}}: \mathcal {U} (t) - \mathcal {V} (t) \cos \Delta > 0; \right. \\ \langle \boldsymbol {b} _ {k} (T _ {\mathrm {I}}), \boldsymbol {x} _ {+} \rangle + \int_ {T _ {\mathrm {I}}} ^ {t} \frac {\sqrt {m}}{\kappa_ {2} m _ {+}} \Big (\mathcal {U} (s) - \mathcal {V} (s) \cos \Delta \Big) \mathrm {d} s > 0, \forall k \in \mathcal {K} _ {+}; \\ \end{array}
$$
$$
\langle \pmb {b} _ {k} (T _ {\mathrm {I}}), \pmb {x} _ {-} \rangle + \int_ {T _ {\mathrm {I}}} ^ {t} \frac {\sqrt {m}}{\kappa_ {2} m _ {+}} \Big (\mathcal {U} (s) \cos \Delta - \mathcal {V} (s) \Big) \mathrm {d} s > 0, \forall k \in \mathcal {K} _ {+} \Big \}.
$$
With the help of Lemma D.4, we know that  $\mathcal{U}(t) - \mathcal{V}(t)\cos \Delta >0$  for any  $t\geq T_{\mathrm{I}}$ . So  $\langle \pmb {b}_k(T_\mathrm{I}),\pmb{x}_{+}\rangle +$ $\int_{T_{\mathrm{I}}}^{t}\frac{\sqrt{m}}{\kappa_{2}m_{+}\Delta}\Bigl (\mathcal{U}(s) - \mathcal{V}(s)\cos \Delta \Bigr)\mathrm{d}s > \langle \pmb {b}_k(T_\mathrm{I}),\pmb{x}_{+}\rangle >0,\forall k\in \mathcal{K}_{+},\forall t\geq T_{\mathrm{I}}$ . Hence, we have the transformation of the hitting time:
$$
\begin{array}{l} T _ {\mathrm {I I}} ^ {*} = \inf  \left\{t \geq T _ {\mathrm {I}}: \exists k \in \mathcal {K} _ {+}, \text {s . t .} \langle \boldsymbol {b} _ {k} (T _ {\mathrm {I}}), \boldsymbol {x} _ {-} \rangle + \int_ {T _ {\mathrm {I}}} ^ {t} \frac {\sqrt {m}}{\kappa_ {2} m _ {+}} \left(\mathcal {U} (s) \cos \Delta - \mathcal {V} (s)\right) \mathrm {d} s \leq 0 \right\}, \\ T _ {\mathrm {I I}} ^ {+} = T _ {\mathrm {I I}} ^ {*}. \\ \end{array}
$$
Lemma D.6 (Time Estimate of Phase II).
$$
T _ {\mathrm {I I}} = T _ {\mathrm {I I}} ^ {+} = T _ {\mathrm {I I}} ^ {*} = \Theta \left(\frac {p ^ {\frac {1}{1 - \alpha \cos \Delta}}}{\kappa_ {2} ^ {2} \Delta^ {2}}\right).
$$
With the help of Theorem C.12 (S1), for any  $k \in \mathcal{K}_+$ , we have:
$$
\begin{array}{l} \frac {4 . 6 6 \sqrt {\kappa_ {1} \kappa_ {2}}}{\sqrt {m}} \leq \rho_ {k} (T _ {1}) \leq \frac {1 2 \sqrt {\kappa_ {1} \kappa_ {2}}}{\sqrt {m}}; \\ \langle \boldsymbol {w} _ {k} (T _ {1}), \boldsymbol {\mu} \rangle \geq \left(1 - 4. 2 \sqrt {\kappa_ {1} \kappa_ {2}}\right) \left(1 - \frac {2}{1 + 0 . 7 \left(1 + 9 . 9 \sqrt {\frac {\kappa_ {2}}{\kappa_ {1}}}\right) ^ {1 . 1 5}}\right) \\ > 1 - 4. 2 \sqrt {\kappa_ {1} \kappa_ {2}} - \frac {2}{1 + 0 . 7 \left(1 + 9 . 9 \sqrt {\frac {\kappa_ {2}}{\kappa_ {1}}}\right) ^ {1 . 1 5}}. \\ \end{array}
$$
With the help of Lemma I.4, we have the estimate of  $\langle \pmb{w}_k(T_1),\pmb{x}_{-}\rangle$
$$
- 2 \sqrt {\epsilon} \sin \Delta - \epsilon \leq \langle \pmb {w} _ {k} (T _ {\mathrm {I}}), \pmb {x} _ {-} \rangle - \frac {p \cos \Delta - 1}{\sqrt {p ^ {2} + 1 - 2 p \cos \Delta}} \leq 2 \sqrt {\epsilon} \sin \Delta ,
$$
$$
\epsilon = 4. 2 \sqrt {\kappa_ {1} \kappa_ {2}} + \frac {2}{1 + 0 . 7 \left(1 + 9 . 9 \sqrt {\frac {\kappa_ {2}}{\kappa_ {1}}}\right) ^ {1 . 1 5}}.
$$
$$
\langle \pmb {b} _ {k} (T _ {\mathrm {I}}), \pmb {x} _ {-} \rangle \leq \frac {1 2 \sqrt {\kappa_ {1} \kappa_ {2}}}{\sqrt {m}} \left(\frac {p \cos \Delta - 1}{\sqrt {p ^ {2} + 1 - 2 p \cos \Delta}} + 2 \sqrt {\epsilon} \sin \Delta\right),
$$
$$
\langle \pmb {b} _ {k} (T _ {1}), \pmb {x} _ {-} \rangle \geq \frac {4 . 6 6 \sqrt {\kappa_ {1} \kappa_ {2}}}{\sqrt {m}} \left(\frac {p \cos \Delta - 1}{\sqrt {p ^ {2} + 1 - 2 p \cos \Delta}} - 2 \sqrt {\epsilon} \sin \Delta - \epsilon\right),
$$
$$
\left\langle \boldsymbol {b} _ {k} (T _ {\mathrm {I}}), \boldsymbol {x} _ {-} \right\rangle = \Theta \left(\frac {\sqrt {\kappa_ {1} \kappa_ {2}}}{\sqrt {m}}\right).
$$
From the dynamics (22), we have
$$
\begin{array}{l} \mathcal {U} (t) \cos \Delta - \mathcal {V} (t) \\ = \frac {m _ {-} \cos \Delta}{m _ {-} + m _ {+}} \left(\mathcal {U} (t) - \mathcal {V} (t) \cos \Delta\right) + \frac {m _ {+}}{m _ {-} + m _ {+}} \left(\mathcal {U} (t) \cos \Delta - \mathcal {V} (t) \left(1 + \alpha \sin^ {2} \Delta\right)\right) \\ \end{array}
$$
$$
= - \frac {m _ {-} \cos \Delta}{m _ {-} + m _ {+}} \frac {\mathrm {d} \mathcal {U} (t)}{\mathcal {U} (t) \mathrm {d} t} + \frac {m _ {+}}{m _ {-} + m _ {+}} \frac {\mathrm {d} \mathcal {V} (t)}{\mathcal {V} (t) \mathrm {d} t},
$$
Taking integral, we obtain:
$$
\begin{array}{l} \int_ {T _ {\mathrm {I}}} ^ {t} \frac {\sqrt {m}}{\kappa_ {2} m _ {+}} \Big (\mathcal {U} (s) \cos \Delta - \mathcal {V} (s) \Big) \mathrm {d} s \\ = \frac {\sqrt {m}}{\kappa_ {2} m _ {+}} \int_ {T _ {\mathrm {I}}} ^ {t} \left(- \frac {m _ {-} \cos \Delta}{m _ {-} + m _ {+}} \frac {\mathrm {d} \mathcal {U} (s)}{\mathcal {U} (s)} + \frac {m _ {+}}{m _ {-} + m _ {+}} \frac {\mathrm {d} \mathcal {V} (s)}{\mathcal {V} (s)}\right) \\ = \frac {\sqrt {m}}{\kappa_ {2} m _ {+}} \left(- \frac {m _ {-} \cos \Delta}{m _ {-} + m _ {+}} \log \left(\frac {\mathcal {U} (t)}{\mathcal {U} (T _ {\mathrm {I}})}\right) + \frac {m _ {+}}{m _ {-} + m _ {+}} \log \left(\frac {\mathcal {V} (t)}{\mathcal {V} (T _ {\mathrm {I}})}\right)\right) \\ = \frac {\sqrt {m}}{\kappa_ {2} m _ {+}} \log \left(\frac {\mathcal {U} (T _ {\mathrm {I}}) ^ {\frac {m _ {-} \cos \Delta}{(m _ {-} + m _ {+})}}}{\mathcal {U} (t) ^ {\frac {m _ {-} \cos \Delta}{(m _ {-} + m _ {+})}}} \cdot \frac {\mathcal {V} (t) ^ {\frac {m _ {+}}{m _ {-} + m _ {+}}}}{\mathcal {V} (T _ {\mathrm {I}}) ^ {\frac {m _ {+}}{m _ {-} + m _ {+}}}}\right). \\ \end{array}
$$
From Lemma D.5, we have:
$$
T _ {\mathrm {I I}} ^ {+} = T _ {\mathrm {I I}} ^ {*} = \inf  \left\{t \geq T _ {\mathrm {I}}: \exists k \in \mathcal {K} _ {+}, \text {s . t .} \langle \boldsymbol {b} _ {k} (T _ {\mathrm {I}}), \boldsymbol {x} _ {-} \rangle + \int_ {T _ {\mathrm {I}}} ^ {t} \frac {\sqrt {m}}{\kappa_ {2} m _ {+}} \Big (\mathcal {U} (s) \cos \Delta - \mathcal {V} (s) \Big) \mathrm {d} s \leq 0 \right\}.
$$