SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
1 <   e ^ {f _ {-} (T _ {\mathrm {I}})} \leq 1 + e ^ {0. 0 4 \log (1. 1)} 3. 8 5 \kappa_ {2} \sqrt {\kappa_ {1} \kappa_ {2}} \leq 1 + 1. 0 0 4 \cdot 3. 8 5 \kappa_ {2} \sqrt {\kappa_ {1} \kappa_ {2}} \leq 1 + 3. 8 7 \kappa_ {2} \sqrt {\kappa_ {1} \kappa_ {2}},
$$
$$
1 > e ^ {- f _ {+} (T _ {\mathrm {I}})} \geq 1 - e ^ {0. 0 4 \log (1. 1)} 3. 8 5 \kappa_ {2} \sqrt {\kappa_ {1} \kappa_ {2}} \geq 1 - 1. 0 0 4 \cdot 3. 8 5 \kappa_ {2} \sqrt {\kappa_ {1} \kappa_ {2}} \geq 1 - 3. 8 7 \kappa_ {2} \sqrt {\kappa_ {1} \kappa_ {2}}.
$$
Notice that  $\mathcal{U}(T_{\mathrm{I}}) = \kappa_2^2\frac{m_+}{m}\frac{p}{1 + p} e^{-f_+(T_{\mathrm{I}})}$  and  $\mathcal{V}(T_{\mathrm{I}}) = \kappa_2^2\frac{m_+}{m}\frac{1}{1 + p} e^{f_-(T_{\mathrm{I}})}$ . Then we have the estimate:
$$
1 - 3. 8 7 \kappa_ {2} \sqrt {\kappa_ {1} \kappa_ {2}} \leq \frac {\mathcal {U} (T _ {\mathrm {I}})}{\kappa_ {2} ^ {2} \frac {m _ {+}}{m} \frac {p}{1 + p}} \leq 1,
$$
$$
1 \leq \frac {\mathcal {V} (T _ {\mathrm {I}})}{\kappa_ {2} ^ {2} \frac {m _ {+}}{m} \frac {1}{1 + p}} \leq 1 + 3. 8 7 \kappa_ {2} \sqrt {\kappa_ {1} \kappa_ {2}}.
$$
From Theorem C.12 (S1)(S2), we have
$$
0. 2 5 8 \leq \frac {0 . 0 7 5 m}{0 . 2 9 m} \leq \alpha \leq \frac {0 . 2 0 5 m}{0 . 2 1 m} \leq 0. 9 7 7. \tag {23}
$$
For  $t = T_{\mathrm{I}}$  , it holds that
$$
\mathcal {U} \left(T _ {\mathrm {I}}\right) \mathcal {V} \left(T _ {\mathrm {I}}\right) \cos \Delta - \mathcal {U} ^ {2} \left(T _ {\mathrm {I}}\right) <   0,
$$
$$
\mathcal {U} \left(T _ {\mathrm {I}}\right) \mathcal {V} \left(T _ {\mathrm {I}}\right) \cos \Delta - \mathcal {V} ^ {2} \left(T _ {\mathrm {I}}\right) (1 + \epsilon) > 0.
$$
Step II. A rough estimate on  $\mathcal{U}(t)$  and  $\mathcal{V}(t)$ . In this step, we aim to prove:
$$
\mathcal {U} (t) > \mathcal {V} (t) > 0, \quad , \mathcal {U} (t) + \mathcal {V} (t) \leq \mathcal {U} \left(T _ {\mathrm {I}}\right) + \mathcal {V} \left(T _ {\mathrm {I}}\right), \quad \forall t \in \left[ T _ {\mathrm {I}}, \infty\right).
$$
First, from the definition of  $\mathcal{U}(t)$  and  $\mathcal{V}(t)$ , we have  $\mathcal{U}(t) > 0$  and  $\mathcal{V}(t) > 0$ .
Then we consider the dynamics of  $\mathcal{U}(t) + \mathcal{V}(t)$ . From
$$
\begin{array}{l} \frac {\mathrm {d}}{\mathrm {d} t} \left(\mathcal {U} (t) + \mathcal {V} (t)\right) = 2 \mathcal {U} (t) \mathcal {V} (t) \cos \Delta - \mathcal {U} ^ {2} (t) - \mathcal {V} ^ {2} (t) (1 + \epsilon) \\ = - \left(\mathcal {U} (t) - \mathcal {V} (t)\right) ^ {2} \cos \Delta - (1 - \cos \Delta) \mathcal {U} ^ {2} (t) - \mathcal {V} ^ {2} (t) (1 + \epsilon - \cos \Delta) <   0, \\ \end{array}
$$
$$
\mathcal {U} (t) + \mathcal {V} (t) \leq \mathcal {U} \left(T _ {\mathrm {I}}\right) + \mathcal {V} \left(T _ {\mathrm {I}}\right), \quad \forall t \geq T _ {\mathrm {I}}.
$$
Then we consider the dynamics of  $\mathcal{U}(t) - \mathcal{V}(t)$ . We define the hitting time
$$
\tau_ {\mathcal {U} - \mathcal {V}} := \inf  \left\{t \geq T _ {\mathrm {I}}: \mathcal {U} (t) \leq \mathcal {V} (t) \right\}.
$$
For any  $t\in [T_1,\tau_{\mathcal{U} - \mathcal{V}})$  , we have:
$$
\begin{array}{l} \frac {\mathrm {d}}{\mathrm {d} t} \Big (\mathcal {U} (t) - \mathcal {V} (t) \Big) = - \mathcal {U} ^ {2} (t) + \mathcal {V} ^ {2} (t) (1 + \epsilon) = - (\mathcal {U} (t) + \mathcal {V} (t)) (\mathcal {U} (t) - \mathcal {V} (t)) + \epsilon \mathcal {V} ^ {2} (t) \\ > - \left(\mathcal {U} (t) + \mathcal {V} (t)\right) \left(\mathcal {U} (t) - \mathcal {V} (t)\right) \geq - \left(\mathcal {U} \left(T _ {\mathrm {I}}\right) + \mathcal {V} \left(T _ {\mathrm {I}}\right)\right) \left(\mathcal {U} (t) - \mathcal {V} (t)\right), \\ \end{array}
$$
We consider the auxiliary ODE:  $\frac{d}{\mathrm{d}t}\mathcal{P}(t) = -(\mathcal{U}(T_{\mathrm{I}}) + \mathcal{V}(T_{\mathrm{I}}))\mathcal{P}(t)$ , where  $\mathcal{P}(T_{\mathrm{I}}) = \mathcal{U}(T_{\mathrm{I}}) - \mathcal{V}(T_{\mathrm{I}}) > 0$ . From the Comparison Principle of ODEs, we have:
$$
\mathcal {U} (t) - \mathcal {V} (t) \geq \mathcal {P} (t) = (\mathcal {U} (T _ {\mathrm {I}}) - \mathcal {V} (T _ {\mathrm {I}})) \exp \Big (- (\mathcal {U} (T _ {\mathrm {I}}) + \mathcal {V} (T _ {\mathrm {I}})) (t - T _ {\mathrm {I}}) \Big) > 0, \forall t \in [ T _ {\mathrm {I}}, \tau_ {\mathcal {U} - \mathcal {V}}).
$$
From the definition of  $\tau_{\mathcal{U} - \nu}$ , we have proved
$$
\tau_ {\mathcal {U} - \nu} = + \infty ;
$$
$$
\mathcal {U} (t) > \mathcal {V} (t), \forall t \in [ T _ {\mathrm {I}}, + \infty).
$$
Step III. Finer estimate in the early Phase  $t \in [T_1, \tau_1]$ . Define the following hitting time
$$
\tau_ {1} := \inf  \left\{t \geq T _ {\mathrm {I}}: \mathcal {U} (t) \cos \Delta \leq \mathcal {V} (t) (1 + \epsilon) \right\}.
$$
From Step I, we know  $\tau_{1}$  exists and  $\tau_{1} > T_{\mathrm{I}}$ . From (22), we have  $\frac{\mathrm{d}\mathcal{U}(t)}{\mathrm{d}t} < 0$  and  $\frac{\mathrm{d}\mathcal{V}(t)}{\mathrm{d}t} > 0$  when  $t \in [T_{\mathrm{I}}, \tau_{1})$ . Moreover, we have the following dynamics for  $t \in [T_{\mathrm{I}}, \tau_{1})$ :
$$
\frac {\mathrm {d} \mathcal {U}}{\mathrm {d} \mathcal {V}} = \frac {\mathcal {U V} \cos \Delta - \mathcal {U} ^ {2}}{\mathcal {U V} \cos \Delta - \mathcal {V} ^ {2} (1 + \epsilon)} = \frac {\frac {\mathcal {U}}{\mathcal {V}} \cos \Delta - \left(\frac {\mathcal {U}}{\mathcal {V}}\right) ^ {2}}{\frac {\mathcal {U}}{\mathcal {V}} \cos \Delta - (1 + \epsilon)}.
$$
If we define  $\mathcal{Z}(t) \coloneqq \frac{\mathcal{U}(t)}{\mathcal{V}(t)}$ , then we have  $\mathrm{d}\mathcal{U} = \mathcal{Z}\mathrm{d}\mathcal{V} + \mathcal{V}\mathrm{d}\mathcal{Z}$ .
The dynamics above can be transformed to:
$$
\mathcal {V} \frac {\mathrm {d} \mathcal {Z}}{\mathrm {d} \mathcal {V}} = \frac {\mathcal {Z} \cos \Delta - \mathcal {Z} ^ {2}}{\mathcal {Z} \cos \Delta - (1 + \epsilon)} - \mathcal {Z},
$$
$$
\frac {1}{\mathcal {V}} \mathrm {d} \mathcal {V} = - \frac {1}{1 + \cos \Delta + \epsilon} \left(\frac {1 + \epsilon}{\mathcal {Z}} + \frac {\sin^ {2} \Delta + \epsilon}{(1 + \cos \Delta + \epsilon) - \mathcal {Z} (1 + \cos \Delta)}\right) \mathrm {d} \mathcal {Z}.
$$
Integrating this equation from  $T_{\mathrm{I}}$  to  $t \in [T_{\mathrm{I}}, \tau_1)$ , we have:
$$
\begin{array}{l} \log \left(\frac {\mathcal {V} (t)}{\mathcal {V} (T _ {\mathrm {I}})}\right) = - \frac {1 + \epsilon}{1 + \cos \Delta + \epsilon} \log \left(\frac {\mathcal {Z} (t)}{\mathcal {Z} (T _ {\mathrm {I}})}\right) \\ + \frac {\sin^ {2} \Delta + \epsilon}{(1 + \cos \Delta + \epsilon) (1 + \cos \Delta)} \log \left(\frac {(1 + \cos \Delta) \mathcal {Z} (t) - (1 + \cos \Delta + \epsilon)}{(1 + \cos \Delta) \mathcal {Z} (T _ {\mathrm {I}}) - (1 + \cos \Delta + \epsilon)}\right), t \in [ T _ {\mathrm {I}}, \tau_ {1}). \tag {24} \\ \end{array}
$$
From the continuity of  $\mathcal{U}(t)$ ,  $\mathcal{V}(t)$  and  $\mathcal{Z}(t)$ , we have
$$
\tau_ {1} = \inf  \left\{t \geq T _ {\mathrm {I}}: \mathcal {Z} (t) \leq \frac {1 + \epsilon}{\cos \Delta} \right\}. \tag {25}
$$
Combining (25) and (24), let  $t \to \tau_1^-$ . Then we have:
$$
\mathcal {Z} (\tau_ {1}) = \frac {1 + \epsilon}{\cos \Delta};
$$
$$
\mathcal {V} (\tau_ {1}) = \mathcal {V} (T _ {\mathrm {I}}) \left(\frac {\mathcal {Z} (\tau_ {1})}{\mathcal {Z} (T _ {\mathrm {I}})}\right) ^ {- \frac {1 + \epsilon}{1 + \cos \Delta + \epsilon}} \left(\frac {(1 + \cos \Delta) \mathcal {Z} (\tau_ {1}) - (1 + \cos \Delta + \epsilon)}{(1 + \cos \Delta) \mathcal {Z} (T _ {\mathrm {I}}) - (1 + \cos \Delta + \epsilon)}\right) ^ {\frac {\sin^ {2} \Delta + \epsilon}{(1 + \cos \Delta + \epsilon) (1 + \cos \Delta)}} > 0,
$$
$$
\left(\frac {1 - 3 . 8 7 \kappa_ {2} \sqrt {\kappa_ {1} \kappa_ {2}}}{1 + 3 . 8 7 \kappa_ {2} \sqrt {\kappa_ {1} \kappa_ {2}}}\right) p \leq \mathcal {Z} (T _ {\mathrm {I}}) = \frac {\mathcal {U} (T _ {\mathrm {I}})}{\mathcal {V} (T _ {\mathrm {I}})} \leq p.
$$
$$
\begin{array}{l} \frac {\mathcal {V} (\tau_ {1})}{\mathcal {V} (T _ {\mathrm {I}})} \leq \left(\frac {p \cos \Delta}{1 + \epsilon}\right) ^ {\frac {1 + \epsilon}{1 + \cos \Delta + \epsilon}} \left(\frac {\sin^ {2} \Delta + \epsilon}{((1 + \cos \Delta) p - (1 + \cos \Delta + \epsilon)) \cos \Delta}\right) ^ {\frac {\sin^ {2} \Delta + \epsilon}{(1 + \cos \Delta + \epsilon) (1 + \cos \Delta)}} \\ \frac {\mathcal {V} (\tau_ {1})}{\mathcal {V} (T _ {\mathrm {I}})} \geq \left(\frac {(1 - 3 . 8 7 \kappa_ {2} \sqrt {\kappa_ {1} \kappa_ {2}}) p \cos \Delta}{(1 + 3 . 8 7 \kappa_ {2} \sqrt {\kappa_ {1} \kappa_ {2}}) (1 + \epsilon)}\right) ^ {\frac {1 + \epsilon}{1 + \cos \Delta + \epsilon}} \left(\frac {\sin^ {2} \Delta + \epsilon}{\left(\frac {(1 - 3 . 8 7 \kappa_ {2} \sqrt {\kappa_ {1} \kappa_ {2}}) (1 + \cos \Delta)}{1 + 3 . 8 7 \kappa_ {2} \sqrt {\kappa_ {1} \kappa_ {2}}} p - (1 + \cos \Delta + \epsilon)\right) \cos \Delta}\right) ^ {\frac {\sin^ {2} \Delta + \epsilon}{(1 + \cos \Delta + \epsilon) (1 + \cos \Delta)}} \\ \end{array}
$$
$$
\mathcal {U} (\tau_ {1}) = \frac {1 + \epsilon}{\cos \Delta} \mathcal {V} (\tau_ {1}),
$$
where  $1 \leq \frac{\mathcal{V}(T_1)}{\kappa_2^{\frac{m_+}{m}}\frac{1}{1 + p}} \leq 1 + 3.87\kappa_2\sqrt{\kappa_1\kappa_2}$  is estimated in Step I.
Step IV. Nearly tight bounds for  $\tau_{1}$
From the definition of  $\tau_{1}$ , we have  $\frac{\mathrm{d}\mathcal{V}(t)}{\mathrm{d}t} > 0$  for any  $t \in [T_{\mathrm{I}}, \tau_{1})$ , thus  $\mathcal{V}(T_{\mathrm{I}}) < \mathcal{V}(t) < \mathcal{V}(\tau_{1}), \forall t \in (T_{\mathrm{I}}, \tau_{1})$ . So we have
$$
\mathcal {U} (t) \mathcal {V} (T _ {\mathrm {I}}) \cos \Delta - \mathcal {U} ^ {2} (t) <   \frac {\mathrm {d} \mathcal {U} (t)}{\mathrm {d} t} <   \mathcal {U} (t) \mathcal {V} (\tau_ {1}) \cos \Delta - \mathcal {U} ^ {2} (t), \forall t \in (T _ {\mathrm {I}}, \tau_ {1}).
$$
We first estimate the upper bound for  $\tau_{1}$ . Consider the following dynamics and the hitting time
$$
\left\{ \begin{array}{l} \frac {\mathrm {d} \phi (t)}{\mathrm {d} t} = \phi (t) \mathcal {V} (\tau_ {1}) \cos \Delta - \phi^ {2} (t), \quad t \geq T _ {\mathrm {I}}, \\ \phi (T _ {\mathrm {I}}) = \mathcal {U} (T _ {\mathrm {I}}). \end{array} \right.
$$
$$
\tau_ {1} ^ {\mathrm {u}} := \inf  \left\{t > T _ {\mathrm {I}}: \phi (t) \leq \mathcal {U} \left(\tau_ {1}\right) \right\}
$$
Then  $\tau_{1} < \tau_{1}^{\mathrm{u}}$ . From the dynamics of  $\phi(t)$ , for any  $t \in (T_{1}, \tau_{1}^{\mathrm{u}}]$ , it holds
$$
\log \left(\frac {\phi (t)}{\phi (t) - \mathcal {V} (\tau_ {1}) \cos \Delta}\right) \bigg | _ {T _ {\mathrm {I}}} ^ {t} = (t - T _ {\mathrm {I}}) \mathcal {V} (\tau_ {1}) \cos \Delta .
$$
$$
\begin{array}{l} \tau_ {1} ^ {\mathrm {u}} - T _ {\mathrm {I}} = \frac {1}{\mathcal {V} (\tau_ {1}) \cos \Delta} \log \left(\frac {\phi \left(\tau_ {1} ^ {\mathrm {u}}\right)}{\phi (T _ {\mathrm {I}})} \frac {\left(\phi \left(T _ {\mathrm {I}}\right) - \mathcal {V} \left(\tau_ {1}\right) \cos \Delta\right)}{\left(\phi \left(\tau_ {1} ^ {\mathrm {u}}\right) - \mathcal {V} \left(\tau_ {1}\right) \cos \Delta\right)}\right) \\ = \frac {1}{\mathcal {V} (\tau_ {1}) \cos \Delta} \log \left(\frac {\mathcal {U} (\tau_ {1})}{\mathcal {U} (T _ {\mathrm {I}})} \frac {(\mathcal {U} (T _ {\mathrm {I}}) - \mathcal {V} (\tau_ {1}) \cos \Delta)}{(\mathcal {U} (\tau_ {1}) - \mathcal {V} (\tau_ {1}) \cos \Delta)}\right). \\ \end{array}
$$
With the help of Theorem C.12, we have  $\frac{m_{+}}{m} = \Theta(1)$  and  $\frac{m_{-}}{m} = \Theta(1)$ . From Step I, we have  $\mathcal{C} = \Theta\left(\frac{\kappa_2^2}{\Delta^2}\right), \mathcal{U}(T_{\mathrm{I}}) = \Theta(\kappa_2^2)$  and  $\mathcal{V}(T_{\mathrm{I}}) = \Theta\left(\frac{\kappa_2^2}{p}\right)$ . Moreover, it holds
$$
\begin{array}{l} \frac {\mathcal {V} (\tau_ {1})}{\mathcal {V} (T _ {\mathrm {I}})} = \Theta \left(p ^ {\frac {1 + \epsilon}{1 + \cos \Delta + \epsilon}} \Big (\frac {\Delta^ {2}}{p} \Big) ^ {\frac {\sin^ {2} \Delta + \epsilon}{(1 + \cos \Delta + \epsilon) (1 + \cos \Delta)}}\right) = \Theta \left(p ^ {\frac {\cos \Delta}{1 + \cos \Delta}} \Delta^ {\frac {2 \sin^ {2} \Delta + \epsilon}{(1 + \cos \Delta + \epsilon) (1 + \cos \Delta)}}\right) = \Theta \left(p ^ {\frac {\cos \Delta}{1 + \cos \Delta}}\right), \\ \mathcal {V} (\tau_ {1}) = \Theta \left(\mathcal {V} (T _ {\mathrm {I}}) p ^ {\frac {\cos \Delta}{1 + \cos \Delta}}\right) = \Theta \left(\kappa_ {2} ^ {2} p ^ {- \frac {1}{1 + \cos \Delta}}\right). \\ \end{array}
$$
It is easy to verify
$$
\frac {\mathcal {U} \left(\tau_ {1}\right)}{\mathcal {U} \left(T _ {\mathrm {I}}\right)} = \frac {1 + \epsilon}{\cos \Delta} \frac {\mathcal {V} \left(\tau_ {1}\right)}{\mathcal {U} \left(T _ {\mathrm {I}}\right)} = \Theta \left(\frac {\mathcal {V} \left(T _ {\mathrm {I}}\right)}{\mathcal {U} \left(T _ {\mathrm {I}}\right)} p ^ {\frac {\cos \Delta}{1 + \cos \Delta}}\right) = \Theta \left(p ^ {- \frac {1}{1 + \cos \Delta}}\right);
$$
$$
\frac {\mathcal {U} (T _ {\mathrm {I}}) - \mathcal {V} (\tau_ {1}) \cos \Delta}{\mathcal {U} (\tau_ {1}) - \mathcal {V} (\tau_ {1}) \cos \Delta} = \Theta \left(\frac {\kappa_ {2} ^ {2} \left(1 - p ^ {- \frac {1}{1 + \cos \Delta}}\right)}{\left(\frac {1 + \epsilon}{\cos \Delta} - \cos \Delta\right) \kappa_ {2} ^ {2} p ^ {- \frac {1}{1 + \cos \Delta}}}\right) = \Theta \left(\frac {\kappa_ {2} ^ {2} \left(1 - p ^ {- \frac {1}{1 + \cos \Delta}}\right)}{\kappa_ {2} ^ {2} \Delta^ {2} p ^ {- \frac {1}{1 + \cos \Delta}}}\right) = \Theta \left(\frac {p ^ {\frac {1}{1 + \cos \Delta}}}{\Delta^ {2}}\right).
$$
Hence, we obtain the upper bound for  $\tau_{1}$ :
$$
\begin{array}{l} \tau_ {1} \leq \tau_ {1} ^ {\mathrm {u}} = T _ {\mathrm {I}} + \Theta \left(\frac {1}{\kappa_ {2} ^ {2} p ^ {- \frac {1}{1 + \cos \Delta}}} \log \left(p ^ {- \frac {1}{1 + \cos \Delta}} \frac {p ^ {\frac {1}{1 + \cos \Delta}}}{\Delta^ {2}}\right)\right) \\ = \mathcal {O} \left(\sqrt {\frac {\kappa_ {1}}{\kappa_ {2}}}\right) + \Theta \left(\frac {p ^ {\frac {1}{1 + \cos \Delta}} \log (1 / \Delta)}{\kappa_ {2} ^ {2}}\right) = \Theta \left(\frac {p ^ {\frac {1}{1 + \cos \Delta}} \log (1 / \Delta)}{\kappa_ {2} ^ {2}}\right). \\ \end{array}
$$
In a similar way, we can derive the lower bound for  $\tau_{1}$ . Consider the following dynamics and the hitting time
$$
\left\{ \begin{array}{l} \frac {\mathrm {d} \psi (t)}{\mathrm {d} t} = \psi (t) \mathcal {V} (T _ {\mathrm {I}}) \cos \Delta - \psi^ {2} (t), \quad t \geq T _ {\mathrm {I}}, \\ \psi (T _ {\mathrm {I}}) = \mathcal {U} (T _ {\mathrm {I}}). \end{array} \right.
$$
$$
\tau_ {1} ^ {1} := \inf  \left\{t > T _ {\mathrm {I}}: \psi (t) \leq \mathcal {U} (\tau_ {1}) \right\}
$$
Then  $\tau_{1} > \tau_{1}^{1}$ . From the dynamics of  $\phi(t)$ , for any  $t \in (T_{\mathrm{I}}, \tau_{1}^{1}]$ , it holds
$$
\log \left(\frac {\psi (t)}{\psi (t) - \mathcal {V} (T _ {\mathrm {I}}) \cos \Delta}\right) \Bigg | _ {T _ {\mathrm {I}}} ^ {t} = (t - T _ {\mathrm {I}}) \mathcal {V} (T _ {\mathrm {I}}) \cos \Delta .
$$
$$
\tau_ {1} ^ {\mathrm {l}} - T _ {\mathrm {I}} = \frac {1}{\mathcal {V} (T _ {\mathrm {I}}) \cos \Delta} \log \left(\frac {\phi (\tau_ {1} ^ {\mathrm {u}})}{\phi (T _ {\mathrm {I}})} \frac {(\phi (T _ {\mathrm {I}}) - \mathcal {V} (T _ {\mathrm {I}}) \cos \Delta)}{(\phi (\tau_ {1} ^ {\mathrm {u}}) - \mathcal {V} (T _ {\mathrm {I}}) \cos \Delta)}\right)
$$
$$
\begin{array}{l} = \frac {1}{\mathcal {V} (T _ {\mathrm {I}}) \cos \Delta} \log \left(\frac {\mathcal {U} \left(\tau_ {1}\right)}{\mathcal {U} (T _ {\mathrm {I}})} \frac {\left(\mathcal {U} \left(T _ {\mathrm {I}}\right) - \mathcal {V} \left(T _ {\mathrm {I}}\right) \cos \Delta\right)}{\left(\mathcal {U} \left(\tau_ {1}\right) - \mathcal {V} \left(T _ {\mathrm {I}}\right) \cos \Delta\right)}\right) \\ = \frac {1}{\mathcal {V} (T _ {\mathrm {I}}) \cos \Delta} \log \left(1 + \frac {\mathcal {V} (T _ {\mathrm {I}})}{\mathcal {U} (T _ {\mathrm {I}})} \frac {\left(\mathcal {U} (T _ {\mathrm {I}}) - \mathcal {U} (\tau_ {1})\right) \cos \Delta}{\left(\mathcal {U} (\tau_ {1}) - \mathcal {V} (T _ {\mathrm {I}}) \cos \Delta\right)}\right). \\ \end{array}
$$
$$
\frac {\mathcal {V} (T _ {\mathrm {I}})}{\mathcal {U} (T _ {\mathrm {I}})} \frac {(\mathcal {U} (T _ {\mathrm {I}}) - \mathcal {U} (\tau_ {1})) \cos \Delta}{(\mathcal {U} (\tau_ {1}) - \mathcal {V} (T _ {\mathrm {I}}) \cos \Delta)} = \Theta \left(\frac {1}{p} \frac {\kappa_ {2} ^ {2} (1 - p ^ {- \frac {1}{1 + \cos \Delta}})}{\kappa_ {2} ^ {2} (p ^ {- \frac {1}{1 + \cos \Delta}} - p ^ {- 1})}\right) = \Theta \left(p ^ {- \frac {\cos \Delta}{1 + \cos \Delta}}\right),
$$
$$
\log \left(1 + \frac {\mathcal {V} (T _ {\mathrm {I}})}{\mathcal {U} (T _ {\mathrm {I}})} \frac {(\mathcal {U} (T _ {\mathrm {I}}) - \mathcal {U} (\tau_ {1})) \cos \Delta}{(\mathcal {U} (\tau_ {1}) - \mathcal {V} (T _ {\mathrm {I}}) \cos \Delta)}\right) = \Theta \left(p ^ {- \frac {\cos \Delta}{1 + \cos \Delta}}\right),
$$
Hence, we obtain the lower bound for  $\tau_{1}$ :
$$
\tau_ {1} \geq \tau_ {1} ^ {1} = T _ {\mathrm {I}} + \Theta \left(\frac {1}{\frac {\kappa_ {2} ^ {2}}{p}} p ^ {- \frac {\cos \Delta}{1 + \cos \Delta}}\right) = \mathcal {O} \left(\sqrt {\frac {\kappa_ {1}}{\kappa_ {2}}}\right) + \Theta \left(\frac {p ^ {\frac {1}{1 + \cos \Delta}}}{\kappa_ {2} ^ {2}}\right) = \Theta \left(\frac {p ^ {\frac {1}{1 + \cos \Delta}}}{\kappa_ {2} ^ {2}}\right).
$$
Step V. Finer Estimate in the late Phase  $t > \tau_{1}$
In this step, we focus on the dynamics when  $t > \tau_{1}$ .
First, we will prove the following nearly tight bound about the ratio of  $\mathcal{U}(t)$  to  $\mathcal{V}(t)$ :
$$
1 + \frac {\epsilon}{2} <   \frac {\mathcal {U} (t)}{\mathcal {V} (t)} <   \frac {1 + 2 \epsilon}{\cos \Delta}, \quad \forall t \in [ \tau_ {1}, + \infty).
$$
For the right inequality, we define the hitting time
$$
\tau_ {\mathcal {U} / \mathcal {V}} ^ {r} := \inf  \left\{t > \tau_ {1}: \mathcal {U} (t) \geq \frac {1 + 2 \epsilon}{\cos \Delta} \mathcal {V} (t) \right\}.
$$
From  $\frac{\mathcal{U}(\tau_1)}{\mathcal{V}(\tau_1)} = \frac{1 + \epsilon}{\cos\Delta} < \frac{1 + 2\epsilon}{\cos\Delta}$ , we know  $\tau_{\mathcal{U} / \mathcal{V}}^r$  exists and  $\tau_{\mathcal{U} / \mathcal{V}}^r > \tau_1$ .
For any  $t\in (\tau_1,\tau_{\mathcal{U} / \mathcal{V}}^r)$  , consider
$$
\begin{array}{l} \frac {\mathrm {d}}{\mathrm {d} t} \left(\mathcal {U} (t) - \frac {1 + 2 \epsilon}{\cos \Delta} \mathcal {V} (t)\right) = \left(1 - \frac {1 + 2 \epsilon}{\cos \Delta}\right) \mathcal {U} (t) \mathcal {V} (t) \cos \Delta - \mathcal {U} ^ {2} (t) + \frac {1 + 2 \epsilon}{\cos \Delta} (1 + \epsilon) \mathcal {V} ^ {2} (t) \\ = - \left(\mathcal {U} (t) - \frac {1 + 2 \epsilon}{\cos \Delta} \mathcal {V} (t)\right) \left(\mathcal {U} (t) + \left((1 + 2 \epsilon) \left(1 + \frac {1}{\cos \Delta}\right) - \cos \Delta\right) \mathcal {V} (t)\right) \\ + \left(\cos \Delta - (1 + 2 \epsilon) \left(1 + \frac {1}{\cos \Delta}\right) + \frac {1 + 2 \epsilon}{\cos \Delta} (1 + \epsilon)\right) \mathcal {V} ^ {2} (t) \\ = - \left(\mathcal {U} (t) - \frac {1 + 2 \epsilon}{\cos \Delta} \mathcal {V} (t)\right) \left(\mathcal {U} (t) + \left((1 + 2 \epsilon) \left(1 + \frac {1}{\cos \Delta}\right) - \cos \Delta\right) \mathcal {V} (t)\right) \\ + \left(\left(\cos \Delta - 1\right) + \left(\frac {1 + 2 \epsilon}{\cos \Delta} \epsilon - 2 \epsilon\right)\right) \mathcal {V} ^ {2} (t) \\ <   - \left(\mathcal {U} (t) - \frac {1 + 2 \epsilon}{\cos \Delta} \mathcal {V} (t)\right) \left(\mathcal {U} (t) + \left((1 + 2 \epsilon) \left(1 + \frac {1}{\cos \Delta}\right) - \cos \Delta\right) \mathcal {V} (t)\right) \\ \stackrel {\text {S t e p I I}} {<  } - \left((1 + 2 \epsilon) (1 + \frac {1}{\cos \Delta}) - \cos \Delta\right) (\mathcal {U} (t) + \mathcal {V} (t)) \left(\mathcal {U} (t) - \frac {1 + 2 \epsilon}{\cos \Delta} \mathcal {V} (t)\right) \\ \stackrel {\text {S t e p I I}} {\leq} - \left((1 + 2 \epsilon) (1 + \frac {1}{\cos \Delta}) - \cos \Delta\right) (\mathcal {U} (T _ {\mathrm {I}}) + \mathcal {V} (T _ {\mathrm {I}})) \left(\mathcal {U} (t) - \frac {1 + 2 \epsilon}{\cos \Delta} \mathcal {V} (t)\right). \\ \end{array}
$$
For simplicity, we denote  $C_1 \coloneqq \left((1 + 2\epsilon)(1 + \frac{1}{\cos\Delta}) - \cos \Delta\right)\left(\mathcal{U}(T_{\mathrm{I}}) + \mathcal{V}(T_{\mathrm{I}})\right) > 0$ . We consider the auxiliary ODE:  $\frac{d}{dt}\mathcal{P}(t) = -C_1\mathcal{P}(t)$ , where  $\mathcal{P}(\tau_1) = \mathcal{U}(\tau_1) - \frac{1 + 2\epsilon}{\cos\Delta}\mathcal{V}(\tau_1) < 0$ . From the Comparison Principle of ODEs, we have:
$$
\mathcal {U} (t) - \frac {1 + 2 \epsilon}{\cos \Delta} \mathcal {V} (t) \leq \mathcal {P} (t) = \mathcal {P} (\tau_ {1}) e ^ {- C _ {1} (t - \tau_ {1})} <   0, \forall t \in (\tau_ {1}, \tau_ {\mathcal {U} / \mathcal {V}} ^ {r}).
$$
From the definition of  $\tau_{\mathcal{U} / \mathcal{V}}^r$ , we have proved
$$
\tau_ {\mathcal {U} / \mathcal {V}} ^ {r} = + \infty ;
$$
$$
\mathcal {U} (t) <   \frac {1 + 2 \epsilon}{\cos \Delta} \mathcal {V} (t), \forall t \in [ \tau_ {1}, + \infty).
$$
In the same way, it can be proved that
$$
\mathcal {U} (t) > \left(1 + \frac {\epsilon}{2}\right) \mathcal {V} (t), \forall t \in [ \tau_ {1}, + \infty).
$$
Moreover, we also need to derive a tight bound for  $\mathcal{U}(t)$  and  $\mathcal{V}(t)$  when  $t > \tau_1$ , respectively.
For any  $t > \tau_{1}$  , we have
$$
\begin{array}{l} \frac {\mathrm {d}}{\mathrm {d} t} \mathcal {V} (t) = \mathcal {V} (t) \left(\mathcal {U} (t) \cos \Delta - (1 + \epsilon) \mathcal {V} (t)\right) \\ > \mathcal {V} (t) \Big ((1 + \frac {\epsilon}{2}) \mathcal {V} (t) \cos \Delta - (1 + \epsilon) \mathcal {V} (t) \Big) = - \Big ((1 + \epsilon) - (1 + \frac {\epsilon}{2}) \cos \Delta \Big) \mathcal {V} ^ {2} (t). \\ \end{array}
$$
We consider the auxiliary ODE:  $\frac{d}{\mathrm{d}t}\mathcal{P}(t) = -(1 + \epsilon) - (1 + \frac{\epsilon}{2})\cos \Delta)\mathcal{P}^2 (t)$ , where  $\mathcal{P}(\tau_1) = \mathcal{V}(\tau_1)$ .
From the Comparison Principle of ODEs, we have the lower bound for  $\mathcal{V}(t)$ :
$$
\mathcal {V} (t) \geq \mathcal {P} (t) = \frac {1}{\frac {1}{\mathcal {V} (\tau_ {1})} + \left((1 + \epsilon) - (1 + \frac {\epsilon}{2}) \cos \Delta\right) (t - \tau_ {1})}, \forall t \in (\tau_ {1}, + \infty).
$$
In the same way, for any  $t > \tau_{1}$ , we have
$$
\begin{array}{l} \frac {\mathrm {d}}{\mathrm {d} t} \mathcal {U} (t) = \mathcal {U} (t) \left(\mathcal {V} (t) \cos \Delta - \mathcal {U} (t)\right) \\ <   \mathcal {U} (t) \left(\mathcal {U} (t) \frac {\cos \Delta}{1 + \frac {\epsilon}{2}} - \mathcal {U} (t)\right) = - \left(1 - \frac {\cos \Delta}{1 + \frac {\epsilon}{2}}\right) \mathcal {U} ^ {2} (t). \\ \end{array}
$$
We consider the auxiliary ODE:  $\frac{d}{dt}\mathcal{P}(t) = -\left(1 - \frac{\cos\Delta}{1 + \frac{\varepsilon}{2}}\right)\mathcal{P}^2 (t)$ , where  $\mathcal{P}(\tau_1) = \mathcal{U}(\tau_1)$ . From the Comparison Principle of ODEs, we have the upper bound for  $\mathcal{U}(t)$ :
$$
\mathcal {U} (t) \leq \mathcal {P} (t) = \frac {1}{\frac {1}{\mathcal {U} (\tau_ {1})} + \left(1 - \frac {\cos \Delta}{1 + \frac {\epsilon}{2}}\right) (t - \tau_ {1})}, \forall t \in (\tau_ {1}, + \infty).
$$
The upper bound for  $\mathcal{V}(t)$  and the lower bound for  $\mathcal{U}(t)$  can be estimated by:
$$
\mathcal {V} (t) <   \frac {\mathcal {U} (t)}{1 + \frac {\epsilon}{2}} \leq \frac {1}{1 + \frac {\epsilon}{2}} \frac {1}{\overline {{\mathcal {U} (\tau_ {1})}} + \left(1 - \frac {\cos \Delta}{1 + \frac {\epsilon}{2}}\right) (t - \tau_ {1})}, \forall t \in (\tau_ {1}, + \infty),
$$
$$
\mathcal {U} (t) > \left(1 + \frac {\epsilon}{2}\right) \mathcal {V} (t) \geq \frac {1 + \frac {\epsilon}{2}}{\frac {1}{\mathcal {V} (\tau_ {1})} + \mathcal {C} \left((1 + \epsilon) - (1 + \frac {\epsilon}{2}) \cos \Delta\right) (t - \tau_ {1})}, \forall t \in (\tau_ {1}, + \infty).
$$
Hence, we obtain the tight bound for  $\mathcal{U}(t)$  and  $\mathcal{V}(t)$ :