SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\left| \Phi \left(w _ {T _ {2}} (\iota)\right) - \Phi \left(w _ {T _ {2}} \left(\iota^ {\prime}\right)\right) \right| \leq 2 \log \frac {1}{\kappa} + \log \frac {1}{\xi} + C \tag {D.205}
$$
Proof of Lemma D.24. Since  $D_{4,T_2} = 0$ , we have  $\mathbb{E}_{w\sim \rho_{T_2}}[P_{4,d}(w)] = \gamma_4$ . Thus, by Eq. (C.4),
$$
\frac {d ^ {2} + 6 d + 8}{d ^ {2} - 1} \underset {w \sim \rho_ {T _ {2}}} {\mathbb {E}} \left[ w ^ {4} \right] - \frac {6 d + 1 2}{d ^ {2} - 1} \underset {w \sim \rho_ {T _ {2}}} {\mathbb {E}} \left[ w ^ {2} \right] + \frac {3}{d ^ {2} - 1} = \gamma_ {4} \tag {D.206}
$$
Since  $|w_{t}(\iota)| \leq 1$  for all  $t \geq 0$  and all  $\iota \in [-1,1]$ , the left-hand side of the above equation is  $\mathbb{E}_{w \sim \rho_{t}}[w^{4}] \pm O\left(\frac{1}{d}\right)$ , and rearranging gives
$$
\underset {w \sim \rho_ {t}} {\mathbb {E}} \left[ w ^ {4} \right] \leq \gamma_ {4} + O \left(\frac {1}{d}\right) \tag {D.207}
$$
By Markov's inequality,
$$
\mathbb {P} _ {w \sim \rho_ {T _ {2}}} (| w | \geq 1 / 2) \lesssim \gamma_ {4} + O \left(\frac {1}{d}\right) \lesssim \gamma_ {4} \tag {D.208}
$$
where the second inequality is because  $d \geq c_{3}$  by Assumption 3.2. Let  $\iota_0$  be such that  $w_{T_2}(\iota_0) = \frac{1}{2}$ . Let  $\iota_{\mathrm{M}} = \min (\iota_0,\iota_{\mathrm{R}})$ . Then,  $\iota_{\mathrm{M}} \in [\iota_{\mathrm{L}},\iota_{\mathrm{R}}]$ , and  $\mathbb{P}_{\iota \sim \rho_0}(|\iota | > \iota_{\mathrm{M}}) \lesssim \gamma_4$ , because  $\mathbb{P}_{\iota \sim \rho_0}(|\iota | > \iota_0) \lesssim \gamma_4$  and  $\mathbb{P}_{\iota \sim \rho_0}(|\iota | > \iota_{\mathrm{R}}) \lesssim O(\kappa)$  by Proposition D.6, meaning that  $\mathbb{P}_{\iota \sim \rho_0}(|\iota | > \iota_{\mathrm{M}}) \lesssim \max (\gamma_4,\kappa) \lesssim \gamma_4$  (where the last inequality is because  $d \geq c_{3}$  by Assumption 3.2).
Now, let  $\iota \in [\iota_{\mathrm{L}},\iota_{\mathrm{M}}]$ . Observe that on the interval  $[0,T_2]$ , for all  $\iota$ ,  $w_{t}(\iota)$  is strictly increasing as a function of  $t$ , for all  $\iota$  (see Lemma D.9). Thus, if  $T_{\mathbb{R}}$  is as defined in Lemma D.7, then for all  $\iota \in [\iota_{\mathrm{L}},\iota_{\mathrm{R}}]$ ,  $|w_{T_{\mathbb{R}}}(\iota)| \gtrsim \xi \kappa^{2}$ . In particular, for all  $\iota \in [\iota_{\mathrm{L}},\iota_{\mathrm{M}}]$ , we can write  $w_{T_2}(\iota) \geq C\xi \kappa^2$  for some universal constant  $C$ . Additionally, for  $\iota \leq \iota_{\mathrm{M}}$ , we have  $w_{T_2}(\iota) \leq \frac{1}{2}$  by Proposition D.37. Thus, since  $\Phi (w)$  is increasing in  $w$ , for  $\iota \in [\iota_{\mathrm{L}},\iota_{\mathrm{M}}]$  we have
$$
\Phi \left(w _ {T _ {2}} (\iota)\right) \geq \Phi \left(C \xi \kappa^ {2}\right) = \log \left(\frac {C \xi \kappa^ {2}}{\sqrt {1 - C ^ {2} \xi^ {2} \kappa^ {4}}}\right) \geq \log \left(C \xi \kappa^ {2}\right) = \log C + \log \xi + 2 \log \kappa \tag {D.209}
$$
$$
\Phi \left(w _ {T _ {2}} (\iota)\right) \leq \Phi (1 / 2) = \log \left(\frac {1 / 2}{\sqrt {3 / 4}}\right) = \log (1 / \sqrt {3}) = - \frac {1}{2} \log 3 \tag {D.210}
$$
Thus, for any  $\iota, \iota' \in [\iota_{\mathrm{L}}, \iota_{\mathrm{M}}]$ ,
$$
\left| \Phi \left(w _ {T _ {2}} (\iota)\right) - \Phi \left(w _ {T _ {2}} \left(\iota^ {\prime}\right)\right) \right| \leq - \frac {1}{2} \log 3 - \left(\log C + \log \xi + 2 \log \kappa\right) = 2 \log \frac {1}{\kappa} + \log \frac {1}{\xi} + C ^ {\prime} \tag {D.211}
$$
where we have defined the constant  $C' = -\frac{1}{2}\log 3 - \log C$
In the next lemma, we show that for  $\iota \in [\iota_{\mathrm{L}},\iota_{\mathrm{M}}]$ , during Phase 3, Case 2,  $w_{t}(\iota)$  can be bounded away from 0 and 1, provided that the conclusion of Lemma D.24 holds at time  $t$ .
Lemma D.25 (Lower and Upper Bounds on Particles Between  $\iota_{\mathrm{L}}$  and  $\iota_{\mathrm{M}}$ ). Suppose that we are in the setting of Theorem 3.3. Let  $s \geq T_2$ , and suppose that for all  $t \in [T_2, s]$ ,  $D_{4,t} \geq 0$  and  $D_{2,t} \leq 0$ . Further suppose that for all  $t \in [T_2, s]$ , the conclusion of Lemma D.24 holds, i.e.
$$
\left| \Phi \left(w _ {s} (\iota)\right) - \Phi \left(w _ {s} \left(\iota^ {\prime}\right)\right) \right| \leq 2 \log \frac {1}{\kappa} + \log \frac {1}{\xi} + C \tag {D.212}
$$
for all  $\iota \in [\iota_{\mathrm{L}},\iota_{\mathrm{M}}]$ , where  $\iota_{\mathrm{M}}$  is such that  $\mathbb{P}_{\iota \sim \rho_0}(|\iota | > \iota_{\mathrm{M}})\lesssim \gamma_4$ , and for some universal constant  $C$ . Then, for any  $t\in [T_2,s]$ , for all  $\iota \geq \iota_{\mathrm{L}}$ ,  $w_{t}(\iota)\geq \xi^{2}\kappa^{2}$ . Furthermore, for all  $t\in [T_2,s]$ , we have  $w_{t}(\iota_{\mathrm{M}})\leq 1 - \kappa^{4}\xi^{3}$ .
Proof of Lemma D.25. Consider some time  $t \in [T_2, s]$ . Suppose that for some  $\iota \in [\iota_{\mathrm{L}}, \iota_{\mathrm{M}}]$ ,  $w_t(\iota) \leq \xi^2 \kappa^2$ . Also, suppose  $\iota' \in [\iota_{\mathrm{L}}, \iota_{\mathrm{M}}]$  such that  $\iota' > \iota$ . Then,
(By Eq. (D.212), and b.c.  $\Phi$  is increasing,  $\iota' > \iota$  and Proposition D.37)
$$
\begin{array}{l} \Phi \left(w _ {t} \left(\iota^ {\prime}\right)\right) \leq \Phi \left(w _ {t} (\iota)\right) + 2 \log \frac {1}{\kappa} + \log \frac {1}{\xi} + C \\ \leq \log \left(\frac {\xi^ {2} \kappa^ {2}}{\sqrt {1 - \xi^ {4} \kappa^ {4}}}\right) + 2 \log \frac {1}{\kappa} + \log \frac {1}{\xi} + C \tag {D.213} \\ \leq \log \left(2 \xi^ {2} \kappa^ {2}\right) + 2 \log \frac {1}{\kappa} + \log \frac {1}{\xi} + C \quad (\text {B . c .} d \geq c _ {3}, \text {A s s u m p t i o n 3 . 2}) \\ = \log \xi + C ^ {\prime} \quad (\text {F o r s o m e u n i v e r s a l c o n s t a n t} C ^ {\prime}) \\ \end{array}
$$
Thus, because  $w \leq \frac{w}{\sqrt{1 - w^2}}$ ,
$$
\log w _ {t} \left(\iota^ {\prime}\right) \leq \log \left(\frac {w _ {t} \left(\iota^ {\prime}\right)}{\sqrt {1 - \left(w _ {t} \left(\iota^ {\prime}\right)\right) ^ {2}}}\right) \leq \log \xi + C ^ {\prime} \tag {D.214}
$$
Thus,  $w_{t}(\iota^{\prime})\lesssim \xi$  and for all  $\iota \in [\iota_{\mathrm{L}},\iota_{\mathrm{M}}]$ , we have shown that  $w_{t}(\iota)\lesssim \xi$ . In addition,  $\mathbb{P}_{\iota \sim \rho_0}(|\iota | > \iota_{\mathrm{M}})\lesssim \gamma_4$ . We can use this to calculate  $D_{2,t}$  and  $D_{4,t}$ :
$$
\begin{array}{l} D _ {2, t} = \underset {w \sim \rho_ {t}} {\mathbb {E}} \left[ \frac {d}{d - 1} w ^ {2} - \frac {1}{d - 1} \right] - \gamma_ {2} \quad (\text {B y}) \\ \leq \underset {w \sim \rho_ {t}} {\mathbb {E}} \left[ w ^ {2} \right] - \gamma_ {2} \quad (\mathrm {B . c .} w \leq 1) \\ \leq \mathbb {P} _ {\iota \sim \rho_ {0}} (| \iota | \leq \iota_ {\mathrm {M}}) \mathbb {E} [ w _ {t} (\iota) ^ {2} \mid | \iota | \leq \iota_ {\mathrm {M}} ] + \mathbb {P} _ {\iota \sim \rho_ {0}} (| \iota | > \iota_ {\mathrm {M}}) \mathbb {E} [ w _ {t} (\iota) ^ {2} \mid | \iota | \geq \iota_ {\mathrm {M}} ] - \gamma_ {2} \tag {D.215} \\ \leq O \left(\xi^ {2}\right) + \gamma_ {4} - \gamma_ {2} \quad \left(\mathrm {B . c .} w _ {t} (\iota) \lesssim \xi \text {f o r} \iota \in \left[ \iota_ {\mathrm {L}}, \iota_ {\mathrm {M}} \right] \text {a n d} \mathbb {P} (| \iota | > \iota_ {\mathrm {M}}) \lesssim \gamma_ {4}\right) \\ \lesssim - \gamma_ {2} \quad (\mathrm {B . c .} \gamma_ {4} \leq c _ {1} \gamma_ {2} ^ {2}, \gamma_ {2} \leq c _ {2} (c _ {2} \text {c h o s e n a f t e r} c _ {1}) \text {a n d} d \geq c _ {3}, \text {A s s u m p t i o n 3 . 2}) \\ \end{array}
$$
(By definition of  $P_{4,d}$ , Eq. (C.4))
$$
\begin{array}{l} D _ {4, t} = \underset {w \sim \rho_ {t}} {\mathbb {E}} \left[ \frac {d ^ {2} + 6 d + 8}{d ^ {2} - 1} w ^ {4} - \frac {6 d + 1 2}{d ^ {2} - 1} w ^ {2} + \frac {3}{d ^ {2} - 1} \right] - \gamma_ {4} \\ = \underset {w \sim \rho_ {t}} {\mathbb {E}} \left[ w ^ {4} \right] - \gamma_ {4} \pm O \left(\frac {1}{d}\right) (D.216) \\ = \mathbb {P} _ {\iota \sim \rho_ {0}} (| \iota | \leq \iota_ {\mathrm {M}}) \mathbb {E} [ w _ {t} (\iota) ^ {4} \mid | \iota | \leq \iota_ {\mathrm {M}} ] + \mathbb {P} _ {\iota \sim \rho_ {0}} (| \iota | > \iota_ {\mathrm {M}}) \mathbb {E} [ w _ {t} (\iota) ^ {4} \mid | \iota | > \iota_ {\mathrm {M}} ] - \gamma_ {4} \pm O \left(\frac {1}{d}\right) (D.217) \\ \leq O \left(\xi^ {4}\right) + O \left(\gamma_ {4}\right) - \gamma_ {4} \pm O \left(\frac {1}{d}\right) \quad (\mathrm {B . c .} w _ {t} (\iota) \lesssim \xi \text {f o r} \iota \in [ \iota_ {\mathrm {L}}, \iota_ {\mathrm {M}} ] \text {a n d} \mathbb {P} (\iota > \iota_ {\mathrm {M}}) \lesssim \gamma_ {4}) \\ \leq O \left(\xi^ {4} + \gamma_ {4}\right) (D.218) \\ \lesssim \gamma_ {4} \quad (\mathrm {B . c .} d \geq c _ {3}, \text {b y A s s u m p t i o n 3 . 2}) \\ \end{array}
$$
To complete the proof, we will show that if these bounds on  $D_{2,t}$  and  $D_{4,t}$  hold, then for any  $w$ ,  $v_{t}(w)\geq 0$ . By Lemma 4.4, it suffices to show that  $P_{t}(w) + Q_{t}(w)\leq 0$  for  $w\geq 0$ , where  $P_{t}$  and  $Q_{t}$  are as defined in Lemma 4.4. If  $C_1$  and  $C_2$  are two appropriately chosen positive universal constants, then by the above upper bounds on  $D_{2,t}$  and  $D_{4,t}$ ,
$$
P _ {t} (w) + Q _ {t} (w) = 2 \hat {\sigma} _ {2, d} ^ {2} D _ {2, t} w + 4 \hat {\sigma} _ {4, d} ^ {2} D _ {4, t} w ^ {3} + \lambda_ {d} ^ {(1)} w + \lambda_ {d} ^ {(3)} w ^ {3} \tag {D.219}
$$
$$
\begin{array}{l} \leq - C _ {1} \hat {\sigma} _ {2, d} ^ {2} \gamma_ {2} w + C _ {2} \hat {\sigma} _ {4, d} ^ {2} \cdot \gamma_ {4} w ^ {3} + O \left(\frac {\hat {\sigma} _ {2 , d} ^ {2} | D _ {2 , t} | + \hat {\sigma} _ {4 , d} ^ {2} | D _ {4 , t} |}{d}\right) w \quad (D. 2 2 0) \\ + O \Big (\frac {\hat {\sigma} _ {4 , d} ^ {2} | D _ {4 , t} |}{d} \Big) w ^ {3} \\ \end{array}
$$
(By definition of  $\lambda_d^{(1)},\lambda_d^{(3)}$  in Lemma 4.4 and above bounds on  $D_{2,t},D_{4,t}$
$$
\leq - C _ {1} ^ {\prime} \hat {\sigma} _ {2, d} ^ {2} \gamma_ {2} w + O \left(\frac {\hat {\sigma} _ {2 , d} ^ {2} | D _ {2 , t} | + \hat {\sigma} _ {4 , d} ^ {2} | D _ {4 , t} |}{d}\right) | w |
$$
(B.c.  $\gamma_4\lesssim \gamma_2^2\leq \gamma_2$  and  $\hat{\sigma}_{4,d}^2\lesssim \hat{\sigma}_{2,d}^2$  by Assumption 3.2, with new constant  $C_1^\prime$
$$
\lesssim - \hat {\sigma} _ {2, d} ^ {2} \gamma_ {2} w \quad (\mathrm {B . c .} d \geq c _ {3}, \text {A s s u m p t i o n 3 . 2})
$$
In summary, if  $w_{t}(\iota) = \xi^{2}\kappa^{2}$  for some  $\iota \in [\iota_{\mathrm{L}},\iota_{\mathrm{M}}]$ , then  $v_{t}(w) > 0$  for all  $w \in [0,1]$ . In addition, at time  $T_{2}$ , we have  $w_{t}(\iota) \gtrsim \xi \kappa^{2}$  for all  $\iota >\iota_{\mathrm{L}}$  by Lemma D.9 and Lemma D.7. Thus, for all  $\iota \geq \iota_{\mathrm{L}}$ ,  $w_{t}(\iota)$  will never go below  $\xi^2\kappa^2$  for  $t \in [T_2,s]$ , as desired.
To finish the proof, we will show the upper bound on  $w_{t}(\iota_{\mathrm{M}})$ . Suppose  $1 - w_{t}(\iota_{\mathrm{M}}) \leq \kappa^{4}\xi^{3}$ . Then,
$$
\begin{array}{l} \Phi \left(w _ {t} \left(\iota_ {\mathrm {M}}\right)\right) = \log \left(\frac {w _ {t} \left(\iota_ {\mathrm {M}}\right)}{\sqrt {1 - w _ {t} \left(\iota_ {\mathrm {M}}\right) ^ {2}}}\right) (D.221) \\ \geq \log \frac {1}{2} - \frac {1}{2} \log \left(1 - w _ {t} \left(\iota_ {\mathrm {M}}\right) ^ {2}\right) \quad (\mathrm {B . c .} w _ {t} \left(\iota_ {\mathrm {M}}\right) \geq \frac {1}{2}) \\ \geq \log \frac {1}{2} - \frac {1}{2} \log \left(1 - \left(1 - \kappa^ {4} \xi^ {3}\right) ^ {2}\right) (D.222) \\ = \log \frac {1}{2} - \frac {1}{2} \log \left(2 \kappa^ {4} \xi^ {3} - \kappa^ {8} \xi^ {6}\right) (D.223) \\ \geq \log \frac {1}{2} - \frac {1}{2} \log \left(\kappa^ {4} \xi^ {3}\right) \quad (\mathrm {B . c .} \kappa , \xi \leq 1 \text {a n d} \kappa^ {4} \xi^ {3} - \left(\kappa^ {4} \xi^ {3}\right) ^ {2} \geq 0) \\ = \log \frac {1}{2} + 2 \log \frac {1}{\kappa} + \frac {3}{2} \log \frac {1}{\xi} (D.224) \\ \end{array}
$$
Thus, for any  $\iota \in [\iota_{\mathrm{L}},\iota_{\mathrm{M}}]$
$$
\Phi \left(w _ {t} (\iota)\right) \geq C + \frac {1}{2} \log \frac {1}{\xi} \tag {D.225}
$$
for some universal constant  $C$ . In other words, letting  $w_{t} \coloneqq w_{t}(\iota)$  and rearranging using the definition of  $\Phi$ , we obtain
$$
\frac {w _ {t}}{\sqrt {1 - w _ {t} ^ {2}}} \gtrsim \frac {1}{\xi^ {1 / 2}} \tag {D.226}
$$
$$
\sqrt {1 - w _ {t} ^ {2}} \lesssim \xi^ {1 / 2} w _ {t} \lesssim \xi^ {1 / 2} \tag {D.227}
$$
which finally gives  $1 - w_{t}^{2}\lesssim \xi$ , or  $w_{t}^{2}\geq 1 - O(\xi)$ . In other words, for all  $\iota \geq \iota_{\mathrm{L}}$ , we have  $w_{t}(\iota)\geq 1 - O(\xi)$ , which implies that
$$
\begin{array}{l} D _ {2, t} = \underset {w \sim \rho_ {t}} {\mathbb {E}} \left[ P _ {2, d} (w) \right] - \gamma_ {2} (D.228) \\ \gtrsim \underset {w \sim \rho_ {t}} {\mathbb {E}} \left[ w ^ {2} \right] - \gamma_ {2} - O \left(\frac {1}{d}\right) (D.229) \\ \geq \mathbb {P} _ {\iota \sim \rho_ {0}} (| \iota | \geq \iota_ {\mathrm {L}}) \underset {w \sim \rho_ {t}} {\mathbb {E}} \left[ w ^ {2} \mid | \iota | \geq \iota_ {\mathrm {L}} \right] - \gamma_ {2} - O \left(\frac {1}{d}\right) (D.230) \\ \geq \left(1 - O (\kappa)\right) \cdot \left(\underset {w \sim \rho_ {t}} {\mathbb {E}} \left[ w ^ {2} \mid | \iota | \geq \iota_ {\mathrm {L}} \right]\right) - \gamma_ {2} - O \left(\frac {1}{d}\right) (ByPropositionD.6) \\ \geq (1 - O (\kappa)) (1 - O (\xi)) - \gamma_ {2} - O \left(\frac {1}{d}\right) \quad \text {(B . c . i f} | \iota | > \iota_ {\mathrm {L}}, \text {t h e n} w _ {t} (\iota) ^ {2} \geq 1 - O (\xi)) \\ \geq 1 - O (\xi + \kappa) - \gamma_ {2} \quad \text {(B . c .} 1 / d \text {i s s m a l l b y A s s u m p t i o n 3 . 2)} \\ > 0 \quad (\text {B y}) \\ \end{array}
$$
and this contradicts the assumption that for all  $t\in [T_2,s]$ $D_{4,t}\geq 0$  and  $D_{2,t}\leq 0$  . Thus, for all  $t\in [T_2,s],1 - w_t(\iota_{\mathrm{M}})\geq \kappa^4\xi^3$  , i.e.  $w_{t}(\iota_{\mathrm{M}})\leq 1 - \kappa^{4}\xi^{3}$
The purpose of the next two lemmas is to show that if Phase 3, Case 2 occurs, then for all  $t \geq T_2$ , we will have  $D_{2,t} \leq 0$  and  $D_{4,t} \geq 0$ , assuming the conclusion of Lemma D.25 holds. We will later put these next two lemmas with the above lemma, Lemma D.25, in order to inductively show that the hypothesis of Lemma D.25 holds for  $t \geq T_2$  (note that Lemma D.25 assumes that the upper bound on the potential difference holds on an interval  $[T_2,s]$ , and more importantly assumes that  $D_{2,t} \leq 0$  and  $D_{4,t} \geq 0$  for all  $t \in [T_2,s]$ ).
Lemma D.26. Suppose we are in the setting of Theorem 3.3, and suppose  $D_{4,t} = 0$  and  $D_{2,t} < 0$  for some  $t \geq 0$ . Additionally, suppose that the conclusion of Lemma D.25 holds at time  $t$ , i.e., for all  $\iota \geq \iota_{\mathrm{L}}$ ,  $w_{t}(\iota) \geq \xi^{2}\kappa^{2}$ , and  $w_{t}(\iota_{\mathrm{M}}) \leq 1 - \kappa^{4}\xi^{3}$ . Then,  $\left.\frac{d}{ds} D_{4,s}\right|_{s=t} > 0$ .
Proof. Recall from Eq. (C.4) that
$$
P _ {4, d} (t) = \frac {d ^ {2} + 6 d + 8}{d ^ {2} - 1} t ^ {4} - \frac {6 d + 1 2}{d ^ {2} - 1} t ^ {2} + \frac {3}{d ^ {2} - 1} \tag {D.231}
$$
$$
P _ {4, d} ^ {\prime} (t) = 4 t ^ {3} \pm O \left(\frac {1}{d}\right) | t | \tag {D.232}
$$
Next we compute the derivative of  $D_{4,t}$ :
$$
\frac {d}{d t} D _ {4, t} = \frac {d}{d t} \underset {w \sim \rho_ {t}} {\mathbb {E}} [ P _ {4, d} (w) ] = \underset {w \sim \rho_ {t}} {\mathbb {E}} [ P _ {4, d} ^ {\prime} (w) \cdot v _ {t} (w) ] \tag {D.233}
$$
For positive particles  $w \gtrsim \frac{1}{\sqrt{d}}$ , we have  $P_{4,d}'(w) \geq 0$ , and for  $w \lesssim \frac{1}{\sqrt{d}}$ , it may be the case that  $P_{4,d}'(w) \leq 0$ . Furthermore, when  $D_{4,t} = 0$  and  $D_{2,t} \leq 0$ , we have
(By Lemma 4.4 and because  $D_{4,t} = 0$ )
$$
\begin{array}{l} v _ {t} (w) = - \left(1 - w ^ {2}\right) \left(2 \hat {\sigma} _ {2, d} ^ {2} D _ {2, t} w + O \left(\frac {\hat {\sigma} _ {2 , d} ^ {2} | D _ {2 , t} |}{d}\right) | w |\right) \\ = \left(1 - w ^ {2}\right) \cdot \left(2 \hat {\sigma} _ {2, d} ^ {2} | D _ {2, t} | w \pm O \left(\frac {\hat {\sigma} _ {2 , d} ^ {2} | D _ {2 , t} |}{d}\right) | w |\right) \quad (\mathrm {B . c .} D _ {2, t} \leq 0) \\ = \left(1 - w ^ {2}\right) \cdot 2 \hat {\sigma} _ {2, d} ^ {2} | D _ {2, t} | w \cdot \left(1 \pm O (1 / d)\right) \tag {D.234} \\ \end{array}
$$
Finally, to lower bound  $\frac{d}{dt} D_{4,t}$ , we must show that there is a significant fraction of particles for which  $w \gtrsim \frac{1}{\sqrt{d}}$  and  $1 - w$  is large. Because we have  $w_t(\iota) \geq \xi^2 \kappa^2$  for all  $\iota \geq \iota_{\mathrm{L}}$ , and  $\mathbb{P}(|\iota| < \iota_{\mathrm{L}}) \lesssim \kappa$  by Proposition D.6, using a union bound we therefore know that
$$
\mathbb {P} _ {w \sim \rho_ {t}} \left(\left| w \right| \in \left[ \xi^ {2} \kappa^ {2}, 1 - \kappa^ {4} \xi^ {3} \right]\right) \geq 1 - O \left(\gamma_ {4}\right) - O (\kappa) \geq 1 - O \left(\gamma_ {4}\right) \tag {D.235}
$$
where the second inequality is by Assumption 3.2 since  $d$  is sufficiently large. Thus,
$$
\frac {d}{d t} D _ {4, t} = \underset {w \sim \rho_ {t}} {\mathbb {E}} \left[ P _ {4, d} ^ {\prime} (w) \cdot v _ {t} (w) \right] \tag {D.236}
$$
$$
\begin{array}{l} = \mathbb {P} _ {w \sim \rho_ {t}} (| w | \lesssim 1 / \sqrt {d}) \underset {w \sim \rho_ {t}} {\mathbb {E}} [ P _ {4, d} ^ {\prime} (w) \cdot v _ {t} (w) \mid | w | \lesssim 1 / \sqrt {d} ] (D.237) \\ + \mathbb {P} _ {w \sim \rho_ {t}} (| w | \gtrsim 1 / \sqrt {d}) \underset {w \sim \rho_ {t}} {\mathbb {E}} \left[ P _ {4, d} ^ {\prime} (w) \cdot v _ {t} (w) \mid | w | \gtrsim 1 / \sqrt {d} \right] (D.238) \\ \end{array}
$$
$$
\geq - O \Big (\frac {1}{d ^ {3 / 2}} \cdot \frac {\hat {\sigma} _ {2 , d} ^ {2} | D _ {2 , t} |}{d ^ {1 / 2}} \Big) + \mathbb {P} _ {w \sim \rho_ {t}} (| w | \gtrsim 1 / \sqrt {d}) \underset {w \sim \rho_ {t}} {\mathbb {E}} [ P _ {4, d} ^ {\prime} (w) \cdot v _ {t} (w) | | w | \gtrsim 1 / \sqrt {d} ]
$$
(By bounding  $|P_{4,d}'(w) \cdot v_t(w)|$  when  $|w| \lesssim 1 / \sqrt{d}$ )
$$
\begin{array}{l} \geq - O \left(\frac {\hat {\sigma} _ {2 , d} ^ {2} | D _ {2 , t} |}{d ^ {2}}\right) + \mathbb {P} _ {w \sim \rho_ {t}} \left(| w | \in \left[ \xi^ {2} \kappa^ {2}, 1 - \kappa^ {4} \xi^ {3} \right]\right) (D.239) \\ \underset {w \sim \rho_ {t}} {\mathbb {E}} \left[ P _ {4, d} ^ {\prime} (w) \cdot v _ {t} (w) \mid | w | \in \left[ \xi^ {2} \kappa^ {2}, 1 - \kappa^ {4} \xi^ {3} \right] \right] (D.240) \\ \end{array}
$$
$$
\begin{array}{l} \geq - O \left(\frac {\hat {\sigma} _ {2 , d} ^ {2} | D _ {2 , t} |}{d ^ {2}}\right) + (1 - O (\gamma_ {4})) \underset {w \sim \rho_ {t}} {\mathbb {E}} \left[ P _ {4, d} ^ {\prime} (w) \cdot v _ {t} (w) \mid | w | \in [ \xi^ {2} \kappa^ {2}, 1 - \kappa^ {4} \xi^ {3} ] \right] \quad (\text {B y E q . (D . 2 3 5)}) \\ \gtrsim - O \left(\frac {\hat {\sigma} _ {2 , d} ^ {2} | D _ {2 , t} |}{d ^ {2}}\right) + (\xi^ {2} \kappa^ {2}) ^ {3} \cdot (1 - (1 - \kappa^ {4} \xi^ {3}) ^ {2}) \hat {\sigma} _ {2, d} ^ {2} | D _ {2, t} | (\xi^ {2} \kappa^ {2}) \tag {D.241} \\ \end{array}
$$
where the last inequality is by Eq. (D.232) and applying the above bound on  $v_{t}(w)$  from Eq. (D.234) for  $|w| \in [\xi^2\kappa^2, 1 - \kappa^4\xi^3]$ , and because  $d \geq c_3$  by Assumption 3.2. Expanding the last line gives
$$
\begin{array}{l} \frac {d}{d t} D _ {4, t} \gtrsim - O \left(\frac {\hat {\sigma} _ {2 , d} ^ {2} | D _ {2 , t} |}{d ^ {2}}\right) + \hat {\sigma} _ {2, d} ^ {2} | D _ {2, t} | \xi^ {6} \kappa^ {6} \kappa^ {4} \xi^ {3} \xi^ {2} \kappa^ {2} (D.242) \\ \gtrsim - O \left(\frac {\hat {\sigma} _ {2 , d} ^ {2} \left| D _ {2 , t} \right|}{d ^ {2}}\right) + \hat {\sigma} _ {2, d} ^ {2} \left| D _ {2, t} \right| \xi^ {1 1} \kappa^ {1 2} (D.243) \\ \end{array}
$$
and the right hand side is strictly positive since  $d$  is sufficiently large by Assumption 3.2.
Lemma D.27. Suppose we are in the setting of Theorem 3.3, and suppose  $D_{4,t} > 0$  and  $D_{2,t} = 0$  for some  $t \geq 0$ . Additionally, suppose that the conclusion of Lemma D.25 holds at time  $t$ , i.e., for all  $\iota \geq \iota_{\mathrm{L}}$ ,  $w_t(\iota) \geq \xi^2 \kappa^2$ , and  $w_t(\iota_{\mathrm{M}}) \leq 1 - \kappa^4 \xi^3$ . Then,  $\left.\frac{d}{ds} D_{2,s}\right|_{s=t} < 0$ .
$$
P _ {2, d} (t) = \frac {d}{d - 1} t ^ {2} - \frac {1}{d - 1} \tag {D.244}
$$
$$
\frac {d}{d t} D _ {2, t} = \frac {d}{d t} \underset {w \sim \rho_ {t}} {\mathbb {E}} [ P _ {2, d} (t) ] = \frac {2 d}{d - 1} \underset {w \sim \rho_ {t}} {\mathbb {E}} [ w \cdot v _ {t} (w) ] \tag {D.245}
$$
Suppose  $D_{2,t} = 0$  and  $D_{4,t} > 0$ . Then, the velocity is
$$
\begin{array}{l} v _ {t} (w) = - \left(1 - w ^ {2}\right) \left(P _ {t} (w) + Q _ {t} (w)\right) (By Lemma 4.4) \\ = - \left(1 - w ^ {2}\right) \left(4 \hat {\sigma} _ {4, d} ^ {2} D _ {4, t} w ^ {3} \pm O \left(\frac {\hat {\sigma} _ {4 , d} ^ {2} | D _ {4 , t} |}{d}\right) | w |\right) (By Lemma 4.4) \\ = - \left(1 - w ^ {2}\right) \cdot 4 \hat {\sigma} _ {4, d} ^ {2} D _ {4, t} \left(w ^ {3} \pm \frac {| w |}{d}\right) (D.246) \\ \end{array}
$$
Thus, for positive particles  $w$ , if  $w \gtrsim \frac{1}{\sqrt{d}}$ ,  $v_{t}(w) \leq 0$ , and for  $w \lesssim \frac{1}{\sqrt{d}}$ , the velocity is potentially positive. Using this, we can upper bound  $\frac{d}{dt} D_{2,t}$  by upper bounding  $\mathbb{E}_{w \sim \rho_t}[w \cdot v_t(w)]$ :
$$
\underset {w \sim \rho_ {t}} {\mathbb {E}} [ w \cdot v _ {t} (w) ] = \mathbb {P} _ {w \sim \rho_ {t}} \left(| w | \lesssim \frac {1}{\sqrt {d}}\right) \underset {w \sim \rho_ {t}} {\mathbb {E}} \left[ w \cdot v _ {t} (w) \mid | w | \lesssim \frac {1}{\sqrt {d}} \right] \tag {D.247}
$$
$$
+ \mathbb {P} _ {w \sim \rho_ {t}} \left(| w | \gtrsim \frac {1}{\sqrt {d}}\right) \underset {w \sim \rho_ {t}} {\mathbb {E}} \left[ w \cdot v _ {t} (w) \mid | w | \gtrsim \frac {1}{\sqrt {d}} \right] \tag {D.248}
$$
$$
\leq O \Big (\frac {1}{d ^ {1 / 2}} \cdot \frac {\hat {\sigma} _ {4 , d} ^ {2} | D _ {4 , t} |}{d ^ {3 / 2}} \Big) + \mathbb {P} _ {w \sim \rho_ {t}} \Big (| w | \gtrsim \frac {1}{\sqrt {d}} \Big) _ {w \sim \rho_ {t}} \mathbb {E} \left[ w \cdot v _ {t} (w) \mid | w | \gtrsim \frac {1}{\sqrt {d}} \right]
$$
(By bounding  $w \cdot v_t(w)$  for  $0 \leq w \lesssim \frac{1}{\sqrt{d}}$ )
$$
\leq O \left(\frac {\hat {\sigma} _ {4 , d} ^ {2} \left| D _ {4 , t} \right|}{d ^ {2}}\right) + \mathbb {P} _ {w \sim \rho_ {t}} \left(\left| w \right| \in \left[ \xi^ {2} \kappa^ {2}, 1 - \kappa^ {4} \xi^ {3} \right]\right) \tag {D.249}
$$
$$
\underset {w \sim \rho_ {t}} {\mathbb {E}} \left[ w \cdot v _ {t} (w) \mid | w | \in [ \xi^ {2} \kappa^ {2}, 1 - \kappa^ {4} \xi^ {3} ] \right]
$$
(B.c. for  $w\gtrsim \frac{1}{\sqrt{d}}$ $v_{t}(w) <   0.$ )
$$
\leq O \left(\frac {\hat {\sigma} _ {4 , d} ^ {2} | D _ {4 , t} |}{d ^ {2}}\right) + (1 - O (\kappa) - O (\gamma_ {4})) \underset {w \sim \rho_ {t}} {\mathbb {E}} \left[ w \cdot v _ {t} (w) \mid | w | \in [ \xi^ {2} \kappa^ {2}, 1 - \kappa^ {4} \xi^ {3} ] \right] \tag {D.250}
$$
where the last inequality is because  $\mathbb{P}(|\iota| > \iota_{\mathrm{M}}) \leq O(\gamma_4)$  (by the definition of  $\iota_{\mathrm{M}}$ , see statement of Lemma D.25) and  $\mathbb{P}(|\iota| < \iota_{\mathrm{L}}) \leq O(\kappa)$  by Proposition D.6. Simplifying further, we obtain,
$$
\underset {w \sim \rho_ {t}} {\mathbb {E}} \left[ w \cdot v _ {t} (w) \right] \leq O \Big (\frac {\hat {\sigma} _ {4 , d} ^ {2} | D _ {4 , t} |}{d ^ {2}} \Big) - (1 - O (\kappa) - O (\gamma_ {4})) (\xi^ {2} \kappa^ {2}) \cdot (1 - (1 - \kappa^ {4} \xi^ {3}) ^ {2}) \hat {\sigma} _ {4, d} ^ {2} D _ {4, t} (\xi^ {2} \kappa^ {2}) ^ {3}
$$
(By Eq. (D.246) and b.c.  $\frac{1}{d}$  is small (Assumption 3.2))
$$
\lesssim O \Big (\frac {\hat {\sigma} _ {4 , d} ^ {2} | D _ {4 , t} |}{d} \Big) - \hat {\sigma} _ {4, d} ^ {2} D _ {4, t} \cdot \xi^ {2} \kappa^ {2} \cdot \kappa^ {4} \xi^ {3} \xi^ {6} \kappa^ {6}
$$
(By Assumption 3.2,  $1 - O(\kappa) - O(\gamma_4)\gtrsim 1$
$$
\lesssim O \left(\frac {\hat {\sigma} _ {4 , d} ^ {2} | D _ {4 , t} |}{d}\right) - \hat {\sigma} _ {4, d} ^ {2} D _ {4, t} \xi^ {1 1} \kappa^ {1 2} \tag {D.251}
$$
and the last expression is strictly negative, since  $d \geq c_3$  by Assumption 3.2 and  $D_{4,t} \geq 0$ . This proves the lemma.