Slack entropy thresholds preserve the quantitative synchronization bound

Put $\delta =(\gamma -1-\epsilon )\mathrm{/}2>0$ and $\beta =1+\epsilon +\delta =(\gamma +1+\epsilon )\mathrm{/}2<\gamma$. By [F1], the $a,b$ in the Statement exist uniquely on the specified branches and satisfy $\gamma I(a)=\gamma I(b)=\beta$. For $X\sim \mathrm{Bin}(n-1,p)$, write $f_j=\mathbb{P}(X=j)$. At $k_{-}=\lfloor a\mu \rfloor$ and $k_{+}=\lceil b\mu \rceil$, [F2] gives $f_{k_{\pm }}=O(n^{-\beta }\mathrm{/}\sqrt{\log n})$. For $1\le j\le k_{-}$ the ratio $f_{j-1}\mathrm{/}{f}_j=j(1-p)\mathrm{/}((n-j)p)$ is at most a fixed ${\rho }_{-}<1$ eventually, since its maximum tends to $a<1$. For $k_{+}\le j<n-1$, $f_{j+1}\mathrm{/}{f}_j=(n-1-j)p\mathrm{/}((j+1)(1-p))$ is at most a fixed ${\rho }_{+}<1$, since its maximum tends to $1\mathrm{/}b<1$. Summing the two geometric bounds gives $\mathbb{P}(X\le k_{-})+\mathbb{P}(X\ge k_{+})=O(n^{-\beta }\mathrm{/}\sqrt{\log n})$.

For any finite family of events, its union indicator is at most the sum of their indicators. Thus [F2] and step 1.1 give $\mathbb{P}(d_{\min }\le k_{-}\text{ or }{d}_{\max }\ge k_{+})=O(n^{1-\beta }\mathrm{/}\sqrt{\log n})=O(n^{-\epsilon })$. On its complement $d_{\min }>a\mu$ and $d_{\max }<b\mu$, with no integer rounding loss. By [F3] the adjacency norm is below $\alpha \mu$ except with probability $2n^{-\gamma }=O(n^{-\epsilon })$. No independence between these three exceptional events is required.

Set ${\tilde{c}}^{-}=a\mathrm{/}2-1>-1$, ${\tilde{c}}^{+}=b$. By [F6], eventually $0<\alpha \le \min \{a\mathrm{/}2,1\mathrm{/}5\}$. On the good event from step 2.1, $(1+{\tilde{c}}^{-}+\alpha )\mu \le a\mu <d_{\min }$ and $d_{\max }<b\mu \le (1+{\tilde{c}}^{+}-\alpha )\mu$. Hence [F4] gives the stated expander. Its reference degree $\mu$ is positive. The coefficient $32(1+4{\tilde{c}}^{+}-3{\tilde{c}}^{-})\mathrm{/}(1+{\tilde{c}}^{-}{)}^2$ is a fixed positive constant. For $0<\alpha \le 1$, $\log ((1+b+\alpha )\mathrm{/}(2\alpha ))\le \log ((b+2)\mathrm{/}2)+\log (1\mathrm{/}\alpha )$; [F6] implies $\alpha \log (1\mathrm{/}\alpha )\to 0$. Thus the strict numerical hypothesis in [F5] holds eventually.

Increase the cutoff to include all supplier cutoffs and the inequalities of step 3.1. Then [F5] gives global synchrony on the same good event. Step 2.1 bounds its complement by $C{n}^{-\epsilon }$ for a constant depending only on the fixed parameters. This proves the complete claimed probability bound, and hence also the weaker $C(\log n{)}^4{n}^{-\epsilon }$ bound for large $n$. All input estimates are independent of the refuted shifted-root proposition.