Finite-sample Borel--Cantelli inequalities under mixing conditions

We prove explicit finite-$N$ lower bounds for $\mathbb P(\bigcup_{k=1}^N A_k)$ when the $\sigma$-algebras generated by an event sequence satisfy quantitative $\varphi$- or $\alpha$-mixing bounds. The main $\varphi$-mixing estimate is obtained by a residue-class blocking argument and a one-sided approximate-independence inequality; it has a free spacing parameter $L\ge0$, spacing coefficient $1/(L+1)$, and residual terms governed by $\varphi(L+1)$. For $\alpha$-mixing families, we derive an additive-correction analogue using strong-mixing covariance control. A windowed rate corollary and a second-order Bonferroni refinement parallel the corresponding $m$-dependent finite-sample results. The coefficient $1/(L+1)$ is sharp as a universal spacing constant only in the zero-residual sense: the full mixing classes contain $L$-dependent block constructions with $\varphi(L+1)=0$ and $\alpha(L+1)=0$ that asymptotically attain the corresponding bound. This sharpness statement does not assert optimality of the residual penalties.