The proof of Lemma 3.4 uses the constant $K=[\epsilon^{-3}]$ which should be your clue. After the definition of $K$ the author covers $(8^t,8^{t+\epsilon})$ (which contains at least $\epsilon 8^n/(2n)$ primes by the choice of $t$ as per previous paragraph and Lemma $3.1$) with $K$ intervals each having at most $O(\epsilon^2 8^n/(n))$ primes by Lemma $3.3$ (see the at most vs at least above)
In particular, if only one (or none) of those intervals contains at least $O(\epsilon^4 8^n/(n))$ primes (so the rest contain at most $o(\epsilon^4 8^n/(n))$ primes) and since there $K \approx \epsilon^{-3}$ such intervals, the total number of primes in $(8^t,8^{t+\epsilon})$ can be at most $$O(\epsilon^2 8^n/n)+\epsilon^{-3}o(\epsilon^4 8^n/n)=o(\epsilon 8^n/(2n))$$ and that is a contradiction with the above. So the claim follows - and actually one gets that at least $\epsilon^{-1}=K^{1/3}$ such nonconsecutive $a,b..$ exist