Worked solution: Maynard's multidimensional sieve: bounded gaps between primes (2013)

For a fixed level of distribution $\theta>0$ and small $\delta>0$, Maynard defines $M_k=\sup_{F\in\mathcal{S}_k}\big(\sum_{m=1}^k J_k^{(m)}(F)\big)/I_k(F)$, the supremum taken over the class $\mathcal{S}_k$ of suitable smooth functions $F$ (Maynard 2013, §4, Proposition 4.2). Setting $\rho=\theta M_k/2-\epsilon$ in the counting sum $S=S_2-\rho S_1$ from Step 2, a short computation with the asymptotics from Step 4 shows $S>0$ for all large $N$, so — following the logic of Step 2 — infinitely many $n$ have at least $r_k=\lceil \theta M_k/2\rceil$ of $n+h_1,\dots,n+h_k$ prime.

This is Maynard's Proposition 4.2, and it is the crux of the whole method: bounded prime gaps of size $r_k$ inside an admissible $\mathcal{H}$ of length $k$ follow automatically once one exhibits, for that $k$, a smooth $F$ making the ratio $\big(\sum_m J_k^{(m)}(F)\big)/I_k(F)$ large enough. In particular $r_k\ge2$ (two or more primes among the $k$ shifts) already gives a bound on $\liminf_n(p_{n+1}-p_n)$, namely the diameter of any admissible $k$-tuple realizing it.

So the entire analytic-number-theory problem has been converted into: find, for a suitable $k$, a good enough $F$. The next step supplies exactly that, via explicit test functions and numerical optimization.