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Statement

Theorem 1.2 (p. 2). Let $m\in\mathbb N$, let $r\in\mathbb N$ be sufficiently large depending on $m$, and let $\mathcal A=\{a_1,a_2,\dots,a_r\}$ be a set of $r$ distinct integers. Then

$$\frac{\#\{\{h_1,\dots,h_m\}\subseteq\mathcal A:\ n+h_1,\dots,n+h_m\text{ are all prime for infinitely many }n\}}{\#\{\{h_1,\dots,h_m\}\subseteq\mathcal A\}}\gg_m1 .$$

The paper reads this (p. 2) as saying that a positive proportion of admissible $m$-tuples satisfy the prime $m$-tuples conjecture for every $m$, "in an appropriate sense". The conjecture, as the paper states it (p. 1): for an admissible set $\{h_1,\dots,h_k\}$ of distinct non-negative integers, one that misses some residue class modulo every prime $p$, there are infinitely many $n$ with all of $n+h_1,\dots,n+h_k$ prime.

Source. J. Maynard, Small gaps between primes, Ann. of Math. (2) 181 (2015), no. 1, 383--413, doi:10.4007/annals.2015.181.1.7, read in the arXiv:1311.4600v3 preprint (28 October 2019) identified on the source card; the pages cited are the preprint's printed pages, not the journal's. Theorem 1.2 on p. 2, the proof on p. 7.

Read depth. Claims checked: the statement and the counting proof on p. 7 were read clause by clause; the result rests on the large-$k$ step whose depth is recorded on Theorem 1.1. Nothing here is independently reviewed.

Proof pointer

P. 7. With $k=\lceil Cm^2e^{4m}\rceil$ as in the proof of Theorem 1.1, every admissible $k$-set contains an $m$-subset all of whose translates are prime infinitely often. Deleting, for each prime $p\le k$ in turn, the sparsest residue class modulo $p$ leaves a subset of $\mathcal A$ of size $\gg_m r$ whose $k$-subsets are all admissible; double counting the pairs of a $k$-subset and a good $m$-subset inside it gives $\gg_m r^m$ good $m$-sets.

Dependencies

Proposition 4.2 with the bound (4.5) from Proposition 4.3 (3), as in the proof of Theorem 1.1.