I'm trying to understand why the Maynard-Tao weights allow one to obtain bounded gaps between primes whereas GPY fail
The usual response is the additional flexibility of allowing the sieve weights to depend on the individual divisors of the translates, rather than the product. But I'm looking for a deeper understanding.
My attempt
Taken the following from: http://www.ams.org/journals/bull/2007-44-01/S0273-0979-06-01142-6/S0273-0979-06-01142-6.pdf
In the GPY case we aim to find a non-negative function $f(n)$ such that $$\sum_{\substack{x \leq n \leq 2x \\ n + h_{j} \text{prime}}}f(n) > \frac{1}{k}\sum_{x \leq n \leq 2x} f(n)$$ where $j$ runs from $1\dots k.$
It turns out that we cannot have this ratio to be greater than $\frac{1}{k}$ and the critical value is $4.$
In the Maynard case, the weights that are used are such that the ratio is about $\frac{\log k}{k}$ and allows the result to be proven.
Therefore in the Maynard-Tao weights a probability density is found such that $\mathbb{P}(n + h_{j}) \asymp \frac{\log k}{k}.$
It's still not clear to me how using the proposed weights by Maynard allows one to obtain this new probability density, and therefore the result.