I'm attempting to understand why the Maynard-Tao weights are successful in proving bounded gaps between primes, but the GPY weights are not.

These two posts do an excellent job in giving an overview of how one arrives at the Maynard-Tao weights and the role of the weights themselves:

- What is the crucial difference the Maynard/Tao approach and Goldston-Pintz-Yildirim that extends to prime k-tuples with $k>2$
- Why do the Maynard-Tao weights work so well?

The additional flexibility in allowing the $\lambda_{d_{1},\dots,d_{k}}$ to depend on the divisors $d$ individually allows the method to be succeed. If possible I'd like a deeper understanding why the additional flexibility allows it to succeed.

Soundararajan's expository paper: https://arxiv.org/pdf/math/0605696.pdf (pages 9 - 14) explain the core ideas of the GPY method and that the ratio of $$\dfrac{\Big(\sum_{\substack{x \leq n \leq 2x} \\ n + h_{j} \text{prime}}a(n)\Big)}{\Big(\sum_{x \leq n \leq 2x}a(n)\Big)}$$ cannot be made greater than $\frac{1}{k}$ and we therefore fail to just prove bounded gaps between primes.

Therefore the GPY method finds a probability distribution such that Probability$(n+h_{i} \text{is prime}) \asymp \frac{1}{k}$ and this differs from the optimal value by a factor of about $k.$

The Maynard-Tao weights clearly give a better weighting so this issue is overcome and find a different probability distribution to enable the inequality to hold but I'm a bit confused. **Why is it that because the $\lambda_{d_{1},\dots,d_{k}}$ to depend on the divisors $d$ individually, this issue is overcome?**

**What is the probability distribution that the Maynard-Tao weights find such that the inequality holds?**

Apologies if this isn't the appropriate place to ask this question. I'm an Undergraduate interested in analytic number theory...