You might warm up to the "here's a tool, what results can we prove with it?" mentality a bit if you think those folks might just be starting at the end of a well-motivated (possibly conjectured) tool. If you think about it, it's not entirely different than "here's a deep theorem/conjecture, what theories can we develop around it?" Methodology is a theory about how-to's after all. I totally understand where you're coming from, though. Sorry for being off-topic!

@Brout Most likely he meant the second MRRW bound, which works only for $q = 2$ but is slightly better when $\delta$ is small; roughly speaking, it beats the first MRRW bound for $\delta < 0.272$ and the two bounds agree for $0.273 \leq \delta \leq 0.5$. I think popular textbooks that mention the first MRRW bound would at least talk about the second MRRW bound. See, for example, the following open access textbook: doi.org/10.1007/978-3-319-51103-0

Ah, there is a typo in my previous comment. But I take your reply as meaning that the undefined set $X$ is actually $X = \{1,\dots,n\}$ (i,e., $\pi = (X, L)$), am I correct? As for the latter half of my previous comment, you define projective by using $E$, which is only defined later on when you introduce the graph $G_{\pi}$. That's why I thought you might mean $e \in L$ instead $e \in E$.

Do you mean "$P(X) = X = \{0,1,\dots\}$"? Also, when you define "projective," do you mean "(unique) $e \in L$"?