It's hard to summarize this in the title, but my question is:
For all positive integral $N \geq M$ for some undetermined constant $M$, are there always at least four distinct naturals $k_1,..., k_4<N$ such that all four of $N+k_1, ..., N+k_4$ are prime?
I wanted to prove this by proving that there are at least 4 primes in the set $[N,2N-1]$, but the few established results I could find about prime gaps didn't seem to help much. I expect I'm missing something relatively basic given that it seems a very easy to do for any particular instance in practice, I just can't prove the general case
Additionally, this is part of a larger problem I'm trying to solve where the value of $M$ becomes relevant, so it'd be helpful to know what if any condition the result implies about it, but it's not necessary to answer. For the purposes of this question, it can be considered as large as needed.
With help from the comments pointing me in the right direction, I think I have found that the answer to my question is given by the Ramanujan primes. Since the sequence $R_n$ is defined as the least prime for which $\pi(x)-\pi(\frac{x}{2}) \ge n$ for all $x>R_n$, then the value $M$ is $\lceil \frac{R_4}{2}\rceil=15$
(Unfortunately, I discovered the theorem I was invoking to use this fact didn't say what I thought it did and this doesn't actually help me, oops)