A tilted residue-class construction for long prime-free intervals

How large can the gap between consecutive primes be, infinitely often? Writing $\log_k$ for the $k$-fold iterated logarithm, Erdős asked (problem #4, a $10,000 prize) whether $p_{n+1}-p_n \gg C\log n\log_2 n\log_4 n/(\log_3 n)^2$ for every $C$; that was settled in 2016, and the record bound since has been Ford-Green-Konyagin-Maynard-Tao's $p_{n+1}-p_n \gg \log n\log_2 n\log_4 n/\log_3 n$. This work claims a stronger bound, $G(T)\gg \log T\log_2 T/\log_4 T$, an improvement by a factor of $\log_3 T/(\log_4 T)^2$, together with $Y(X)\gg X\log X/\log_3 X$ for the covering problem behind it.

Claims $G(T)\gg\log T\log_2 T/\log_4 T$ against FGKMT's $\log T\log_2 T\log_4 T/\log_3 T$, a gain of $\log_3 T/(\log_4 T)^2$, together with $Y(X)\gg X\log X/\log_3 X$ for the covering problem behind it. Ben Green resists calling it incremental: "The sieving procedure is different to the Erdos-Rankin one which underpinned all bounds on the problem since 1938, and it wins $\log_3 N$ over that procedure. The [FGKMT] paper also wins a $\log_3 N$. These wins are essentially independent of one another so one now wins basically $(\log_3 N)^2$ over Rankin's 1938 bound." He adds that the new sieve "by itself could have claimed the Erdos 10000 dollars for this question". What is new is narrow: only the intermediate sieve is replaced, its hard cutoff smoothed into a probabilistic tilt; the hypergraph covering theorem and Maynard weight come from FGKMT. Readers on the thread note that several later sections reproduce FGKMT with no new content, and Green calls the exposition "horrific".