Improved maximal prime-gap lower bound

The paper proves $$G(X)\gg \frac{\log X\,(\log_2 X)^2\,\log_4 X}{(\log_3 X)^2}$$ for all sufficiently large $X$. Its main new ingredient is a short-translates theorem: for any sufficiently small set $S\subseteq[1,H]$ with $|S|\le\delta x$, one can find a short translate making every corresponding linear form composite. Combining this with an Erdős--Rankin covering argument produces prime-free intervals of the claimed length. This directly and asymptotically improves the August 2026 GPT-5.6 Sol bound $$G(X)\gg\frac{\log X\log_2 X}{\log_4 X}$$ by the unbounded factor $$\frac{\log_2 X(\log_4 X)^2}{(\log_3 X)^2}.$$

The paper proves $$G(X)\gg \frac{\log X\,(\log_2 X)^2\,\log_4 X}{(\log_3 X)^2}$$ for all sufficiently large $X$. Its main new ingredient is a short-translates theorem: for any sufficiently small set $S\subseteq[1,H]$ with $|S|\le\delta x$, one can find a short translate making every corresponding linear form composite. Combining this with an Erdős--Rankin covering argument produces prime-free intervals of the claimed length. This directly and asymptotically improves the August 2026 GPT-5.6 Sol bound $$G(X)\gg\frac{\log X\log_2 X}{\log_4 X}$$ by the unbounded factor $$\frac{\log_2 X(\log_4 X)^2}{(\log_3 X)^2}.$$