Dense binary $PG(t-1,2)$-free matroids have critical number $t-1$ or $t$

The critical threshold of a (simple binary) matroid $N$ is the infimum over all $\rho$ such that any $N$-free matroid $M$ with $|M|>\rho2^{r(M)}$ has bounded critical number. In this paper, we resolve two conjectures of Geelen and Nelson, showing that the critical threshold of the projective geometry $PG(t-1,2)$ is $1-3\cdot2^{-t}$. We do so by proving the following stronger statement: if $M$ is $PG(t-1,2)$-free with $|M|>(1-3\cdot2^{-t})2^{r(M)}$, then the critical number of $M$ is $t-1$ or $t$. Together with earlier results of Geelen and Nelson [GN14] and Govaerts and Storme [GS06], this completes the classification of dense $PG(t-1,2)$-free matroids.