Observe that it is possible to extend an arrangement to one that is `quadrilateralating away from infinity' by superimposing a sufficiently large square mesh (two perpendicular families of parallel lines), where the squares are sufficiently small that all lines that intersect the square pass through a single point.

Oops, there are non-Abelian groups with Abelian automorphism groups: mathoverflow.net/questions/9944/when-is-autg-abelian That kills my entire argument.

Vaqhpgvba ba gur ahzore bs ebcrf: Jvgu n ebcr wbvavat cbvagf N naq O, lbh erzbir gur ebcr, phg gur ynqqref ng N naq O naq vagrepunatr gur gbcf, erfhygvat va na rdhvinyrag fvghngvba jvgu bar srjre ebcr.

I don't think that the denominators increase sufficiently rapidly for the usual Liouville-esque proof to apply, without assuming Lenstra-Pomerance-Wagstaff. What argument for irrationality are you suggesting?

Oh, yes, I meant Mersenne primes. Thanks. I'll edit the comment accordingly.