Maximally symmetric $({23}_4)$ configurations and the end of the existence problem for geometric $(n_4)$ configurations

A geometric $(n_4)$ configuration is a set of $n$ points and $n$ lines in the real projective plane such that every point lies on exactly four of the lines and every line contains exactly four of the points. The question of which $n$ admit such a configuration was posed by Grünbaum in 2000, following his 1990 paper with Rigby that gave the first geometric $4$-configuration, a $({21}_4)$. Successive work of Bokowski-Schewe, of Bokowski, Grünbaum and Schewe, of Bokowski-Pilaud, and of Cuntz settled every case but one. Cuntz’s 2018 paper, which constructed $({22}_4)$ and $({26}_4)$ configurations, states that only $n=23$ remained open. We settle this last case. We prove that for every prime $n\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$, in particular for $n=23$, the group of projective symmetries of a geometric $(n_4)$ configuration has order at most four, which explains why $n=23$ resisted the symmetric constructions that settled the other values. Reducing the case of a Klein four-group of symmetries to a few thousand bilinear polynomial systems in $18$ unknowns, and solving all of them exactly with Gröbner bases and saturation, we show that exactly two geometric $({23}_4)$ configurations reach this bound, up to projective equivalence, and that there are no others even over $\mathbb{C}$. Both are self-polar with respect to a conic. The first has integer homogeneous coordinates; the second is defined over $\mathbb{Q}(\sqrt{17})$, but not over $\mathbb{Q}$. Consequently geometric $(n_4)$ configurations exist if and only if $n\ge 18$ and $n\mathrm{\ne }19$. A third configuration, self-polar and defined over a real cubic field, has only a single symmetry of order two. The same classification at $n=22$ and $n=26$ recovers Cuntz’s configurations and finds three new combinatorial types at $n=22$ and $94$ at $n=26$, among them a $({26}_4)$ configuration that is self-polar with respect to a conic without real points.