Non-regular $\mathrm{\mid }2\mathrm{\mid }$-graded geometries I: general theory

This paper analyses non-regular $\mathrm{\mid }2\mathrm{\mid }$-graded geometries, and show that they share many of the properties of regular geometries – the existence of a unique normal Cartan connection encoding the structure, the harmonic curvature as obstruction to flatness of the geometry, the existence of the first two BGG splitting operators and of (in most cases) invariant prolongations for the standard Tractor bundle $\text{mc}T$. Finally, it investigates whether these geometries are determined entirely by the distribution $H=T_{-1}$ and concludes that this is generically the case, up to a finite choice, whenever $H^1(\text{mf}{g}^1,\text{mf}g)$ vanishes in non-negative homogeneity.