Introduction and scope

An observation boundary specifies which state data a controller may use. It does not specify which geometric outputs are admissible. A geometric coupling therefore requires two independent pieces: a declared accessible input and a complete admissibility certificate for its output.

This paper supplies a finite matching contract for one declared spinless conical model. The notation ${\mathsf{Ein}}_0$, retained from the VIII programme, denotes that contract. It is not a classification of arbitrary Einstein initial data and is not an unstated continuum Einstein equation.

The classical geometric motivation is that source-free $2+1$-dimensional Einstein geometry with zero cosmological constant is locally flat DJH1984. Polygonal and holonomy descriptions have a first-order-gravity interpretation Kadar2005. These references motivate the dictionary. The dictionary itself is stated explicitly below, and no unquoted constraint-algebra theorem is used.

Our polygonal descriptions need not belong to a particular trivalent canonical polygon gauge. Nonconvex faces and regular vertices of valence four are allowed. We prove matching statements for the interface actually defined here.