Proposition: Ball sections are compatible with confluence

There is an admissible record network in two spatial dimensions whose one-step spatial reachability section is exactly a closed Euclidean unit ball for the metric restricted to a lattice.

Proof. Set

and put

Time strictly increases along edges. The indegree is at most five and the outdegree is five. Each principal past is finite, since there are finitely many step sequences of each possible length. Theorem thm:network applies.

On the next slice, the reachable sites are

The one-step section is the discrete Euclidean unit ball; Minkowski iteration produces the discrete diamond $\Vert z{\Vert }_1\le n$, as proved below and illustrated in Figure fig:stencil. Away from the initial layer every event has the same input and output arities.