Manuscript En05-B@1 · En05-B.tex · lemma · 8b626460
Lemma: Metric substrate
The substrate: increasing directions and separated ports › Hubs, lanes, and junction cliques · lines 464–481 · proof included or linked
En05-B@1#lem:substrate
The graph is countable and connected. Its degree is finite and bounded at each fixed , and its metric balls are finite. Every junction clique is finite and has diameter one.
Proof. There are countably many labels. An interior lane site has degree two. A junction has sites, so its sites have at most neighbors, including a possible lane neighbor. The presence of connects all hubs, and every lane site connects to its endpoint hubs.
A bounded-degree graph has finite balls by the finite branching bound on walks of any fixed length. The clique assertion follows from its definition.