The valence of a vertex is the number of edges connected to that vertex. In Arnold, the maximum valence is 255 (that’s because of the data type we use to store the valence; we minimize the memory requirements since this is stored per-vertex).
If a mesh in the scene has a vertex with more than 255 edges, you’ll get a WARNING like this:
[polymesh] example_mesh: mesh has at least a vertex with valence higher than 255, disabling subdivision
But if the adaptive subdivision results in a vertex with too many edges, you’ll get an ERROR:
ERROR: [arnold] [subdiv] example_mesh: edge (144578,287741) in face 263433 has a vertex that exceeded the max valence limit of 255