Update: this figure shows the numbering of vertices(red), edges(green), and faces(blue). The arrows show the direction of the X-(red), Y-(green), and Z-axes(blue).
Here the sides of the cube are not generated with Marching-Cubes, they are just extracted directly from the octree. Nodes are subdivided whenever the signed distance-field of the cutter indicates that the surface is contained within the node, i.e. the distance-field evaluates to both positive and negative at the eight
fourvertices of a node. This apparently leads to transient holes in the surface when the cutter is just about to enter a coarse node which hasn't been subdivided very far yet. It should be possible to adjust the subdivision criterion so that the octree 'anticipates' the cutter slightly and subdivides ahead of the actual cutter surface.