Once surjectivity has been established, there is no doubt that the uniform multiplicity of counterimages is precisely what would be expected from the size of the partial neighborhoods bounding a finite configuration. Ways exist through which the multiplicity may be rendered ineffective; one is to terminate the configuration with quiescent neighborhoods, another is to close them with a cyclic connection. Fixed boundaries and phase relations in the closure (e.g. the left boundary is the complement of the right boundary) are other alternatives which could be considered.

All the foregoing mechanisms involve selections from a naturally occurring set of boundaries, but there is also the possibility that the boundaries don't really matter. In other words, aside from a finite boundary layer which would be arbitrarily remote from the center of a very long configuration, the different ancestors could just happen to agree. Surely this is what is happening in shift rules or the identity map, and is clearly provided for in rules of Fredkin's type.

Boundaryless injectivity, or one might say, injectivity with a natural boundary, can be tested in one of the pair diagrams. Since the ordered pair diagram is coincidentally the second moment matrix, the condition of zero variance imposes a direct condition. Just as the empty set, implying a Garden of Eden, must be met in a limited number of steps (if at all) in the subset diagram, a pair of ancestors must either coalesce within the confines of the pair diagram or encounter some means of coexisting.

Apparently the decision to use a boundary condition should be postponed as long as possible; the eventual choice may be that much simpler.

Harold V. McIntosh