Some consequences of zero variance are easily described. First and
foremost, no path beginning in the subset 
can ever return,
because that would violate the reasoning leading up to Eq. 27.
In other words, if two ancestors of a configuration initially coincide
--- even for a single partial neighborhood --- and later diverge, they
can never recombine. Seen from another point of view, no product of the
matrices 
can ever have a matrix element as large as 2 while
the rule has zero variance.
A similar prohibition applies to a pair of isomorphic ancestors; it is
only necessary to replace 
by the set 
.
It is also necessary that the matrix S corresponding to the subset
not have an eigenvalue exceeding that of a de Bruijn matrix,
which can be checked in any particular case, but not as easily when it
is a general proposition. Essentially, the requirement is that mixed
ancestors not proliferate more rapidly than single ancestors as the
length of a configuration increases, which would clearly be possible if
sections of an ancestor could be freely substituted for one another.