It is evidently possible to check the powers of more and more complex
words to see whether there is a possibility of inversion; If 
admits inversion for some particular values of n, 
can be
tested, then 
, and so on, always noting the least common
multiple of the admissible values. Eventually enough combinations may
be examined to reach a decision: some word is impossible to resolve, or
enough words have been analyzed to know that there is a length of
neighborhood for which the ancestor is unconditionally unique, and the
inverse Rule can be read off according to the sequences of 
s
and 
s in which the unique values occur---e.g., if the middle
letter in each element of the symbolic matrix 
is 0,
for inverse evolutionary function 
.
Fortunately there is an upper bound to the lengths of words which must
be tested, making eventuality a certainty. Consider the matrix product
corresponding to a particular counterimage; given
that the pth letter is the same for all the words in the full
product, let 
be the pth factor. Decompose 
into a sum
in which 
contains the symbols which will
be lost, and 
those which will survive; 
must vanish leaving 
. There may
be symbols common to 
and 
, but 
must contain
one and only one of them, 0 say.
We are interested in showing that an excessively long 
or
can be shortened; such a demonstration would be akin to
shrinking a long path through the vector subset diagram by excising
loops, whereby a loop free path would be minimal. But that is not the
way to our proof, which is independent of the diagram. Rather, note
that our matrices are sparse, in their numerical version all matrix
elements are zeroes or ones, with exactly k ones per matrix. All
their possible products are postulated to have the same property; yet
there can only be just so many such matrices of a given dimension.
Some of them are original de Bruijn fragments, others exist only as
products, but each of them will have a product of shortest length
generating it. Let N be the maximum length of the shortest products
generating the multiplicative closure of the de Bruijn fragments. Now
replace both 
and 
by minimal equivalents through the
process of converting them to numerical form, finding the minimal
product, and forming the symbolic product all over again. This will
naturally shorten and alter any ancestors being represented; call the
new products 
and 
.
The important property of these new chains is
since the vanishing of the products depends on the placement of their
nonzero elements and not their value, even when it is symbolic. In turn
new position 
is guaranteed for which every matrix element
has the same symbol for its 
th letter, all within a neighborhood
of no greater length than 2N+1.
It may still happen that every ancestor has a position with a unique ancestor, but that the positions do not coincide. But then padding may be inserted to extend the neighborhood either to the right or to the left in such a way as to give them all a common centering. Should this be necessary one will simply have an inverse rule which is insensitive to the values of certain cells.
In practice none of this rearrangement has to be carried out; the possibility that it can be done simply shows that there is nothing in a very long or irregular neighborhood whose equivalent would not already have been encountered in a systematic sweep through short neighborhoods.