In response to a note of mine,
A search treegiving distances from Start calculates d(I,IY) for all
positions IY in the domain of inquiry. With an X-rooted tree,
distances are of the form d(X,XZ) for all positions XZ in the domain
of inquiry. In general, it is not the case that d(I,IY)=d(X,XY).
whereupon what's-his-name :-) responds
In this notation, it is certainly true that
d(<id>,<h>) = d(<g>,<g><h>). This is because each process that
transforms <id> to the state <h>, will also transform <g> to <g><h>,
and likewise each process that transforms <g> to <g><h> will also
transform <id> to <h>.
This is what I was trying to say in the message that started this: that
one is building a tree of all move sequences no longer than N, which is
to say a certain subset of permutations of the cube. But these
permutations can be applied to arbitrary positions just as well as as
they can be to Start. Any Cubist knows this; it's the basis for many
of the common solving macros: that a process that (say) swaps RF and
RB, and TF and TB, can be used to swap whatever cubies happen to be in
those cubicles, even if they aren't the RF/RB/TF/TB cubies.