Thanks for taking the time to clarify, Gareth. 4. Whatever that reduction is, we can imagine doing it
one square (and one unit of winding for that square) at a time.
This is not obvious to me, but I can prove it. (You have to decrement the larger numbers first.)
When we reduce one PSWN by 1, the number of path components cannot reduce by more than one -- for we cannot have components of more than two disconnected paths adjacent to a single square.
This lemma (if I'm understanding it correctly) cannot hold in full generality. Consider the PSWN array 010 101 010 (four separate loops) and the PSWN array 010 111 010 (a single loop); when we turn the former into the latter, a single PSWN changes by 1, but the number of components changes by 3. I'll mention Allan Wechsler here, for no reason other than the fact that I seem to keep misspelling his first name and could use some practice spelling it right. Allan, Allan, Allan. There! Let's see if that helps. :-) Jim