why is the NL solution PATH negated unsufficient to prove unreachability
maybe i was a bit to specific, i had a general misunderstanding in how non-deterministic machines work. Here is what i figured:

an accepting nondeterministic turing machine always gives the right answer ("found something, therefore a solution exists",
on the other hand, a reject means "did not find anything" which is not the same as "there is no solution"

therefore a reject can't be trusted and any algorithm which uses a reject from a non deterministic (Existance) machine has to double check. Also this implies, that a non-deterministic machine could give a false negative.