is $P_{CTC} = BPP_{path}$?
I believe I have found the answer: The proof is wrong. BPP_PATH is not in P_CTC because P_CTC is required to give a single definite answer, while BPP_PATH is a probabilistic algorithm, so the second reduction does not work. For it to work, one would have to use the time-travel information to count the number of successes of the probabilistic algorithm vs the number of its failures. I have no idea how to do this, or if it even can be done (probably not).