You’re right, after rereading the proofs and talking with Evan, the only way for H to be polynomial time is to get oracle access to M. Which is slightly unintuitive, but makes sense because the part of the computation that depends on H and not on M is indeed polynomial time.
You’re right, after rereading the proofs and talking with Evan, the only way for H to be polynomial time is to get oracle access to M. Which is slightly unintuitive, but makes sense because the part of the computation that depends on H and not on M is indeed polynomial time.