I read a previous question, Is a secondary TM sufficient to detect all loops?, which asks if there is a TM1 that takes the description and input of any other TM2 to detect if TM2 will loop for the input.
One answer was to run two copies of TM2, the second at 1/2 the speed.If the second machine ever reaches the same configuration as the first, a loop was detected. Seems to me that the process requires a third entity to do the comparison at every step. Since neither copy output their complete configuration, it is not obvious this third entity and the complete system of all three entity can be a TM. Another easy way is for a second TM2 to keep a list of all configuration of TM1 and check to if there any repeats. This can surely work. But again, what is not obvious is if this complete system of the checking process on TM2, the running of TM1 and outputting TM1's complete state configurations to TM2 can be realized as a TM. TM1 can't output its own states to TM2, so TM2 somehow has to grab these from TM1, not just its output but TM1's complete state information or there has to be a third entity that does the communication.
My question: Can this process be implemented as a Turing machine?