In the episode, there's this great big problem about letting the agent interfere and whether that will impact Earth's history or not, and Kirk and Spock are in the dilemma of not knowing whether to let the agent proceed with his work or not. In the end, everything works out the way it was supposed to according to their records - or so they say. When did Kirk and Spock actually get the chance to check their records before the detonation of the missile? My understanding was that they didn't check their historical records before they entered the building the final time and I couldn't see any opportunity for them to check the records again. Yet, the records show what happened was 'supposed' to happen. But if they checked after the detonation, the records would have shown that the missile was supposed to detonate at 104 miles, not what was 'supposed' to happen. Hence, my question is do we know if Kirk and Spock checked the historical records before the detonation?
By the time Spock told Gary Seven about the records some time had passed. Kirk and Spock had returned to Enterprise; we can tell because they are out of the 1960's clothes and back into their uniforms. So they must have checked the record tapes after detonation. This is a classic pre-destination paradox where the time travellers got just the information they needed to cause events not to diverge from what was known to have occurred.