Continuity mistake: When Mr. Scott asks Watkins to check the bypass valve, Watkins is facing the engineering boards with both hands on the panel. When we cut to a different angle, however, he's suddenly jumped to a position facing Scott with only one hand on the panel.
Plot hole: Scotty is receiving images of the launch, supposedly from an orbiting satellite. But many of the camera views are obviously from ground level, and couldn't possibly have been taken from orbit. It's not a rebroadcast from a ground camera, either. The implication is clearly that Scotty is is picking up a live satellite feed.
Continuity mistake: In Kirk's log entry, which he somehow makes from jail without a tricorder, he states that 5 witnesses heard him speak to the "spirits." This isn't true, and Kirk ought to know it. The crowd rushed in through an archway after he spoke to Spock and McCoy, and they were all too far away before to hear him.
Factual error: If Miri's planet is a "duplicate" (meaning identical) Earth, it should have clouds. It doesn't. This remains a mistake because "duplicate" means "exactly the same," and thus the clouds should be there. The special effects crew forgot to put them in. Noteworthy: the very first thing fixed in the digitally enhanced version of this episode was the duplicate Earth. It has clouds now.
Plot hole: Kirk knows that Lazarus is insane and that he wants the Enterprise dilithium crystals. Yet he's not restrained in sickbay and is, in fact, given free run of the ship so that he can knock out the crew in engineering and steal the crystals. Other than to further a woefully weak plotline, this makes no sense whatsoever.
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