I swear there's a line of dialogue during the scene where the senor staff are discussing Gary Mitchell, that confirms Spock is the Science Officer. It goes "Pleasant or unpleasant, it's my science officer's job to see that I'm provided with that." Kirk then urges him to continue speaking despite Dr. Denher's outburst.
Kirk does refer to Spock as his Science Officer during the briefing room scene before the others leave. Spock still refers to himself as Second Officer in a log entry as late as "Dagger of the Mind."
This idea originated in Vonda McIntyre's Enterprise: The First Adventure. Mitchell's still on medical leave after saving Kirk's ass during their time on the Lydia Sutherland (Kirk's prior command where he got the medal for Ghioghe).
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