I will systematically prove that all Star Trek series are in the same universe. That universe being the Universe of TOS (disregarding trips to parallel universe or the occasional time travel adventure that has always been shown to be reversed):
First, let's define a Star Trek series. There are seven main series: TOS, TAS, TNG, DS9, Voyager, Enterprise, and Discovery. There's also Short Treks, but those are unrelated shorts I will consider on par with the films.
So, I will ask a question and elaborate for each of the sequels to TOS.
2. Star Trek: The Animated Series (TAS)
Is this a continuation of TOS? Yes. Same characters, references to events of TOS. Easiest of the bunch.
3. Star Trek: The Next Generation (TNG)
Is this a continuation of TOS? Yes. Spock shows up, referencing TOS. So does Sarek. Scotty shows up and sits on the bridge of the original Enterprise. Same universe, no doubt.
4. Star Trek: Deep Space Nine (DS9)
Is this a continuation of TOS? Yes. They time travel into a TOS episode, and comment on it being a real historical event.
5. Star Trek: Voyager (Voyager)
Is this a continuation of TOS? Yes. Tuvok flashbacks to hanging out with Sulu. They talk about Kirk and crew and the historical five-year mission.
6. Enterprise
Is this a continuation of TOS? Kinda, as it's a prequel. But it's a continuation of Star Trek: First Contact, a TNG movie, and we see the USS Defiant from TOS show up (albeit in an alternate universe) with information on Archer and crew as historical figures in the TOS era.
7. Star Trek: Discovery (Discovery)
Is this a continuation of TOS? Well, it's also a prequel, except in one respect: it features a direct sequel episode to The Cage (TOS's unsold pilot), itself canonized by its inclusion in The Menagerie. So, Yes, same universe. No doubt. Err, uh, not much doubt.
All the movies feature characters from the series (TOS and TNG, intricately tied together by Relics and Generations). The Kelvin universe films is an alternate timeline caused by a Spock who is from the TOS films era per a photograph in Star Trek Beyond (and therefore, via Relics and Generations, the original TOS).
The Short Treks are all outgrowths of Discovery, or Discovery's interpretation of the Pike-era Enterprise, and are therefore in-continuity with each other. We can argue about Harry Mudd or Calypso, but it's getting a little pedantic at that point.
You can say something is a 'continuation' but there is a lot of material that contradicts it being a continuation. Intent vs. what shows up on screen are two distinct things. Also, since someone like Archer exists in one universe, it doesn't preclude him from existing up in multiple universes.
For me, I'd say their little sidetrip to "The Cage" did more to damage their case about it being one universe than helped it.
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