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I just don't see how a Stargate could affect a star at all. Stargates are TINY. I guess the sun is super-high-pressure on the inside, but it's still like trying to drain the ocean through a hole 14 centimeters across. (if I did the math right)

Well, the other side did have a black hole sucking away at the star, if I remember my Stargate trivia correctly. If that still wouldn't drain the mass suddenly enough, maybe the insane gravity and time dilation caused the star to destabilize?
 
I concede that point. The Stefan–Boltzmann law shows it would take a hundred thousand years for our sun to cool by 1% if fusion were to suddenly stop. I'm still not seeing how sudden mass loss would cause a super nova though. What we're really talking about I guess, is having a star that has more explosive outward force than inwards gravitational force. Which is what happens the moment a star starts fusing a higher element. When our sun starts fusing helium, it'll rapidly expand and become a red giant, ultimately shedding one third of it's mass as the outer layers reach escape velocity. Isn't that what would happen in the case of draining mass with a star gate? What am I missing that could trigger a super nova?
technically it wouldn't be a supernova, it would simply fly apart but the destructive potential would be enough for any practical purposes.
 
Nitpicking here, why would the ringworld consume all planets in the system but the Dyson sphere wouldn't? Based on my (likely off) calculations, the surface area for a ringworld one AU in radius and 6 million km in width (which is pretty wide for a ringworld, I think Niven's was around 1 million km) would be about the same as Dyson sphere practically on the surface of the sun. In reality the sphere would probably have to be some distance away, making it much larger.