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If Jupiter starts going significantly downwards by the time when it'll be just a fem million years until Sun goes supernova, we (whoever that will be) will have to hurry and flee anyway. It it involves crashing Jupiter into the Sun, and likely wreaking complete havoc in the inner planets' orbits, so be it; the system will become unusable anyway.

But until starfaring is developed enough, we should be careful with any gigaton-scale spacecraft doing gravity assists!

You answered your own question. I’m not going to do the math you you would probably need trillions of flybys before anything observable happens.

xkcd did a What-If on this: https://what-if.xkcd.com/146/

TL;DR: It will not happen. You would need to throw on the order of a thousand Earth-masses to even get close to neutralizing Jupiter's orbit.

All processes in this world are unsustainable, but only some matter.

Seems only one person got the joke :-)

lol