I also don't see the compressive forces for a rotational system here. The "moon" object would create some, but we haven't specified how large a moon it is, so those can be negligible compared to the momentum of the planets.

@J... the planet is light with respect to the massive bodies, and so its affects are small. By tuning the mass ratio, we can make its influence sufficiently small to make the assemblage stable over a timescale of our choosing

@Clumsycat the compressive forces are on braces between the four massive bodies which will naturally attract each other. The compressive forces in the braces resist that attraction and keep the bodies in position

of course, the simpler solution to all of this would be to give the massive bodies station-keeping engines with sufficient fuel to last for whatever timescale we choose

@Tristan If the system is rotating at least as fast as these objects require to remain in orbit about their centre of gravity then we can keep everything under tension and there are no compressive forces.

@Tristan Planets aren't light. Even the smallest dwarf is still 10^22kg. The mass ratio is irrelevant, and the larger you make massive bodies the more impossible your support structure becomes. There is no known configuration of matter which can withstand these forces. You could only ever make it work with objects MUCH smaller than planets. @Clumsycat The system cannot be allowed to rotate or it doesn't work. There is also no way to stop the system from rotating, so it is impossible.

@J... Well clearly they have to rotate in the same direction, or over time the forces that lead things to become tidally locked will destroy their orbits. Rotating in the same direction means an effectively steeper cornering angle, but I see no other change needed.

@Clumsycat Because you haven't fully thought it through. Consider the rate of rotation required to relieve the system of stress (ie :four co-orbiting massive bodies). Now you don't need a ridiculous strut system between them. Consider now what the orbiting velocity of the small planet would need to be for the squircle orbit and notice how everything falls apart.

@Clumsycat If you slow the rotation to a speed that does allow your "bent squircle" idea, you're back to fighting gravity with materials from science-fiction. It is utterly impossible. I defy you to come up with numbers and materials that satisfy the laws of physics and this system at the same time.

@J... what issue do you see in a fast moon orbit?

Also "Because you haven't fully thought it through." is rude and unconstructive.

Consider that if the four bodies are static the system is impossible because no structure can hold them there. If we allow them to rotate they must rotate at whatever co-orbiting velocity the universe demands for their size and distance. At that rotational speed, a planet trying to orbit the four OUTSIDE their co-orbit cannot go faster than the four themselves. There is no way to make this work. One way or another it requires magic somewhere.

@J... The moon isn't in a Newtonian orbit, it's slingshotting 4 times. So the angle of approach between the moon and the planet will change the acceleration that it gets.

@Clumsycat If you can show math that makes sense and proves your point I'll believe it. I'm convinced it doesn't exist. Otherwise, you're arguing a dead case. I will never tell you that this proposal makes sense, so don't bother trying to make a case based on intuition. The system will fall apart no matter how you slice it. It's neither stable nor physically possible. If you change the angle of approach and exit the orbit furthermore ceases to be a squircle. It's now a spirograph...and unstable.