[terminalbraid]

If changing definitions and adding dimensions are fair game, try it for a damped oscillator and tell me where whatever dimension sphere is in there.

If you understood what you were trying to push now you'd recognize that adding in your time dimension to force an argument involving a circle is just a matter of the sign in the 2nd order differential equation. This also falls apart if that sign changes and you get hyperbolic solutions. and I'm sure you'll just say that's just another kind of sphere with negative curvature