Constant pure four-force obstruction
= Constant pure four-force obstruction
A nonzero <four-force> cannot have all components constant in an inertial frame along an entire constant-mass trajectory. If $U=U_0+F\tau/m$, preserving $U^2=c^2$ for all proper times requires both $U_0\cdot F=0$ and $F^2=0$. The orthogonal complement of timelike $U_0$ is negative definite, forcing $F=0$. A constant laboratory three-force or a <four-force> of constant invariant magnitude is a different, consistent condition.