> If you are presupposing external perturbations or collisions, it's not an N=3 system... we're talking about the three body problem.

Let me clarify something. A "three body problem" system is any orbital system with more than two bodies. The term "three-body problem" certainly doesn't mean systems with only three bodies.

> A tidally locked system with periodic resonance is permanently stable in the absence of external forces.

No. In an orbital system with more than two bodies, external forces are the name of the game. For such a system, the expression "permanently stable" cannot apply. Such a system is not open to a closed-form solution and all such systems must be modeled numerically.

Closed-form solutions are available for orbits with two bodies, and can sometimes approximate the behavior of systems with more than two, but the reliability of such a model degrades rapidly as time increases, until the predictions become meaningless.

From https://en.wikipedia.org/wiki/Orbit_of_the_Moon : "The properties of the orbit described in this section are approximations. The Moon's orbit around Earth has many variations (perturbations) due to the gravitational attraction of the Sun and planets, the study of which [ ... ] has a long history."