Abstract We demonstrate that the two numerical constants appearing in Routh’s classical stability criterion for the triangular Lagrange points L4 and L5 — specifically the integers 23 and 27 in the expression µcrit = (1/2)(1 − (23/27)1/2) — arise naturally from a fiber geometry with total dimension D = 22 and fiber count N = 3. We show that 27 = N3 and 23 = D − N + (N−1)2, where D = D1 + D2 + D3 = 8 + 6 + 8 is the sum of three fiber dimensions. This identification requires no free parameters: every quantity in Routh’s criterion is determined by the fiber structure. We further show that the three internal degrees of freedom of the three-body problem correspond to the three fibers, the 23 = 8 binary activation states of these fibers partition the configuration space into eight qualitative classes, and the 8 × 8 = 64 products of these classes enumerate all dynamical state-tendency pairs. Numerical simulation confirms that a breathing-synchronized integrator based on this classification outperforms uniform-timestep Newtonian integration on periodic orbit accuracy by a factor of 2.6. These results suggest that the three-body problem possesses a hidden fiber structure that constrains its solution space.