A Numerically Resolved Free-Boundary MOTS Saddle-Node in Dynamical Braneworld Collapse

We numerically resolve the critical structure of a brane-terminating marginally outer trapped-surface (MOTS) pair in the SO(3)-symmetric sector of five-dimensional Einstein–anti-de Sitter gravity on a Goldberger–Wise-stabilized compact interval. An independently constructed parent sequence contains no accepted upper-brane-capped MOTS at t/ℓ = 0.001 and exactly two at t/ℓ = 0.0015 on three successive spacetime grids. Pseudo-arclength continuation of the coupled surface–time problem locates a common critical time t∗/ℓ ≃ 1.17254 × 10−3 in the chosen foliation. The principal stability eigenvalue crosses zero with a real, sign-definite mode, while adjoint projections give nonzero near-critical transversality and quadratic coefficients under a fully specified discrete normalization. The invariant one-sided proper-area separation follows the expected square-root law. An independent G10 half-timestep evolution repeats the parent and continuation, yields an overlapping critical-time bracket, and gives a free exponent p ≃ 0.498. These results numerically resolve the established saddle-node mechanism in a dynamical free-boundary MOTS problem rather than inferring it only from pair anatomy. The claim is finite-resolution and quasi-local: it is not a continuum theorem, event-horizon reconstruction, or phase diagram.