Curvature–Authority Thresholds and Robust Stabilization of Null Congruences

This work reformulates the twist-free null Raychaudhuri equation as a nonlinear control system in which beyond-Einstein corrections are treated as bounded defocusing authority. Under assumptions of bounded Ricci focusing, bounded Weyl electric amplitude, and bounded corrective contribution, we derive a sharp sufficient condition for the existence of a forward-invariant region in the expansion–shear phase plane. The resulting curvature–authority threshold u_max > R_max + √2 E_max provides a model-agnostic diagnostic for strong-field stabilization of null congruences. We further prove that no bounded authority can globally prevent focusing from sufficiently negative expansion. Applications to Schwarzschild and Kerr scaling are presented, together with an explicit worked example in f(R) gravity illustrating the difficulty of satisfying the threshold in astrophysical regimes. The result is conservative with respect to twist and worst-case Weyl alignment and is intended as a diagnostic framework rather than a claim of singularity avoidance.