Nonlinear Thrust-Direction Rigidity Near Fixed-Axis Radiating Thin Shells

Can a radiating shell enclosing a flat cavity change its thrust direction while firing? This paper proves local nonlinear rigidity for timelike first-junction matchings with a Kinnersley exterior. Fix a smooth exact axial reference in a specified small C⁴ box about a positive-mass round static shell, with round static end collars and thrust nonzero throughout one open burn. Require every departure to vanish on both reference collars after constant ambient normalization and axis alignment. Let η be the non-axisymmetric part of the full departure. For every finite nonnegative B, there is an H¹⁰ neighborhood, depending on the reference and B, in which every exact competitor satisfying ‖η‖H¹³ ≤ B‖η‖H¹⁰ is axial. The norms cover thrust, mass, radius, and all four cavity embedding components. The conclusion extends to round, thrust-free collars with changed radius and mass histories and a free outgoing embedding. The proof combines the complete transverse linear kernel, energy estimates, and compactness through an exact nonlinear averaging identity, and controls transverse departures by the transverse H⁹ metric residual. Sufficiently small fixed-axis pulses constructed by Le (2026) supply the references. This paper also proves azimuthal-order-one linear rigidity for arbitrary massive axial backgrounds. Unrestricted rigidity under absolute smallness remains Conjecture C.