Complete-Cancellation Thresholds and Action-Based Design of Nonlinear-Kicker Injection in Diffraction-Limited Storage Rings

Nonlinear-kicker (NLK) injection aims to reduce injected-beam oscillations while minimizing disturbance to the stored beam. An action-based Universal Hardware Equation (UHE) relates the nonlinear kick profile, initial centroid action, and local optics to the minimum offset required for complete normalized-momentum cancellation. This condition provides a common reference for comparing NLK designs under current, trajectory-accessibility, and stored-beam constraints, while remaining distinct from unrestricted action minimization and beam capture. The cubic-quintic model describes nonlinear field shaping, while the cubic model extends the analysis to staged NLK correction, quantifying current sharing and diminishing action reduction. Comparisons of two reported eight-wire geometries demonstrate how stored-beam limits and accessible offsets influence geometry selection and operating points. Sampled-field calculations verify the UHE implementation, while a 0.50 m uniform-body model changes the screened reference action by only 0.14%. Vertical-kick analysis identifies a characteristic centering scale of approximately 42 μm for an assumed 0.10 nm centroid-action budget under the adopted stored-beam kick limits and zero incoming normalized vertical momentum. The framework thus provides a rapid, reproducible basis for screening NLK injection configurations before machine-specific optimization.