Angular-Momentum Loss and Radiated Lorentz Charge in Kerr Scattering at Quadratic Spin

We present a complete derivation of the radiated Lorentz charge in the scattering of two Kerr black holes at third post-Minkowskian order and quadratic spin. Starting from the leading Kerr waveform, reverse unitarity gives the connected finite-frequency contribution. It includes all quadratic self- and mixed-spin terms for generic spin magnitudes and orientations, with exact dependence on the masses and velocity. We then derive the required zero-frequency term from the integrated leading-soft Lorentz charge and show that it is fixed by the complete on-shell quadratic-spin 2PM impulse; stationary Kerr data introduce no further quadratic-spin surface datum. The resulting six global charges comprise the three components of angular-momentum loss and the three components of radiative mass dipole. We keep ordinary-origin, Bondi-frame, and spin-supplementary-condition transformations as separate layers and provide the full analytic tensor in machine-readable form. A source-multipole calculation reproduces the post-Newtonian expansion of all 126 connected coefficients through fourth order in relative momentum within an explicitly tested mass class. Finally, the completed loss determines a fourth-PM contribution to the aligned-spin scattering angle that is linear in radiation. In the equal-mass anti-aligned sector its quadratic-spin coefficient has a unique simple zero at p∞ = 0.5286907017... through a cancellation between nonzero finite-frequency and mechanical-endpoint terms. We give its exact logarithmic equation and root-count proof, identify the associated null spin direction, and derive its local continuation under mass and spin imbalance. The standalone manuscript is available as a separate PDF for online preview. The accompanying ZIP contains the same PDF, the electronic Supplemental Material in Kerr_G3S2_Coefficients.wl and Kerr_G3S2_PN_Coefficients.wl, and a README with usage instructions. The two Wolfram Language files provide the exact coefficients and separately named completion and response layers, independently source-reconstructed post-Newtonian coefficients, and the deposited checks and benchmark records described in the manuscript.