Fixed Points III: Disturbance–Damping Balance and Stability

We develop disturbance–damping stability on the joint spectral decomposition of the Fixed Points II internal operators. For each noncoherent joint mode, its one-sided nonlinear damping margin is the linear damping rate minus the nonlinear Lipschitz allowance. Deterministic force amplitude and stochastic noise power are treated as distinct quantities. We prove separate deterministic input-to-state and stochastic second-moment floors whenever the damping margin is positive. Gaussian invariant laws and necessity of the sign condition are asserted only for exact Ornstein–Uhlenbeck dynamics or robust worst-case stability. Bundlewise results use separate stochastic trace and deterministic weighted-series conditions. Finally, deterministic homogenization is made conditional on an enhanced functional central-limit theorem, tightness, and rough-path convergence, with the effective covariance given by the symmetrized Green–Kubo integral.