Endpoint well-posedness for the Boussinesq equation

We prove small-data global well-posedness for the quadratic good Boussinesq equation on the real line and the circle at the endpoint $H^{-1/2}\times H^{-3/2}$. On each domain, a small invariant spectral set supports a continuous global group, with uniqueness in a fixed local integral class. In the periodic case, both components have zero mean and the flow is Lipschitz on bounded time intervals. The a priori bounds follow from a balanced perturbation determinant for the third-order Lax operator. Averaging two real spectral branches yields coercivity and frequency-tail control. On the line, we construct the endpoint flow by commuting Hamiltonians and Green-field estimates. On the circle, discrete spectral bounds and Kishimoto's endpoint local theory give global well-posedness.