A parametric logarithmic improvement of the critical Hardy inequality and stability of the deficit

We address a classification problem for critical one-dimensional Hardy forms perturbed by a logarithmic remainder. On $(0,1)$, with the gauge $\log \frac{e}{x}=\log (e/x)$, we construct an explicit one-parameter family of Riccati weights that yields an identity-level ground-state representation. This produces a continuum of logarithmic improvements of the critical Hardy inequality with a computable remainder coefficient and an explicit positive ground state. We then derive a quantitative interior stability estimate: The Hardy deficit controls the distance to the associated ground state on every interior subinterval. We further classify the constant-coefficient logarithmic remainder class by reducing the associated ground-state ordinary differential equation (ODE) to a Euler equation in the logarithmic variable, and we obtain an interior compactness statement for sequences with vanishing deficit. As an application, we prove positivity and a priori bounds for a class of Dirichlet Schrödinger problems with critical singular potentials.