Backward uniqueness with critical covariant drift and a quantitative background slice theorem

We establish a Carleman estimate for backward parabolic operators on R3 whose principalpart is of (covariant) Laplace type and whose lower order terms include a scale invariantdrift in a parabolic Morrey class. As consequences we obtain an optimal three cylinderinequality, backward uniqueness from a terminal time slice, and an ancient-solution rigidityprinciple. These tools are designed to be robust under the covariant reductions that occur inbackground or DeTurck gauge formulations of Yang–Mills type parabolic flows, where thegauge fixing transport produces precisely such critical drifts.As a complementary elliptic input we prove a quantitative local slice theorem around asmooth background connection, with explicit constants controlled by an accretivity marginfor the mixed Faddeev–Popov form. This yields coercivity on the L2-orthogonal complementof gauge directions and a quantitative description of a first background Gribov region.Together, the analytic and geometric modules provide a self-contained PDE framework forstudying linearized, gauge-fixed Yang–Mills heat-type evolutions at the level of scale criticalcoefficients.